*DECK DLSODE
SUBROUTINE DLSODE (F, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
1 ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JAC, MF)
EXTERNAL F, JAC
INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK, LIW, MF
DOUBLE PRECISION Y, T, TOUT, RTOL, ATOL, RWORK
DIMENSION NEQ(*), Y(*), RTOL(*), ATOL(*), RWORK(LRW), IWORK(LIW)
C***BEGIN PROLOGUE DLSODE
C***PURPOSE Livermore Solver for Ordinary Differential Equations.
C DLSODE solves the initial-value problem for stiff or
C nonstiff systems of first-order ODE's,
C dy/dt = f(t,y), or, in component form,
C dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(N)), i=1,...,N.
C***CATEGORY I1A
C***TYPE DOUBLE PRECISION (SLSODE-S, DLSODE-D)
C***KEYWORDS ORDINARY DIFFERENTIAL EQUATIONS, INITIAL VALUE PROBLEM,
C STIFF, NONSTIFF
C***AUTHOR Hindmarsh, Alan C., (LLNL)
C Center for Applied Scientific Computing, L-561
C Lawrence Livermore National Laboratory
C Livermore, CA 94551.
C***DESCRIPTION
C
C NOTE: The "Usage" and "Arguments" sections treat only a subset of
C available options, in condensed fashion. The options
C covered and the information supplied will support most
C standard uses of DLSODE.
C
C For more sophisticated uses, full details on all options are
C given in the concluding section, headed "Long Description."
C A synopsis of the DLSODE Long Description is provided at the
C beginning of that section; general topics covered are:
C - Elements of the call sequence; optional input and output
C - Optional supplemental routines in the DLSODE package
C - internal COMMON block
C
C *Usage:
C Communication between the user and the DLSODE package, for normal
C situations, is summarized here. This summary describes a subset
C of the available options. See "Long Description" for complete
C details, including optional communication, nonstandard options,
C and instructions for special situations.
C
C A sample program is given in the "Examples" section.
C
C Refer to the argument descriptions for the definitions of the
C quantities that appear in the following sample declarations.
C
C For MF = 10,
C PARAMETER (LRW = 20 + 16*NEQ, LIW = 20)
C For MF = 21 or 22,
C PARAMETER (LRW = 22 + 9*NEQ + NEQ**2, LIW = 20 + NEQ)
C For MF = 24 or 25,
C PARAMETER (LRW = 22 + 10*NEQ + (2*ML+MU)*NEQ,
C * LIW = 20 + NEQ)
C
C EXTERNAL F, JAC
C INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK(LIW),
C * LIW, MF
C DOUBLE PRECISION Y(NEQ), T, TOUT, RTOL, ATOL(ntol), RWORK(LRW)
C
C CALL DLSODE (F, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
C * ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JAC, MF)
C
C *Arguments:
C F :EXT Name of subroutine for right-hand-side vector f.
C This name must be declared EXTERNAL in calling
C program. The form of F must be:
C
C SUBROUTINE F (NEQ, T, Y, YDOT)
C INTEGER NEQ
C DOUBLE PRECISION T, Y(*), YDOT(*)
C
C The inputs are NEQ, T, Y. F is to set
C
C YDOT(i) = f(i,T,Y(1),Y(2),...,Y(NEQ)),
C i = 1, ..., NEQ .
C
C NEQ :IN Number of first-order ODE's.
C
C Y :INOUT Array of values of the y(t) vector, of length NEQ.
C Input: For the first call, Y should contain the
C values of y(t) at t = T. (Y is an input
C variable only if ISTATE = 1.)
C Output: On return, Y will contain the values at the
C new t-value.
C
C T :INOUT Value of the independent variable. On return it
C will be the current value of t (normally TOUT).
C
C TOUT :IN Next point where output is desired (.NE. T).
C
C ITOL :IN 1 or 2 according as ATOL (below) is a scalar or
C an array.
C
C RTOL :IN Relative tolerance parameter (scalar).
C
C ATOL :IN Absolute tolerance parameter (scalar or array).
C If ITOL = 1, ATOL need not be dimensioned.
C If ITOL = 2, ATOL must be dimensioned at least NEQ.
C
C The estimated local error in Y(i) will be controlled
C so as to be roughly less (in magnitude) than
C
C EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
C EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
C
C Thus the local error test passes if, in each
C component, either the absolute error is less than
C ATOL (or ATOL(i)), or the relative error is less
C than RTOL.
C
C Use RTOL = 0.0 for pure absolute error control, and
C use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative
C error control. Caution: Actual (global) errors may
C exceed these local tolerances, so choose them
C conservatively.
C
C ITASK :IN Flag indicating the task DLSODE is to perform.
C Use ITASK = 1 for normal computation of output
C values of y at t = TOUT.
C
C ISTATE:INOUT Index used for input and output to specify the state
C of the calculation.
C Input:
C 1 This is the first call for a problem.
C 2 This is a subsequent call.
C Output:
C 1 Nothing was done, because TOUT was equal to T.
C 2 DLSODE was successful (otherwise, negative).
C Note that ISTATE need not be modified after a
C successful return.
C -1 Excess work done on this call (perhaps wrong
C MF).
C -2 Excess accuracy requested (tolerances too
C small).
C -3 Illegal input detected (see printed message).
C -4 Repeated error test failures (check all
C inputs).
C -5 Repeated convergence failures (perhaps bad
C Jacobian supplied or wrong choice of MF or
C tolerances).
C -6 Error weight became zero during problem
C (solution component i vanished, and ATOL or
C ATOL(i) = 0.).
C
C IOPT :IN Flag indicating whether optional inputs are used:
C 0 No.
C 1 Yes. (See "Optional inputs" under "Long
C Description," Part 1.)
C
C RWORK :WORK Real work array of length at least:
C 20 + 16*NEQ for MF = 10,
C 22 + 9*NEQ + NEQ**2 for MF = 21 or 22,
C 22 + 10*NEQ + (2*ML + MU)*NEQ for MF = 24 or 25.
C
C LRW :IN Declared length of RWORK (in user's DIMENSION
C statement).
C
C IWORK :WORK Integer work array of length at least:
C 20 for MF = 10,
C 20 + NEQ for MF = 21, 22, 24, or 25.
C
C If MF = 24 or 25, input in IWORK(1),IWORK(2) the
C lower and upper Jacobian half-bandwidths ML,MU.
C
C On return, IWORK contains information that may be
C of interest to the user:
C
C Name Location Meaning
C ----- --------- -----------------------------------------
C NST IWORK(11) Number of steps taken for the problem so
C far.
C NFE IWORK(12) Number of f evaluations for the problem
C so far.
C NJE IWORK(13) Number of Jacobian evaluations (and of
C matrix LU decompositions) for the problem
C so far.
C NQU IWORK(14) Method order last used (successfully).
C LENRW IWORK(17) Length of RWORK actually required. This
C is defined on normal returns and on an
C illegal input return for insufficient
C storage.
C LENIW IWORK(18) Length of IWORK actually required. This
C is defined on normal returns and on an
C illegal input return for insufficient
C storage.
C
C LIW :IN Declared length of IWORK (in user's DIMENSION
C statement).
C
C JAC :EXT Name of subroutine for Jacobian matrix (MF =
C 21 or 24). If used, this name must be declared
C EXTERNAL in calling program. If not used, pass a
C dummy name. The form of JAC must be:
C
C SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD)
C INTEGER NEQ, ML, MU, NROWPD
C DOUBLE PRECISION T, Y(*), PD(NROWPD,*)
C
C See item c, under "Description" below for more
C information about JAC.
C
C MF :IN Method flag. Standard values are:
C 10 Nonstiff (Adams) method, no Jacobian used.
C 21 Stiff (BDF) method, user-supplied full Jacobian.
C 22 Stiff method, internally generated full
C Jacobian.
C 24 Stiff method, user-supplied banded Jacobian.
C 25 Stiff method, internally generated banded
C Jacobian.
C
C *Description:
C DLSODE solves the initial value problem for stiff or nonstiff
C systems of first-order ODE's,
C
C dy/dt = f(t,y) ,
C
C or, in component form,
C
C dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(NEQ))
C (i = 1, ..., NEQ) .
C
C DLSODE is a package based on the GEAR and GEARB packages, and on
C the October 23, 1978, version of the tentative ODEPACK user
C interface standard, with minor modifications.
C
C The steps in solving such a problem are as follows.
C
C a. First write a subroutine of the form
C
C SUBROUTINE F (NEQ, T, Y, YDOT)
C INTEGER NEQ
C DOUBLE PRECISION T, Y(*), YDOT(*)
C
C which supplies the vector function f by loading YDOT(i) with
C f(i).
C
C b. Next determine (or guess) whether or not the problem is stiff.
C Stiffness occurs when the Jacobian matrix df/dy has an
C eigenvalue whose real part is negative and large in magnitude
C compared to the reciprocal of the t span of interest. If the
C problem is nonstiff, use method flag MF = 10. If it is stiff,
C there are four standard choices for MF, and DLSODE requires the
C Jacobian matrix in some form. This matrix is regarded either
C as full (MF = 21 or 22), or banded (MF = 24 or 25). In the
C banded case, DLSODE requires two half-bandwidth parameters ML
C and MU. These are, respectively, the widths of the lower and
C upper parts of the band, excluding the main diagonal. Thus the
C band consists of the locations (i,j) with
C
C i - ML <= j <= i + MU ,
C
C and the full bandwidth is ML + MU + 1 .
C
C c. If the problem is stiff, you are encouraged to supply the
C Jacobian directly (MF = 21 or 24), but if this is not feasible,
C DLSODE will compute it internally by difference quotients (MF =
C 22 or 25). If you are supplying the Jacobian, write a
C subroutine of the form
C
C SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD)
C INTEGER NEQ, ML, MU, NRWOPD
C DOUBLE PRECISION T, Y(*), PD(NROWPD,*)
C
C which provides df/dy by loading PD as follows:
C - For a full Jacobian (MF = 21), load PD(i,j) with df(i)/dy(j),
C the partial derivative of f(i) with respect to y(j). (Ignore
C the ML and MU arguments in this case.)
C - For a banded Jacobian (MF = 24), load PD(i-j+MU+1,j) with
C df(i)/dy(j); i.e., load the diagonal lines of df/dy into the
C rows of PD from the top down.
C - In either case, only nonzero elements need be loaded.
C
C d. Write a main program that calls subroutine DLSODE once for each
C point at which answers are desired. This should also provide
C for possible use of logical unit 6 for output of error messages
C by DLSODE.
C
C Before the first call to DLSODE, set ISTATE = 1, set Y and T to
C the initial values, and set TOUT to the first output point. To
C continue the integration after a successful return, simply
C reset TOUT and call DLSODE again. No other parameters need be
C reset.
C
C *Examples:
C The following is a simple example problem, with the coding needed
C for its solution by DLSODE. The problem is from chemical kinetics,
C and consists of the following three rate equations:
C
C dy1/dt = -.04*y1 + 1.E4*y2*y3
C dy2/dt = .04*y1 - 1.E4*y2*y3 - 3.E7*y2**2
C dy3/dt = 3.E7*y2**2
C
C on the interval from t = 0.0 to t = 4.E10, with initial conditions
C y1 = 1.0, y2 = y3 = 0. The problem is stiff.
C
C The following coding solves this problem with DLSODE, using
C MF = 21 and printing results at t = .4, 4., ..., 4.E10. It uses
C ITOL = 2 and ATOL much smaller for y2 than for y1 or y3 because y2
C has much smaller values. At the end of the run, statistical
C quantities of interest are printed.
C
C EXTERNAL FEX, JEX
C INTEGER IOPT, IOUT, ISTATE, ITASK, ITOL, IWORK(23), LIW, LRW,
C * MF, NEQ
C DOUBLE PRECISION ATOL(3), RTOL, RWORK(58), T, TOUT, Y(3)
C NEQ = 3
C Y(1) = 1.D0
C Y(2) = 0.D0
C Y(3) = 0.D0
C T = 0.D0
C TOUT = .4D0
C ITOL = 2
C RTOL = 1.D-4
C ATOL(1) = 1.D-6
C ATOL(2) = 1.D-10
C ATOL(3) = 1.D-6
C ITASK = 1
C ISTATE = 1
C IOPT = 0
C LRW = 58
C LIW = 23
C MF = 21
C DO 40 IOUT = 1,12
C CALL DLSODE (FEX, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
C * ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JEX, MF)
C WRITE(6,20) T, Y(1), Y(2), Y(3)
C 20 FORMAT(' At t =',D12.4,' y =',3D14.6)
C IF (ISTATE .LT. 0) GO TO 80
C 40 TOUT = TOUT*10.D0
C WRITE(6,60) IWORK(11), IWORK(12), IWORK(13)
C 60 FORMAT(/' No. steps =',i4,', No. f-s =',i4,', No. J-s =',i4)
C STOP
C 80 WRITE(6,90) ISTATE
C 90 FORMAT(///' Error halt.. ISTATE =',I3)
C STOP
C END
C
C SUBROUTINE FEX (NEQ, T, Y, YDOT)
C INTEGER NEQ
C DOUBLE PRECISION T, Y(3), YDOT(3)
C YDOT(1) = -.04D0*Y(1) + 1.D4*Y(2)*Y(3)
C YDOT(3) = 3.D7*Y(2)*Y(2)
C YDOT(2) = -YDOT(1) - YDOT(3)
C RETURN
C END
C
C SUBROUTINE JEX (NEQ, T, Y, ML, MU, PD, NRPD)
C INTEGER NEQ, ML, MU, NRPD
C DOUBLE PRECISION T, Y(3), PD(NRPD,3)
C PD(1,1) = -.04D0
C PD(1,2) = 1.D4*Y(3)
C PD(1,3) = 1.D4*Y(2)
C PD(2,1) = .04D0
C PD(2,3) = -PD(1,3)
C PD(3,2) = 6.D7*Y(2)
C PD(2,2) = -PD(1,2) - PD(3,2)
C RETURN
C END
C
C The output from this program (on a Cray-1 in single precision)
C is as follows.
C
C At t = 4.0000e-01 y = 9.851726e-01 3.386406e-05 1.479357e-02
C At t = 4.0000e+00 y = 9.055142e-01 2.240418e-05 9.446344e-02
C At t = 4.0000e+01 y = 7.158050e-01 9.184616e-06 2.841858e-01
C At t = 4.0000e+02 y = 4.504846e-01 3.222434e-06 5.495122e-01
C At t = 4.0000e+03 y = 1.831701e-01 8.940379e-07 8.168290e-01
C At t = 4.0000e+04 y = 3.897016e-02 1.621193e-07 9.610297e-01
C At t = 4.0000e+05 y = 4.935213e-03 1.983756e-08 9.950648e-01
C At t = 4.0000e+06 y = 5.159269e-04 2.064759e-09 9.994841e-01
C At t = 4.0000e+07 y = 5.306413e-05 2.122677e-10 9.999469e-01
C At t = 4.0000e+08 y = 5.494530e-06 2.197825e-11 9.999945e-01
C At t = 4.0000e+09 y = 5.129458e-07 2.051784e-12 9.999995e-01
C At t = 4.0000e+10 y = -7.170603e-08 -2.868241e-13 1.000000e+00
C
C No. steps = 330, No. f-s = 405, No. J-s = 69
C
C *Accuracy:
C The accuracy of the solution depends on the choice of tolerances
C RTOL and ATOL. Actual (global) errors may exceed these local
C tolerances, so choose them conservatively.
C
C *Cautions:
C The work arrays should not be altered between calls to DLSODE for
C the same problem, except possibly for the conditional and optional
C inputs.
C
C *Portability:
C Since NEQ is dimensioned inside DLSODE, some compilers may object
C to a call to DLSODE with NEQ a scalar variable. In this event,
C use DIMENSION NEQ(1). Similar remarks apply to RTOL and ATOL.
C
C Note to Cray users:
C For maximum efficiency, use the CFT77 compiler. Appropriate
C compiler optimization directives have been inserted for CFT77.
C
C *Reference:
C Alan C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE
C Solvers," in Scientific Computing, R. S. Stepleman, et al., Eds.
C (North-Holland, Amsterdam, 1983), pp. 55-64.
C
C *Long Description:
C The following complete description of the user interface to
C DLSODE consists of four parts:
C
C 1. The call sequence to subroutine DLSODE, which is a driver
C routine for the solver. This includes descriptions of both
C the call sequence arguments and user-supplied routines.
C Following these descriptions is a description of optional
C inputs available through the call sequence, and then a
C description of optional outputs in the work arrays.
C
C 2. Descriptions of other routines in the DLSODE package that may
C be (optionally) called by the user. These provide the ability
C to alter error message handling, save and restore the internal
C COMMON, and obtain specified derivatives of the solution y(t).
C
C 3. Descriptions of COMMON block to be declared in overlay or
C similar environments, or to be saved when doing an interrupt
C of the problem and continued solution later.
C
C 4. Description of two routines in the DLSODE package, either of
C which the user may replace with his own version, if desired.
C These relate to the measurement of errors.
C
C
C Part 1. Call Sequence
C ----------------------
C
C Arguments
C ---------
C The call sequence parameters used for input only are
C
C F, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK, IOPT, LRW, LIW, JAC, MF,
C
C and those used for both input and output are
C
C Y, T, ISTATE.
C
C The work arrays RWORK and IWORK are also used for conditional and
C optional inputs and optional outputs. (The term output here
C refers to the return from subroutine DLSODE to the user's calling
C program.)
C
C The legality of input parameters will be thoroughly checked on the
C initial call for the problem, but not checked thereafter unless a
C change in input parameters is flagged by ISTATE = 3 on input.
C
C The descriptions of the call arguments are as follows.
C
C F The name of the user-supplied subroutine defining the ODE
C system. The system must be put in the first-order form
C dy/dt = f(t,y), where f is a vector-valued function of
C the scalar t and the vector y. Subroutine F is to compute
C the function f. It is to have the form
C
C SUBROUTINE F (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y(*), YDOT(*)
C
C where NEQ, T, and Y are input, and the array YDOT =
C f(T,Y) is output. Y and YDOT are arrays of length NEQ.
C Subroutine F should not alter Y(1),...,Y(NEQ). F must be
C declared EXTERNAL in the calling program.
C
C Subroutine F may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),..., if NEQ is an array
C (dimensioned in F) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y below.
C
C If quantities computed in the F routine are needed
C externally to DLSODE, an extra call to F should be made
C for this purpose, for consistent and accurate results.
C If only the derivative dy/dt is needed, use DINTDY
C instead.
C
C NEQ The size of the ODE system (number of first-order
C ordinary differential equations). Used only for input.
C NEQ may be decreased, but not increased, during the
C problem. If NEQ is decreased (with ISTATE = 3 on input),
C the remaining components of Y should be left undisturbed,
C if these are to be accessed in F and/or JAC.
C
C Normally, NEQ is a scalar, and it is generally referred
C to as a scalar in this user interface description.
C However, NEQ may be an array, with NEQ(1) set to the
C system size. (The DLSODE package accesses only NEQ(1).)
C In either case, this parameter is passed as the NEQ
C argument in all calls to F and JAC. Hence, if it is an
C array, locations NEQ(2),... may be used to store other
C integer data and pass it to F and/or JAC. Subroutines
C F and/or JAC must include NEQ in a DIMENSION statement
C in that case.
C
C Y A real array for the vector of dependent variables, of
C length NEQ or more. Used for both input and output on
C the first call (ISTATE = 1), and only for output on
C other calls. On the first call, Y must contain the
C vector of initial values. On output, Y contains the
C computed solution vector, evaluated at T. If desired,
C the Y array may be used for other purposes between
C calls to the solver.
C
C This array is passed as the Y argument in all calls to F
C and JAC. Hence its length may exceed NEQ, and locations
C Y(NEQ+1),... may be used to store other real data and
C pass it to F and/or JAC. (The DLSODE package accesses
C only Y(1),...,Y(NEQ).)
C
C T The independent variable. On input, T is used only on
C the first call, as the initial point of the integration.
C On output, after each call, T is the value at which a
C computed solution Y is evaluated (usually the same as
C TOUT). On an error return, T is the farthest point
C reached.
C
C TOUT The next value of T at which a computed solution is
C desired. Used only for input.
C
C When starting the problem (ISTATE = 1), TOUT may be equal
C to T for one call, then should not equal T for the next
C call. For the initial T, an input value of TOUT .NE. T
C is used in order to determine the direction of the
C integration (i.e., the algebraic sign of the step sizes)
C and the rough scale of the problem. Integration in
C either direction (forward or backward in T) is permitted.
C
C If ITASK = 2 or 5 (one-step modes), TOUT is ignored
C after the first call (i.e., the first call with
C TOUT .NE. T). Otherwise, TOUT is required on every call.
C
C If ITASK = 1, 3, or 4, the values of TOUT need not be
C monotone, but a value of TOUT which backs up is limited
C to the current internal T interval, whose endpoints are
C TCUR - HU and TCUR. (See "Optional Outputs" below for
C TCUR and HU.)
C
C
C ITOL An indicator for the type of error control. See
C description below under ATOL. Used only for input.
C
C RTOL A relative error tolerance parameter, either a scalar or
C an array of length NEQ. See description below under
C ATOL. Input only.
C
C ATOL An absolute error tolerance parameter, either a scalar or
C an array of length NEQ. Input only.
C
C The input parameters ITOL, RTOL, and ATOL determine the
C error control performed by the solver. The solver will
C control the vector e = (e(i)) of estimated local errors
C in Y, according to an inequality of the form
C
C rms-norm of ( e(i)/EWT(i) ) <= 1,
C
C where
C
C EWT(i) = RTOL(i)*ABS(Y(i)) + ATOL(i),
C
C and the rms-norm (root-mean-square norm) here is
C
C rms-norm(v) = SQRT(sum v(i)**2 / NEQ).
C
C Here EWT = (EWT(i)) is a vector of weights which must
C always be positive, and the values of RTOL and ATOL
C should all be nonnegative. The following table gives the
C types (scalar/array) of RTOL and ATOL, and the
C corresponding form of EWT(i).
C
C ITOL RTOL ATOL EWT(i)
C ---- ------ ------ -----------------------------
C 1 scalar scalar RTOL*ABS(Y(i)) + ATOL
C 2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
C 3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
C 4 array array RTOL(i)*ABS(Y(i)) + ATOL(i)
C
C When either of these parameters is a scalar, it need not
C be dimensioned in the user's calling program.
C
C If none of the above choices (with ITOL, RTOL, and ATOL
C fixed throughout the problem) is suitable, more general
C error controls can be obtained by substituting
C user-supplied routines for the setting of EWT and/or for
C the norm calculation. See Part 4 below.
C
C If global errors are to be estimated by making a repeated
C run on the same problem with smaller tolerances, then all
C components of RTOL and ATOL (i.e., of EWT) should be
C scaled down uniformly.
C
C ITASK An index specifying the task to be performed. Input
C only. ITASK has the following values and meanings:
C 1 Normal computation of output values of y(t) at
C t = TOUT (by overshooting and interpolating).
C 2 Take one step only and return.
C 3 Stop at the first internal mesh point at or beyond
C t = TOUT and return.
C 4 Normal computation of output values of y(t) at
C t = TOUT but without overshooting t = TCRIT. TCRIT
C must be input as RWORK(1). TCRIT may be equal to or
C beyond TOUT, but not behind it in the direction of
C integration. This option is useful if the problem
C has a singularity at or beyond t = TCRIT.
C 5 Take one step, without passing TCRIT, and return.
C TCRIT must be input as RWORK(1).
C
C Note: If ITASK = 4 or 5 and the solver reaches TCRIT
C (within roundoff), it will return T = TCRIT (exactly) to
C indicate this (unless ITASK = 4 and TOUT comes before
C TCRIT, in which case answers at T = TOUT are returned
C first).
C
C ISTATE An index used for input and output to specify the state
C of the calculation.
C
C On input, the values of ISTATE are as follows:
C 1 This is the first call for the problem
C (initializations will be done). See "Note" below.
C 2 This is not the first call, and the calculation is to
C continue normally, with no change in any input
C parameters except possibly TOUT and ITASK. (If ITOL,
C RTOL, and/or ATOL are changed between calls with
C ISTATE = 2, the new values will be used but not
C tested for legality.)
C 3 This is not the first call, and the calculation is to
C continue normally, but with a change in input
C parameters other than TOUT and ITASK. Changes are
C allowed in NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, MF,
C ML, MU, and any of the optional inputs except H0.
C (See IWORK description for ML and MU.)
C
C Note: A preliminary call with TOUT = T is not counted as
C a first call here, as no initialization or checking of
C input is done. (Such a call is sometimes useful for the
C purpose of outputting the initial conditions.) Thus the
C first call for which TOUT .NE. T requires ISTATE = 1 on
C input.
C
C On output, ISTATE has the following values and meanings:
C 1 Nothing was done, as TOUT was equal to T with
C ISTATE = 1 on input.
C 2 The integration was performed successfully.
C -1 An excessive amount of work (more than MXSTEP steps)
C was done on this call, before completing the
C requested task, but the integration was otherwise
C successful as far as T. (MXSTEP is an optional input
C and is normally 500.) To continue, the user may
C simply reset ISTATE to a value >1 and call again (the
C excess work step counter will be reset to 0). In
C addition, the user may increase MXSTEP to avoid this
C error return; see "Optional Inputs" below.
C -2 Too much accuracy was requested for the precision of
C the machine being used. This was detected before
C completing the requested task, but the integration
C was successful as far as T. To continue, the
C tolerance parameters must be reset, and ISTATE must
C be set to 3. The optional output TOLSF may be used
C for this purpose. (Note: If this condition is
C detected before taking any steps, then an illegal
C input return (ISTATE = -3) occurs instead.)
C -3 Illegal input was detected, before taking any
C integration steps. See written message for details.
C (Note: If the solver detects an infinite loop of
C calls to the solver with illegal input, it will cause
C the run to stop.)
C -4 There were repeated error-test failures on one
C attempted step, before completing the requested task,
C but the integration was successful as far as T. The
C problem may have a singularity, or the input may be
C inappropriate.
C -5 There were repeated convergence-test failures on one
C attempted step, before completing the requested task,
C but the integration was successful as far as T. This
C may be caused by an inaccurate Jacobian matrix, if
C one is being used.
C -6 EWT(i) became zero for some i during the integration.
C Pure relative error control (ATOL(i)=0.0) was
C requested on a variable which has now vanished. The
C integration was successful as far as T.
C
C Note: Since the normal output value of ISTATE is 2, it
C does not need to be reset for normal continuation. Also,
C since a negative input value of ISTATE will be regarded
C as illegal, a negative output value requires the user to
C change it, and possibly other inputs, before calling the
C solver again.
C
C IOPT An integer flag to specify whether any optional inputs
C are being used on this call. Input only. The optional
C inputs are listed under a separate heading below.
C 0 No optional inputs are being used. Default values
C will be used in all cases.
C 1 One or more optional inputs are being used.
C
C RWORK A real working array (double precision). The length of
C RWORK must be at least
C
C 20 + NYH*(MAXORD + 1) + 3*NEQ + LWM
C
C where
C NYH = the initial value of NEQ,
C MAXORD = 12 (if METH = 1) or 5 (if METH = 2) (unless a
C smaller value is given as an optional input),
C LWM = 0 if MITER = 0,
C LWM = NEQ**2 + 2 if MITER = 1 or 2,
C LWM = NEQ + 2 if MITER = 3, and
C LWM = (2*ML + MU + 1)*NEQ + 2
C if MITER = 4 or 5.
C (See the MF description below for METH and MITER.)
C
C Thus if MAXORD has its default value and NEQ is constant,
C this length is:
C 20 + 16*NEQ for MF = 10,
C 22 + 16*NEQ + NEQ**2 for MF = 11 or 12,
C 22 + 17*NEQ for MF = 13,
C 22 + 17*NEQ + (2*ML + MU)*NEQ for MF = 14 or 15,
C 20 + 9*NEQ for MF = 20,
C 22 + 9*NEQ + NEQ**2 for MF = 21 or 22,
C 22 + 10*NEQ for MF = 23,
C 22 + 10*NEQ + (2*ML + MU)*NEQ for MF = 24 or 25.
C
C The first 20 words of RWORK are reserved for conditional
C and optional inputs and optional outputs.
C
C The following word in RWORK is a conditional input:
C RWORK(1) = TCRIT, the critical value of t which the
C solver is not to overshoot. Required if ITASK
C is 4 or 5, and ignored otherwise. See ITASK.
C
C LRW The length of the array RWORK, as declared by the user.
C (This will be checked by the solver.)
C
C IWORK An integer work array. Its length must be at least
C 20 if MITER = 0 or 3 (MF = 10, 13, 20, 23), or
C 20 + NEQ otherwise (MF = 11, 12, 14, 15, 21, 22, 24, 25).
C (See the MF description below for MITER.) The first few
C words of IWORK are used for conditional and optional
C inputs and optional outputs.
C
C The following two words in IWORK are conditional inputs:
C IWORK(1) = ML These are the lower and upper half-
C IWORK(2) = MU bandwidths, respectively, of the banded
C Jacobian, excluding the main diagonal.
C The band is defined by the matrix locations
C (i,j) with i - ML <= j <= i + MU. ML and MU
C must satisfy 0 <= ML,MU <= NEQ - 1. These are
C required if MITER is 4 or 5, and ignored
C otherwise. ML and MU may in fact be the band
C parameters for a matrix to which df/dy is only
C approximately equal.
C
C LIW The length of the array IWORK, as declared by the user.
C (This will be checked by the solver.)
C
C Note: The work arrays must not be altered between calls to DLSODE
C for the same problem, except possibly for the conditional and
C optional inputs, and except for the last 3*NEQ words of RWORK.
C The latter space is used for internal scratch space, and so is
C available for use by the user outside DLSODE between calls, if
C desired (but not for use by F or JAC).
C
C JAC The name of the user-supplied routine (MITER = 1 or 4) to
C compute the Jacobian matrix, df/dy, as a function of the
C scalar t and the vector y. (See the MF description below
C for MITER.) It is to have the form
C
C SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD)
C DOUBLE PRECISION T, Y(*), PD(NROWPD,*)
C
C where NEQ, T, Y, ML, MU, and NROWPD are input and the
C array PD is to be loaded with partial derivatives
C (elements of the Jacobian matrix) on output. PD must be
C given a first dimension of NROWPD. T and Y have the same
C meaning as in subroutine F.
C
C In the full matrix case (MITER = 1), ML and MU are
C ignored, and the Jacobian is to be loaded into PD in
C columnwise manner, with df(i)/dy(j) loaded into PD(i,j).
C
C In the band matrix case (MITER = 4), the elements within
C the band are to be loaded into PD in columnwise manner,
C with diagonal lines of df/dy loaded into the rows of PD.
C Thus df(i)/dy(j) is to be loaded into PD(i-j+MU+1,j). ML
C and MU are the half-bandwidth parameters (see IWORK).
C The locations in PD in the two triangular areas which
C correspond to nonexistent matrix elements can be ignored
C or loaded arbitrarily, as they are overwritten by DLSODE.
C
C JAC need not provide df/dy exactly. A crude approximation
C (possibly with a smaller bandwidth) will do.
C
C In either case, PD is preset to zero by the solver, so
C that only the nonzero elements need be loaded by JAC.
C Each call to JAC is preceded by a call to F with the same
C arguments NEQ, T, and Y. Thus to gain some efficiency,
C intermediate quantities shared by both calculations may
C be saved in a user COMMON block by F and not recomputed
C by JAC, if desired. Also, JAC may alter the Y array, if
C desired. JAC must be declared EXTERNAL in the calling
C program.
C
C Subroutine JAC may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in JAC) and/or Y has length exceeding
C NEQ(1). See the descriptions of NEQ and Y above.
C
C MF The method flag. Used only for input. The legal values
C of MF are 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24,
C and 25. MF has decimal digits METH and MITER:
C MF = 10*METH + MITER .
C
C METH indicates the basic linear multistep method:
C 1 Implicit Adams method.
C 2 Method based on backward differentiation formulas
C (BDF's).
C
C MITER indicates the corrector iteration method:
C 0 Functional iteration (no Jacobian matrix is
C involved).
C 1 Chord iteration with a user-supplied full (NEQ by
C NEQ) Jacobian.
C 2 Chord iteration with an internally generated
C (difference quotient) full Jacobian (using NEQ
C extra calls to F per df/dy value).
C 3 Chord iteration with an internally generated
C diagonal Jacobian approximation (using one extra call
C to F per df/dy evaluation).
C 4 Chord iteration with a user-supplied banded Jacobian.
C 5 Chord iteration with an internally generated banded
C Jacobian (using ML + MU + 1 extra calls to F per
C df/dy evaluation).
C
C If MITER = 1 or 4, the user must supply a subroutine JAC
C (the name is arbitrary) as described above under JAC.
C For other values of MITER, a dummy argument can be used.
C
C Optional Inputs
C ---------------
C The following is a list of the optional inputs provided for in the
C call sequence. (See also Part 2.) For each such input variable,
C this table lists its name as used in this documentation, its
C location in the call sequence, its meaning, and the default value.
C The use of any of these inputs requires IOPT = 1, and in that case
C all of these inputs are examined. A value of zero for any of
C these optional inputs will cause the default value to be used.
C Thus to use a subset of the optional inputs, simply preload
C locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively,
C and then set those of interest to nonzero values.
C
C Name Location Meaning and default value
C ------ --------- -----------------------------------------------
C H0 RWORK(5) Step size to be attempted on the first step.
C The default value is determined by the solver.
C HMAX RWORK(6) Maximum absolute step size allowed. The
C default value is infinite.
C HMIN RWORK(7) Minimum absolute step size allowed. The
C default value is 0. (This lower bound is not
C enforced on the final step before reaching
C TCRIT when ITASK = 4 or 5.)
C MAXORD IWORK(5) Maximum order to be allowed. The default value
C is 12 if METH = 1, and 5 if METH = 2. (See the
C MF description above for METH.) If MAXORD
C exceeds the default value, it will be reduced
C to the default value. If MAXORD is changed
C during the problem, it may cause the current
C order to be reduced.
C MXSTEP IWORK(6) Maximum number of (internally defined) steps
C allowed during one call to the solver. The
C default value is 500.
C MXHNIL IWORK(7) Maximum number of messages printed (per
C problem) warning that T + H = T on a step
C (H = step size). This must be positive to
C result in a nondefault value. The default
C value is 10.
C
C Optional Outputs
C ----------------
C As optional additional output from DLSODE, the variables listed
C below are quantities related to the performance of DLSODE which
C are available to the user. These are communicated by way of the
C work arrays, but also have internal mnemonic names as shown.
C Except where stated otherwise, all of these outputs are defined on
C any successful return from DLSODE, and on any return with ISTATE =
C -1, -2, -4, -5, or -6. On an illegal input return (ISTATE = -3),
C they will be unchanged from their existing values (if any), except
C possibly for TOLSF, LENRW, and LENIW. On any error return,
C outputs relevant to the error will be defined, as noted below.
C
C Name Location Meaning
C ----- --------- ------------------------------------------------
C HU RWORK(11) Step size in t last used (successfully).
C HCUR RWORK(12) Step size to be attempted on the next step.
C TCUR RWORK(13) Current value of the independent variable which
C the solver has actually reached, i.e., the
C current internal mesh point in t. On output,
C TCUR will always be at least as far as the
C argument T, but may be farther (if interpolation
C was done).
C TOLSF RWORK(14) Tolerance scale factor, greater than 1.0,
C computed when a request for too much accuracy
C was detected (ISTATE = -3 if detected at the
C start of the problem, ISTATE = -2 otherwise).
C If ITOL is left unaltered but RTOL and ATOL are
C uniformly scaled up by a factor of TOLSF for the
C next call, then the solver is deemed likely to
C succeed. (The user may also ignore TOLSF and
C alter the tolerance parameters in any other way
C appropriate.)
C NST IWORK(11) Number of steps taken for the problem so far.
C NFE IWORK(12) Number of F evaluations for the problem so far.
C NJE IWORK(13) Number of Jacobian evaluations (and of matrix LU
C decompositions) for the problem so far.
C NQU IWORK(14) Method order last used (successfully).
C NQCUR IWORK(15) Order to be attempted on the next step.
C IMXER IWORK(16) Index of the component of largest magnitude in
C the weighted local error vector ( e(i)/EWT(i) ),
C on an error return with ISTATE = -4 or -5.
C LENRW IWORK(17) Length of RWORK actually required. This is
C defined on normal returns and on an illegal
C input return for insufficient storage.
C LENIW IWORK(18) Length of IWORK actually required. This is
C defined on normal returns and on an illegal
C input return for insufficient storage.
C
C The following two arrays are segments of the RWORK array which may
C also be of interest to the user as optional outputs. For each
C array, the table below gives its internal name, its base address
C in RWORK, and its description.
C
C Name Base address Description
C ---- ------------ ----------------------------------------------
C YH 21 The Nordsieck history array, of size NYH by
C (NQCUR + 1), where NYH is the initial value of
C NEQ. For j = 0,1,...,NQCUR, column j + 1 of
C YH contains HCUR**j/factorial(j) times the jth
C derivative of the interpolating polynomial
C currently representing the solution, evaluated
C at t = TCUR.
C ACOR LENRW-NEQ+1 Array of size NEQ used for the accumulated
C corrections on each step, scaled on output to
C represent the estimated local error in Y on
C the last step. This is the vector e in the
C description of the error control. It is
C defined only on successful return from DLSODE.
C
C
C Part 2. Other Callable Routines
C --------------------------------
C
C The following are optional calls which the user may make to gain
C additional capabilities in conjunction with DLSODE.
C
C Form of call Function
C ------------------------ ----------------------------------------
C CALL XSETUN(LUN) Set the logical unit number, LUN, for
C output of messages from DLSODE, if the
C default is not desired. The default
C value of LUN is 6. This call may be made
C at any time and will take effect
C immediately.
C CALL XSETF(MFLAG) Set a flag to control the printing of
C messages by DLSODE. MFLAG = 0 means do
C not print. (Danger: this risks losing
C valuable information.) MFLAG = 1 means
C print (the default). This call may be
C made at any time and will take effect
C immediately.
C CALL DSRCOM(RSAV,ISAV,JOB) Saves and restores the contents of the
C internal COMMON blocks used by DLSODE
C (see Part 3 below). RSAV must be a
C real array of length 218 or more, and
C ISAV must be an integer array of length
C 37 or more. JOB = 1 means save COMMON
C into RSAV/ISAV. JOB = 2 means restore
C COMMON from same. DSRCOM is useful if
C one is interrupting a run and restarting
C later, or alternating between two or
C more problems solved with DLSODE.
C CALL DINTDY(,,,,,) Provide derivatives of y, of various
C (see below) orders, at a specified point t, if
C desired. It may be called only after a
C successful return from DLSODE. Detailed
C instructions follow.
C
C Detailed instructions for using DINTDY
C --------------------------------------
C The form of the CALL is:
C
C CALL DINTDY (T, K, RWORK(21), NYH, DKY, IFLAG)
C
C The input parameters are:
C
C T Value of independent variable where answers are
C desired (normally the same as the T last returned by
C DLSODE). For valid results, T must lie between
C TCUR - HU and TCUR. (See "Optional Outputs" above
C for TCUR and HU.)
C K Integer order of the derivative desired. K must
C satisfy 0 <= K <= NQCUR, where NQCUR is the current
C order (see "Optional Outputs"). The capability
C corresponding to K = 0, i.e., computing y(t), is
C already provided by DLSODE directly. Since
C NQCUR >= 1, the first derivative dy/dt is always
C available with DINTDY.
C RWORK(21) The base address of the history array YH.
C NYH Column length of YH, equal to the initial value of NEQ.
C
C The output parameters are:
C
C DKY Real array of length NEQ containing the computed value
C of the Kth derivative of y(t).
C IFLAG Integer flag, returned as 0 if K and T were legal,
C -1 if K was illegal, and -2 if T was illegal.
C On an error return, a message is also written.
C
C
C Part 3. Common Blocks
C ----------------------
C
C If DLSODE is to be used in an overlay situation, the user must
C declare, in the primary overlay, the variables in:
C (1) the call sequence to DLSODE,
C (2) the internal COMMON block /DLS001/, of length 255
C (218 double precision words followed by 37 integer words).
C
C If DLSODE is used on a system in which the contents of internal
C COMMON blocks are not preserved between calls, the user should
C declare the above COMMON block in his main program to insure that
C its contents are preserved.
C
C If the solution of a given problem by DLSODE is to be interrupted
C and then later continued, as when restarting an interrupted run or
C alternating between two or more problems, the user should save,
C following the return from the last DLSODE call prior to the
C interruption, the contents of the call sequence variables and the
C internal COMMON block, and later restore these values before the
C next DLSODE call for that problem. In addition, if XSETUN and/or
C XSETF was called for non-default handling of error messages, then
C these calls must be repeated. To save and restore the COMMON
C block, use subroutine DSRCOM (see Part 2 above).
C
C
C Part 4. Optionally Replaceable Solver Routines
C -----------------------------------------------
C
C Below are descriptions of two routines in the DLSODE package which
C relate to the measurement of errors. Either routine can be
C replaced by a user-supplied version, if desired. However, since
C such a replacement may have a major impact on performance, it
C should be done only when absolutely necessary, and only with great
C caution. (Note: The means by which the package version of a
C routine is superseded by the user's version may be system-
C dependent.)
C
C DEWSET
C ------
C The following subroutine is called just before each internal
C integration step, and sets the array of error weights, EWT, as
C described under ITOL/RTOL/ATOL above:
C
C SUBROUTINE DEWSET (NEQ, ITOL, RTOL, ATOL, YCUR, EWT)
C
C where NEQ, ITOL, RTOL, and ATOL are as in the DLSODE call
C sequence, YCUR contains the current dependent variable vector,
C and EWT is the array of weights set by DEWSET.
C
C If the user supplies this subroutine, it must return in EWT(i)
C (i = 1,...,NEQ) a positive quantity suitable for comparing errors
C in Y(i) to. The EWT array returned by DEWSET is passed to the
C DVNORM routine (see below), and also used by DLSODE in the
C computation of the optional output IMXER, the diagonal Jacobian
C approximation, and the increments for difference quotient
C Jacobians.
C
C In the user-supplied version of DEWSET, it may be desirable to use
C the current values of derivatives of y. Derivatives up to order NQ
C are available from the history array YH, described above under
C optional outputs. In DEWSET, YH is identical to the YCUR array,
C extended to NQ + 1 columns with a column length of NYH and scale
C factors of H**j/factorial(j). On the first call for the problem,
C given by NST = 0, NQ is 1 and H is temporarily set to 1.0.
C NYH is the initial value of NEQ. The quantities NQ, H, and NST
C can be obtained by including in SEWSET the statements:
C DOUBLE PRECISION RLS
C COMMON /DLS001/ RLS(218),ILS(37)
C NQ = ILS(33)
C NST = ILS(34)
C H = RLS(212)
C Thus, for example, the current value of dy/dt can be obtained as
C YCUR(NYH+i)/H (i=1,...,NEQ) (and the division by H is unnecessary
C when NST = 0).
C
C DVNORM
C ------
C DVNORM is a real function routine which computes the weighted
C root-mean-square norm of a vector v:
C
C d = DVNORM (n, v, w)
C
C where:
C n = the length of the vector,
C v = real array of length n containing the vector,
C w = real array of length n containing weights,
C d = SQRT( (1/n) * sum(v(i)*w(i))**2 ).
C
C DVNORM is called with n = NEQ and with w(i) = 1.0/EWT(i), where
C EWT is as set by subroutine DEWSET.
C
C If the user supplies this function, it should return a nonnegative
C value of DVNORM suitable for use in the error control in DLSODE.
C None of the arguments should be altered by DVNORM. For example, a
C user-supplied DVNORM routine might:
C - Substitute a max-norm of (v(i)*w(i)) for the rms-norm, or
C - Ignore some components of v in the norm, with the effect of
C suppressing the error control on those components of Y.
C ---------------------------------------------------------------------
C***ROUTINES CALLED DEWSET, DINTDY, DUMACH, DSTODE, DVNORM, XERRWD
C***COMMON BLOCKS DLS001
C***REVISION HISTORY (YYYYMMDD)
C 19791129 DATE WRITTEN
C 19791213 Minor changes to declarations; DELP init. in STODE.
C 19800118 Treat NEQ as array; integer declarations added throughout;
C minor changes to prologue.
C 19800306 Corrected TESCO(1,NQP1) setting in CFODE.
C 19800519 Corrected access of YH on forced order reduction;
C numerous corrections to prologues and other comments.
C 19800617 In main driver, added loading of SQRT(UROUND) in RWORK;
C minor corrections to main prologue.
C 19800923 Added zero initialization of HU and NQU.
C 19801218 Revised XERRWD routine; minor corrections to main prologue.
C 19810401 Minor changes to comments and an error message.
C 19810814 Numerous revisions: replaced EWT by 1/EWT; used flags
C JCUR, ICF, IERPJ, IERSL between STODE and subordinates;
C added tuning parameters CCMAX, MAXCOR, MSBP, MXNCF;
C reorganized returns from STODE; reorganized type decls.;
C fixed message length in XERRWD; changed default LUNIT to 6;
C changed Common lengths; changed comments throughout.
C 19870330 Major update by ACH: corrected comments throughout;
C removed TRET from Common; rewrote EWSET with 4 loops;
C fixed t test in INTDY; added Cray directives in STODE;
C in STODE, fixed DELP init. and logic around PJAC call;
C combined routines to save/restore Common;
C passed LEVEL = 0 in error message calls (except run abort).
C 19890426 Modified prologue to SLATEC/LDOC format. (FNF)
C 19890501 Many improvements to prologue. (FNF)
C 19890503 A few final corrections to prologue. (FNF)
C 19890504 Minor cosmetic changes. (FNF)
C 19890510 Corrected description of Y in Arguments section. (FNF)
C 19890517 Minor corrections to prologue. (FNF)
C 19920514 Updated with prologue edited 891025 by G. Shaw for manual.
C 19920515 Converted source lines to upper case. (FNF)
C 19920603 Revised XERRWD calls using mixed upper-lower case. (ACH)
C 19920616 Revised prologue comment regarding CFT. (ACH)
C 19921116 Revised prologue comments regarding Common. (ACH).
C 19930326 Added comment about non-reentrancy. (FNF)
C 19930723 Changed D1MACH to DUMACH. (FNF)
C 19930801 Removed ILLIN and NTREP from Common (affects driver logic);
C minor changes to prologue and internal comments;
C changed Hollerith strings to quoted strings;
C changed internal comments to mixed case;
C replaced XERRWD with new version using character type;
C changed dummy dimensions from 1 to *. (ACH)
C 19930809 Changed to generic intrinsic names; changed names of
C subprograms and Common blocks to DLSODE etc. (ACH)
C 19930929 Eliminated use of REAL intrinsic; other minor changes. (ACH)
C 20010412 Removed all 'own' variables from Common block /DLS001/
C (affects declarations in 6 routines). (ACH)
C 20010509 Minor corrections to prologue. (ACH)
C 20031105 Restored 'own' variables to Common block /DLS001/, to
C enable interrupt/restart feature. (ACH)
C 20031112 Added SAVE statements for data-loaded constants.
C
C***END PROLOGUE DLSODE
C
C*Internal Notes:
C
C Other Routines in the DLSODE Package.
C
C In addition to Subroutine DLSODE, the DLSODE package includes the
C following subroutines and function routines:
C DINTDY computes an interpolated value of the y vector at t = TOUT.
C DSTODE is the core integrator, which does one step of the
C integration and the associated error control.
C DCFODE sets all method coefficients and test constants.
C DPREPJ computes and preprocesses the Jacobian matrix J = df/dy
C and the Newton iteration matrix P = I - h*l0*J.
C DSOLSY manages solution of linear system in chord iteration.
C DEWSET sets the error weight vector EWT before each step.
C DVNORM computes the weighted R.M.S. norm of a vector.
C DSRCOM is a user-callable routine to save and restore
C the contents of the internal Common block.
C DGEFA and DGESL are routines from LINPACK for solving full
C systems of linear algebraic equations.
C DGBFA and DGBSL are routines from LINPACK for solving banded
C linear systems.
C DUMACH computes the unit roundoff in a machine-independent manner.
C XERRWD, XSETUN, XSETF, IXSAV, IUMACH handle the printing of all
C error messages and warnings. XERRWD is machine-dependent.
C Note: DVNORM, DUMACH, IXSAV, and IUMACH are function routines.
C All the others are subroutines.
C
C**End
C
C Declare externals.
EXTERNAL DPREPJ, DSOLSY
DOUBLE PRECISION DUMACH, DVNORM
C
C Declare all other variables.
INTEGER INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER I, I1, I2, IFLAG, IMXER, KGO, LF0,
1 LENIW, LENRW, LENWM, ML, MORD, MU, MXHNL0, MXSTP0
DOUBLE PRECISION ROWNS,
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
DOUBLE PRECISION ATOLI, AYI, BIG, EWTI, H0, HMAX, HMX, RH, RTOLI,
1 TCRIT, TDIST, TNEXT, TOL, TOLSF, TP, SIZE, SUM, W0
DIMENSION MORD(2)
LOGICAL IHIT
CHARACTER*80 MSG
SAVE MORD, MXSTP0, MXHNL0
C-----------------------------------------------------------------------
C The following internal Common block contains
C (a) variables which are local to any subroutine but whose values must
C be preserved between calls to the routine ("own" variables), and
C (b) variables which are communicated between subroutines.
C The block DLS001 is declared in subroutines DLSODE, DINTDY, DSTODE,
C DPREPJ, and DSOLSY.
C Groups of variables are replaced by dummy arrays in the Common
C declarations in routines where those variables are not used.
C-----------------------------------------------------------------------
COMMON /DLS001/ ROWNS(209),
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
2 INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS(6),
3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
DATA MORD(1),MORD(2)/12,5/, MXSTP0/500/, MXHNL0/10/
C-----------------------------------------------------------------------
C Block A.
C This code block is executed on every call.
C It tests ISTATE and ITASK for legality and branches appropriately.
C If ISTATE .GT. 1 but the flag INIT shows that initialization has
C not yet been done, an error return occurs.
C If ISTATE = 1 and TOUT = T, return immediately.
C-----------------------------------------------------------------------
C
C***FIRST EXECUTABLE STATEMENT DLSODE
IF (ISTATE .LT. 1 .OR. ISTATE .GT. 3) GO TO 601
IF (ITASK .LT. 1 .OR. ITASK .GT. 5) GO TO 602
IF (ISTATE .EQ. 1) GO TO 10
IF (INIT .EQ. 0) GO TO 603
IF (ISTATE .EQ. 2) GO TO 200
GO TO 20
10 INIT = 0
IF (TOUT .EQ. T) RETURN
C-----------------------------------------------------------------------
C Block B.
C The next code block is executed for the initial call (ISTATE = 1),
C or for a continuation call with parameter changes (ISTATE = 3).
C It contains checking of all inputs and various initializations.
C
C First check legality of the non-optional inputs NEQ, ITOL, IOPT,
C MF, ML, and MU.
C-----------------------------------------------------------------------
20 IF (NEQ(1) .LE. 0) GO TO 604
IF (ISTATE .EQ. 1) GO TO 25
IF (NEQ(1) .GT. N) GO TO 605
25 N = NEQ(1)
IF (ITOL .LT. 1 .OR. ITOL .GT. 4) GO TO 606
IF (IOPT .LT. 0 .OR. IOPT .GT. 1) GO TO 607
METH = MF/10
MITER = MF - 10*METH
IF (METH .LT. 1 .OR. METH .GT. 2) GO TO 608
IF (MITER .LT. 0 .OR. MITER .GT. 5) GO TO 608
IF (MITER .LE. 3) GO TO 30
ML = IWORK(1)
MU = IWORK(2)
IF (ML .LT. 0 .OR. ML .GE. N) GO TO 609
IF (MU .LT. 0 .OR. MU .GE. N) GO TO 610
30 CONTINUE
C Next process and check the optional inputs. --------------------------
IF (IOPT .EQ. 1) GO TO 40
MAXORD = MORD(METH)
MXSTEP = MXSTP0
MXHNIL = MXHNL0
IF (ISTATE .EQ. 1) H0 = 0.0D0
HMXI = 0.0D0
HMIN = 0.0D0
GO TO 60
40 MAXORD = IWORK(5)
IF (MAXORD .LT. 0) GO TO 611
IF (MAXORD .EQ. 0) MAXORD = 100
MAXORD = MIN(MAXORD,MORD(METH))
MXSTEP = IWORK(6)
IF (MXSTEP .LT. 0) GO TO 612
IF (MXSTEP .EQ. 0) MXSTEP = MXSTP0
MXHNIL = IWORK(7)
IF (MXHNIL .LT. 0) GO TO 613
IF (MXHNIL .EQ. 0) MXHNIL = MXHNL0
IF (ISTATE .NE. 1) GO TO 50
H0 = RWORK(5)
IF ((TOUT - T)*H0 .LT. 0.0D0) GO TO 614
50 HMAX = RWORK(6)
IF (HMAX .LT. 0.0D0) GO TO 615
HMXI = 0.0D0
IF (HMAX .GT. 0.0D0) HMXI = 1.0D0/HMAX
HMIN = RWORK(7)
IF (HMIN .LT. 0.0D0) GO TO 616
C-----------------------------------------------------------------------
C Set work array pointers and check lengths LRW and LIW.
C Pointers to segments of RWORK and IWORK are named by prefixing L to
C the name of the segment. E.g., the segment YH starts at RWORK(LYH).
C Segments of RWORK (in order) are denoted YH, WM, EWT, SAVF, ACOR.
C-----------------------------------------------------------------------
60 LYH = 21
IF (ISTATE .EQ. 1) NYH = N
LWM = LYH + (MAXORD + 1)*NYH
IF (MITER .EQ. 0) LENWM = 0
IF (MITER .EQ. 1 .OR. MITER .EQ. 2) LENWM = N*N + 2
IF (MITER .EQ. 3) LENWM = N + 2
IF (MITER .GE. 4) LENWM = (2*ML + MU + 1)*N + 2
LEWT = LWM + LENWM
LSAVF = LEWT + N
LACOR = LSAVF + N
LENRW = LACOR + N - 1
IWORK(17) = LENRW
LIWM = 1
LENIW = 20 + N
IF (MITER .EQ. 0 .OR. MITER .EQ. 3) LENIW = 20
IWORK(18) = LENIW
IF (LENRW .GT. LRW) GO TO 617
IF (LENIW .GT. LIW) GO TO 618
C Check RTOL and ATOL for legality. ------------------------------------
RTOLI = RTOL(1)
ATOLI = ATOL(1)
DO 70 I = 1,N
IF (ITOL .GE. 3) RTOLI = RTOL(I)
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
IF (RTOLI .LT. 0.0D0) GO TO 619
IF (ATOLI .LT. 0.0D0) GO TO 620
70 CONTINUE
IF (ISTATE .EQ. 1) GO TO 100
C If ISTATE = 3, set flag to signal parameter changes to DSTODE. -------
JSTART = -1
IF (NQ .LE. MAXORD) GO TO 90
C MAXORD was reduced below NQ. Copy YH(*,MAXORD+2) into SAVF. ---------
DO 80 I = 1,N
80 RWORK(I+LSAVF-1) = RWORK(I+LWM-1)
C Reload WM(1) = RWORK(LWM), since LWM may have changed. ---------------
90 IF (MITER .GT. 0) RWORK(LWM) = SQRT(UROUND)
IF (N .EQ. NYH) GO TO 200
C NEQ was reduced. Zero part of YH to avoid undefined references. -----
I1 = LYH + L*NYH
I2 = LYH + (MAXORD + 1)*NYH - 1
IF (I1 .GT. I2) GO TO 200
DO 95 I = I1,I2
95 RWORK(I) = 0.0D0
GO TO 200
C-----------------------------------------------------------------------
C Block C.
C The next block is for the initial call only (ISTATE = 1).
C It contains all remaining initializations, the initial call to F,
C and the calculation of the initial step size.
C The error weights in EWT are inverted after being loaded.
C-----------------------------------------------------------------------
100 UROUND = DUMACH()
TN = T
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 110
TCRIT = RWORK(1)
IF ((TCRIT - TOUT)*(TOUT - T) .LT. 0.0D0) GO TO 625
IF (H0 .NE. 0.0D0 .AND. (T + H0 - TCRIT)*H0 .GT. 0.0D0)
1 H0 = TCRIT - T
110 JSTART = 0
IF (MITER .GT. 0) RWORK(LWM) = SQRT(UROUND)
NHNIL = 0
NST = 0
NJE = 0
NSLAST = 0
HU = 0.0D0
NQU = 0
CCMAX = 0.3D0
MAXCOR = 3
MSBP = 20
MXNCF = 10
C Initial call to F. (LF0 points to YH(*,2).) -------------------------
LF0 = LYH + NYH
CALL F (NEQ, T, Y, RWORK(LF0))
NFE = 1
C Load the initial value vector in YH. ---------------------------------
DO 115 I = 1,N
115 RWORK(I+LYH-1) = Y(I)
C Load and invert the EWT array. (H is temporarily set to 1.0.) -------
NQ = 1
H = 1.0D0
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 120 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 621
120 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
C-----------------------------------------------------------------------
C The coding below computes the step size, H0, to be attempted on the
C first step, unless the user has supplied a value for this.
C First check that TOUT - T differs significantly from zero.
C A scalar tolerance quantity TOL is computed, as MAX(RTOL(I))
C if this is positive, or MAX(ATOL(I)/ABS(Y(I))) otherwise, adjusted
C so as to be between 100*UROUND and 1.0E-3.
C Then the computed value H0 is given by..
C NEQ
C H0**2 = TOL / ( w0**-2 + (1/NEQ) * SUM ( f(i)/ywt(i) )**2 )
C 1
C where w0 = MAX ( ABS(T), ABS(TOUT) ),
C f(i) = i-th component of initial value of f,
C ywt(i) = EWT(i)/TOL (a weight for y(i)).
C The sign of H0 is inferred from the initial values of TOUT and T.
C-----------------------------------------------------------------------
IF (H0 .NE. 0.0D0) GO TO 180
TDIST = ABS(TOUT - T)
W0 = MAX(ABS(T),ABS(TOUT))
IF (TDIST .LT. 2.0D0*UROUND*W0) GO TO 622
TOL = RTOL(1)
IF (ITOL .LE. 2) GO TO 140
DO 130 I = 1,N
130 TOL = MAX(TOL,RTOL(I))
140 IF (TOL .GT. 0.0D0) GO TO 160
ATOLI = ATOL(1)
DO 150 I = 1,N
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
AYI = ABS(Y(I))
IF (AYI .NE. 0.0D0) TOL = MAX(TOL,ATOLI/AYI)
150 CONTINUE
160 TOL = MAX(TOL,100.0D0*UROUND)
TOL = MIN(TOL,0.001D0)
SUM = DVNORM (N, RWORK(LF0), RWORK(LEWT))
SUM = 1.0D0/(TOL*W0*W0) + TOL*SUM**2
H0 = 1.0D0/SQRT(SUM)
H0 = MIN(H0,TDIST)
H0 = SIGN(H0,TOUT-T)
C Adjust H0 if necessary to meet HMAX bound. ---------------------------
180 RH = ABS(H0)*HMXI
IF (RH .GT. 1.0D0) H0 = H0/RH
C Load H with H0 and scale YH(*,2) by H0. ------------------------------
H = H0
DO 190 I = 1,N
190 RWORK(I+LF0-1) = H0*RWORK(I+LF0-1)
GO TO 270
C-----------------------------------------------------------------------
C Block D.
C The next code block is for continuation calls only (ISTATE = 2 or 3)
C and is to check stop conditions before taking a step.
C-----------------------------------------------------------------------
200 NSLAST = NST
GO TO (210, 250, 220, 230, 240), ITASK
210 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
220 TP = TN - HU*(1.0D0 + 100.0D0*UROUND)
IF ((TP - TOUT)*H .GT. 0.0D0) GO TO 623
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
GO TO 400
230 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
IF ((TCRIT - TOUT)*H .LT. 0.0D0) GO TO 625
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 245
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
240 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
245 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
IF (ISTATE .EQ. 2) JSTART = -2
C-----------------------------------------------------------------------
C Block E.
C The next block is normally executed for all calls and contains
C the call to the one-step core integrator DSTODE.
C
C This is a looping point for the integration steps.
C
C First check for too many steps being taken, update EWT (if not at
C start of problem), check for too much accuracy being requested, and
C check for H below the roundoff level in T.
C-----------------------------------------------------------------------
250 CONTINUE
IF ((NST-NSLAST) .GE. MXSTEP) GO TO 500
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 260 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 510
260 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
270 TOLSF = UROUND*DVNORM (N, RWORK(LYH), RWORK(LEWT))
IF (TOLSF .LE. 1.0D0) GO TO 280
TOLSF = TOLSF*2.0D0
IF (NST .EQ. 0) GO TO 626
GO TO 520
280 IF ((TN + H) .NE. TN) GO TO 290
NHNIL = NHNIL + 1
IF (NHNIL .GT. MXHNIL) GO TO 290
MSG = 'DLSODE- Warning..internal T (=R1) and H (=R2) are'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' such that in the machine, T + H = T on the next step '
CALL XERRWD (MSG, 60, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' (H = step size). Solver will continue anyway'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 2, TN, H)
IF (NHNIL .LT. MXHNIL) GO TO 290
MSG = 'DLSODE- Above warning has been issued I1 times. '
CALL XERRWD (MSG, 50, 102, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' It will not be issued again for this problem'
CALL XERRWD (MSG, 50, 102, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
290 CONTINUE
C-----------------------------------------------------------------------
C CALL DSTODE(NEQ,Y,YH,NYH,YH,EWT,SAVF,ACOR,WM,IWM,F,JAC,DPREPJ,DSOLSY)
C-----------------------------------------------------------------------
CALL DSTODE (NEQ, Y, RWORK(LYH), NYH, RWORK(LYH), RWORK(LEWT),
1 RWORK(LSAVF), RWORK(LACOR), RWORK(LWM), IWORK(LIWM),
2 F, JAC, DPREPJ, DSOLSY)
KGO = 1 - KFLAG
GO TO (300, 530, 540), KGO
C-----------------------------------------------------------------------
C Block F.
C The following block handles the case of a successful return from the
C core integrator (KFLAG = 0). Test for stop conditions.
C-----------------------------------------------------------------------
300 INIT = 1
GO TO (310, 400, 330, 340, 350), ITASK
C ITASK = 1. If TOUT has been reached, interpolate. -------------------
310 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
C ITASK = 3. Jump to exit if TOUT was reached. ------------------------
330 IF ((TN - TOUT)*H .GE. 0.0D0) GO TO 400
GO TO 250
C ITASK = 4. See if TOUT or TCRIT was reached. Adjust H if necessary.
340 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 345
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
345 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
JSTART = -2
GO TO 250
C ITASK = 5. See if TCRIT was reached and jump to exit. ---------------
350 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
C-----------------------------------------------------------------------
C Block G.
C The following block handles all successful returns from DLSODE.
C If ITASK .NE. 1, Y is loaded from YH and T is set accordingly.
C ISTATE is set to 2, and the optional outputs are loaded into the
C work arrays before returning.
C-----------------------------------------------------------------------
400 DO 410 I = 1,N
410 Y(I) = RWORK(I+LYH-1)
T = TN
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 420
IF (IHIT) T = TCRIT
420 ISTATE = 2
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
RETURN
C-----------------------------------------------------------------------
C Block H.
C The following block handles all unsuccessful returns other than
C those for illegal input. First the error message routine is called.
C If there was an error test or convergence test failure, IMXER is set.
C Then Y is loaded from YH and T is set to TN. The optional outputs
C are loaded into the work arrays before returning.
C-----------------------------------------------------------------------
C The maximum number of steps was taken before reaching TOUT. ----------
500 MSG = 'DLSODE- At current T (=R1), MXSTEP (=I1) steps '
CALL XERRWD (MSG, 50, 201, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' taken on this call before reaching TOUT '
CALL XERRWD (MSG, 50, 201, 0, 1, MXSTEP, 0, 1, TN, 0.0D0)
ISTATE = -1
GO TO 580
C EWT(I) .LE. 0.0 for some I (not at start of problem). ----------------
510 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODE- At T (=R1), EWT(I1) has become R2 .LE. 0.'
CALL XERRWD (MSG, 50, 202, 0, 1, I, 0, 2, TN, EWTI)
ISTATE = -6
GO TO 580
C Too much accuracy requested for machine precision. -------------------
520 MSG = 'DLSODE- At T (=R1), too much accuracy requested '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' for precision of machine.. see TOLSF (=R2) '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 2, TN, TOLSF)
RWORK(14) = TOLSF
ISTATE = -2
GO TO 580
C KFLAG = -1. Error test failed repeatedly or with ABS(H) = HMIN. -----
530 MSG = 'DLSODE- At T(=R1) and step size H(=R2), the error'
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' test failed repeatedly or with ABS(H) = HMIN'
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 2, TN, H)
ISTATE = -4
GO TO 560
C KFLAG = -2. Convergence failed repeatedly or with ABS(H) = HMIN. ----
540 MSG = 'DLSODE- At T (=R1) and step size H (=R2), the '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' corrector convergence failed repeatedly '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' or with ABS(H) = HMIN '
CALL XERRWD (MSG, 30, 205, 0, 0, 0, 0, 2, TN, H)
ISTATE = -5
C Compute IMXER if relevant. -------------------------------------------
560 BIG = 0.0D0
IMXER = 1
DO 570 I = 1,N
SIZE = ABS(RWORK(I+LACOR-1)*RWORK(I+LEWT-1))
IF (BIG .GE. SIZE) GO TO 570
BIG = SIZE
IMXER = I
570 CONTINUE
IWORK(16) = IMXER
C Set Y vector, T, and optional outputs. -------------------------------
580 DO 590 I = 1,N
590 Y(I) = RWORK(I+LYH-1)
T = TN
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
RETURN
C-----------------------------------------------------------------------
C Block I.
C The following block handles all error returns due to illegal input
C (ISTATE = -3), as detected before calling the core integrator.
C First the error message routine is called. If the illegal input
C is a negative ISTATE, the run is aborted (apparent infinite loop).
C-----------------------------------------------------------------------
601 MSG = 'DLSODE- ISTATE (=I1) illegal '
CALL XERRWD (MSG, 30, 1, 0, 1, ISTATE, 0, 0, 0.0D0, 0.0D0)
IF (ISTATE .LT. 0) GO TO 800
GO TO 700
602 MSG = 'DLSODE- ITASK (=I1) illegal '
CALL XERRWD (MSG, 30, 2, 0, 1, ITASK, 0, 0, 0.0D0, 0.0D0)
GO TO 700
603 MSG = 'DLSODE- ISTATE .GT. 1 but DLSODE not initialized '
CALL XERRWD (MSG, 50, 3, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
GO TO 700
604 MSG = 'DLSODE- NEQ (=I1) .LT. 1 '
CALL XERRWD (MSG, 30, 4, 0, 1, NEQ(1), 0, 0, 0.0D0, 0.0D0)
GO TO 700
605 MSG = 'DLSODE- ISTATE = 3 and NEQ increased (I1 to I2) '
CALL XERRWD (MSG, 50, 5, 0, 2, N, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
606 MSG = 'DLSODE- ITOL (=I1) illegal '
CALL XERRWD (MSG, 30, 6, 0, 1, ITOL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
607 MSG = 'DLSODE- IOPT (=I1) illegal '
CALL XERRWD (MSG, 30, 7, 0, 1, IOPT, 0, 0, 0.0D0, 0.0D0)
GO TO 700
608 MSG = 'DLSODE- MF (=I1) illegal '
CALL XERRWD (MSG, 30, 8, 0, 1, MF, 0, 0, 0.0D0, 0.0D0)
GO TO 700
609 MSG = 'DLSODE- ML (=I1) illegal.. .LT.0 or .GE.NEQ (=I2)'
CALL XERRWD (MSG, 50, 9, 0, 2, ML, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
610 MSG = 'DLSODE- MU (=I1) illegal.. .LT.0 or .GE.NEQ (=I2)'
CALL XERRWD (MSG, 50, 10, 0, 2, MU, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
611 MSG = 'DLSODE- MAXORD (=I1) .LT. 0 '
CALL XERRWD (MSG, 30, 11, 0, 1, MAXORD, 0, 0, 0.0D0, 0.0D0)
GO TO 700
612 MSG = 'DLSODE- MXSTEP (=I1) .LT. 0 '
CALL XERRWD (MSG, 30, 12, 0, 1, MXSTEP, 0, 0, 0.0D0, 0.0D0)
GO TO 700
613 MSG = 'DLSODE- MXHNIL (=I1) .LT. 0 '
CALL XERRWD (MSG, 30, 13, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
614 MSG = 'DLSODE- TOUT (=R1) behind T (=R2) '
CALL XERRWD (MSG, 40, 14, 0, 0, 0, 0, 2, TOUT, T)
MSG = ' Integration direction is given by H0 (=R1) '
CALL XERRWD (MSG, 50, 14, 0, 0, 0, 0, 1, H0, 0.0D0)
GO TO 700
615 MSG = 'DLSODE- HMAX (=R1) .LT. 0.0 '
CALL XERRWD (MSG, 30, 15, 0, 0, 0, 0, 1, HMAX, 0.0D0)
GO TO 700
616 MSG = 'DLSODE- HMIN (=R1) .LT. 0.0 '
CALL XERRWD (MSG, 30, 16, 0, 0, 0, 0, 1, HMIN, 0.0D0)
GO TO 700
617 CONTINUE
MSG='DLSODE- RWORK length needed, LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 17, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
618 CONTINUE
MSG='DLSODE- IWORK length needed, LENIW (=I1), exceeds LIW (=I2)'
CALL XERRWD (MSG, 60, 18, 0, 2, LENIW, LIW, 0, 0.0D0, 0.0D0)
GO TO 700
619 MSG = 'DLSODE- RTOL(I1) is R1 .LT. 0.0 '
CALL XERRWD (MSG, 40, 19, 0, 1, I, 0, 1, RTOLI, 0.0D0)
GO TO 700
620 MSG = 'DLSODE- ATOL(I1) is R1 .LT. 0.0 '
CALL XERRWD (MSG, 40, 20, 0, 1, I, 0, 1, ATOLI, 0.0D0)
GO TO 700
621 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODE- EWT(I1) is R1 .LE. 0.0 '
CALL XERRWD (MSG, 40, 21, 0, 1, I, 0, 1, EWTI, 0.0D0)
GO TO 700
622 CONTINUE
MSG='DLSODE- TOUT (=R1) too close to T(=R2) to start integration'
CALL XERRWD (MSG, 60, 22, 0, 0, 0, 0, 2, TOUT, T)
GO TO 700
623 CONTINUE
MSG='DLSODE- ITASK = I1 and TOUT (=R1) behind TCUR - HU (= R2) '
CALL XERRWD (MSG, 60, 23, 0, 1, ITASK, 0, 2, TOUT, TP)
GO TO 700
624 CONTINUE
MSG='DLSODE- ITASK = 4 OR 5 and TCRIT (=R1) behind TCUR (=R2) '
CALL XERRWD (MSG, 60, 24, 0, 0, 0, 0, 2, TCRIT, TN)
GO TO 700
625 CONTINUE
MSG='DLSODE- ITASK = 4 or 5 and TCRIT (=R1) behind TOUT (=R2) '
CALL XERRWD (MSG, 60, 25, 0, 0, 0, 0, 2, TCRIT, TOUT)
GO TO 700
626 MSG = 'DLSODE- At start of problem, too much accuracy '
CALL XERRWD (MSG, 50, 26, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' requested for precision of machine.. See TOLSF (=R1) '
CALL XERRWD (MSG, 60, 26, 0, 0, 0, 0, 1, TOLSF, 0.0D0)
RWORK(14) = TOLSF
GO TO 700
627 MSG = 'DLSODE- Trouble in DINTDY. ITASK = I1, TOUT = R1'
CALL XERRWD (MSG, 50, 27, 0, 1, ITASK, 0, 1, TOUT, 0.0D0)
C
700 ISTATE = -3
RETURN
C
800 MSG = 'DLSODE- Run aborted.. apparent infinite loop '
CALL XERRWD (MSG, 50, 303, 2, 0, 0, 0, 0, 0.0D0, 0.0D0)
RETURN
C----------------------- END OF SUBROUTINE DLSODE ----------------------
END
*DECK DLSODES
SUBROUTINE DLSODES (F, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
1 ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JAC, MF)
EXTERNAL F, JAC
INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK, LIW, MF
DOUBLE PRECISION Y, T, TOUT, RTOL, ATOL, RWORK
DIMENSION NEQ(*), Y(*), RTOL(*), ATOL(*), RWORK(LRW), IWORK(LIW)
C-----------------------------------------------------------------------
C This is the 12 November 2003 version of
C DLSODES: Livermore Solver for Ordinary Differential Equations
C with general Sparse Jacobian matrix.
C
C This version is in double precision.
C
C DLSODES solves the initial value problem for stiff or nonstiff
C systems of first order ODEs,
C dy/dt = f(t,y) , or, in component form,
C dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(NEQ)) (i = 1,...,NEQ).
C DLSODES is a variant of the DLSODE package, and is intended for
C problems in which the Jacobian matrix df/dy has an arbitrary
C sparse structure (when the problem is stiff).
C
C Authors: Alan C. Hindmarsh
C Center for Applied Scientific Computing, L-561
C Lawrence Livermore National Laboratory
C Livermore, CA 94551
C and
C Andrew H. Sherman
C J. S. Nolen and Associates
C Houston, TX 77084
C-----------------------------------------------------------------------
C References:
C 1. Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE
C Solvers, in Scientific Computing, R. S. Stepleman et al. (Eds.),
C North-Holland, Amsterdam, 1983, pp. 55-64.
C
C 2. S. C. Eisenstat, M. C. Gursky, M. H. Schultz, and A. H. Sherman,
C Yale Sparse Matrix Package: I. The Symmetric Codes,
C Int. J. Num. Meth. Eng., 18 (1982), pp. 1145-1151.
C
C 3. S. C. Eisenstat, M. C. Gursky, M. H. Schultz, and A. H. Sherman,
C Yale Sparse Matrix Package: II. The Nonsymmetric Codes,
C Research Report No. 114, Dept. of Computer Sciences, Yale
C University, 1977.
C-----------------------------------------------------------------------
C Summary of Usage.
C
C Communication between the user and the DLSODES package, for normal
C situations, is summarized here. This summary describes only a subset
C of the full set of options available. See the full description for
C details, including optional communication, nonstandard options,
C and instructions for special situations. See also the example
C problem (with program and output) following this summary.
C
C A. First provide a subroutine of the form:
C SUBROUTINE F (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y(*), YDOT(*)
C which supplies the vector function f by loading YDOT(i) with f(i).
C
C B. Next determine (or guess) whether or not the problem is stiff.
C Stiffness occurs when the Jacobian matrix df/dy has an eigenvalue
C whose real part is negative and large in magnitude, compared to the
C reciprocal of the t span of interest. If the problem is nonstiff,
C use a method flag MF = 10. If it is stiff, there are two standard
C choices for the method flag, MF = 121 and MF = 222. In both cases,
C DLSODES requires the Jacobian matrix in some form, and it treats this
C matrix in general sparse form, with sparsity structure determined
C internally. (For options where the user supplies the sparsity
C structure, see the full description of MF below.)
C
C C. If the problem is stiff, you are encouraged to supply the Jacobian
C directly (MF = 121), but if this is not feasible, DLSODES will
C compute it internally by difference quotients (MF = 222).
C If you are supplying the Jacobian, provide a subroutine of the form:
C SUBROUTINE JAC (NEQ, T, Y, J, IAN, JAN, PDJ)
C DOUBLE PRECISION T, Y(*), IAN(*), JAN(*), PDJ(*)
C Here NEQ, T, Y, and J are input arguments, and the JAC routine is to
C load the array PDJ (of length NEQ) with the J-th column of df/dy.
C I.e., load PDJ(i) with df(i)/dy(J) for all relevant values of i.
C The arguments IAN and JAN should be ignored for normal situations.
C DLSODES will call the JAC routine with J = 1,2,...,NEQ.
C Only nonzero elements need be loaded. Usually, a crude approximation
C to df/dy, possibly with fewer nonzero elements, will suffice.
C
C D. Write a main program which calls Subroutine DLSODES once for
C each point at which answers are desired. This should also provide
C for possible use of logical unit 6 for output of error messages by
C DLSODES. On the first call to DLSODES, supply arguments as follows:
C F = name of subroutine for right-hand side vector f.
C This name must be declared External in calling program.
C NEQ = number of first order ODEs.
C Y = array of initial values, of length NEQ.
C T = the initial value of the independent variable t.
C TOUT = first point where output is desired (.ne. T).
C ITOL = 1 or 2 according as ATOL (below) is a scalar or array.
C RTOL = relative tolerance parameter (scalar).
C ATOL = absolute tolerance parameter (scalar or array).
C The estimated local error in Y(i) will be controlled so as
C to be roughly less (in magnitude) than
C EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
C EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
C Thus the local error test passes if, in each component,
C either the absolute error is less than ATOL (or ATOL(i)),
C or the relative error is less than RTOL.
C Use RTOL = 0.0 for pure absolute error control, and
C use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative error
C control. Caution: actual (global) errors may exceed these
C local tolerances, so choose them conservatively.
C ITASK = 1 for normal computation of output values of Y at t = TOUT.
C ISTATE = integer flag (input and output). Set ISTATE = 1.
C IOPT = 0 to indicate no optional inputs used.
C RWORK = real work array of length at least:
C 20 + 16*NEQ for MF = 10,
C 20 + (2 + 1./LENRAT)*NNZ + (11 + 9./LENRAT)*NEQ
C for MF = 121 or 222,
C where:
C NNZ = the number of nonzero elements in the sparse
C Jacobian (if this is unknown, use an estimate), and
C LENRAT = the real to integer wordlength ratio (usually 1 in
C single precision and 2 in double precision).
C In any case, the required size of RWORK cannot generally
C be predicted in advance if MF = 121 or 222, and the value
C above is a rough estimate of a crude lower bound. Some
C experimentation with this size may be necessary.
C (When known, the correct required length is an optional
C output, available in IWORK(17).)
C LRW = declared length of RWORK (in user dimension).
C IWORK = integer work array of length at least 30.
C LIW = declared length of IWORK (in user dimension).
C JAC = name of subroutine for Jacobian matrix (MF = 121).
C If used, this name must be declared External in calling
C program. If not used, pass a dummy name.
C MF = method flag. Standard values are:
C 10 for nonstiff (Adams) method, no Jacobian used
C 121 for stiff (BDF) method, user-supplied sparse Jacobian
C 222 for stiff method, internally generated sparse Jacobian
C Note that the main program must declare arrays Y, RWORK, IWORK,
C and possibly ATOL.
C
C E. The output from the first call (or any call) is:
C Y = array of computed values of y(t) vector.
C T = corresponding value of independent variable (normally TOUT).
C ISTATE = 2 if DLSODES was successful, negative otherwise.
C -1 means excess work done on this call (perhaps wrong MF).
C -2 means excess accuracy requested (tolerances too small).
C -3 means illegal input detected (see printed message).
C -4 means repeated error test failures (check all inputs).
C -5 means repeated convergence failures (perhaps bad Jacobian
C supplied or wrong choice of MF or tolerances).
C -6 means error weight became zero during problem. (Solution
C component i vanished, and ATOL or ATOL(i) = 0.)
C -7 means a fatal error return flag came from sparse solver
C CDRV by way of DPRJS or DSOLSS. Should never happen.
C A return with ISTATE = -1, -4, or -5 may result from using
C an inappropriate sparsity structure, one that is quite
C different from the initial structure. Consider calling
C DLSODES again with ISTATE = 3 to force the structure to be
C reevaluated. See the full description of ISTATE below.
C
C F. To continue the integration after a successful return, simply
C reset TOUT and call DLSODES again. No other parameters need be reset.
C
C-----------------------------------------------------------------------
C Example Problem.
C
C The following is a simple example problem, with the coding
C needed for its solution by DLSODES. The problem is from chemical
C kinetics, and consists of the following 12 rate equations:
C dy1/dt = -rk1*y1
C dy2/dt = rk1*y1 + rk11*rk14*y4 + rk19*rk14*y5
C - rk3*y2*y3 - rk15*y2*y12 - rk2*y2
C dy3/dt = rk2*y2 - rk5*y3 - rk3*y2*y3 - rk7*y10*y3
C + rk11*rk14*y4 + rk12*rk14*y6
C dy4/dt = rk3*y2*y3 - rk11*rk14*y4 - rk4*y4
C dy5/dt = rk15*y2*y12 - rk19*rk14*y5 - rk16*y5
C dy6/dt = rk7*y10*y3 - rk12*rk14*y6 - rk8*y6
C dy7/dt = rk17*y10*y12 - rk20*rk14*y7 - rk18*y7
C dy8/dt = rk9*y10 - rk13*rk14*y8 - rk10*y8
C dy9/dt = rk4*y4 + rk16*y5 + rk8*y6 + rk18*y7
C dy10/dt = rk5*y3 + rk12*rk14*y6 + rk20*rk14*y7
C + rk13*rk14*y8 - rk7*y10*y3 - rk17*y10*y12
C - rk6*y10 - rk9*y10
C dy11/dt = rk10*y8
C dy12/dt = rk6*y10 + rk19*rk14*y5 + rk20*rk14*y7
C - rk15*y2*y12 - rk17*y10*y12
C
C with rk1 = rk5 = 0.1, rk4 = rk8 = rk16 = rk18 = 2.5,
C rk10 = 5.0, rk2 = rk6 = 10.0, rk14 = 30.0,
C rk3 = rk7 = rk9 = rk11 = rk12 = rk13 = rk19 = rk20 = 50.0,
C rk15 = rk17 = 100.0.
C
C The t interval is from 0 to 1000, and the initial conditions
C are y1 = 1, y2 = y3 = ... = y12 = 0. The problem is stiff.
C
C The following coding solves this problem with DLSODES, using MF = 121
C and printing results at t = .1, 1., 10., 100., 1000. It uses
C ITOL = 1 and mixed relative/absolute tolerance controls.
C During the run and at the end, statistical quantities of interest
C are printed (see optional outputs in the full description below).
C
C EXTERNAL FEX, JEX
C DOUBLE PRECISION ATOL, RTOL, RWORK, T, TOUT, Y
C DIMENSION Y(12), RWORK(500), IWORK(30)
C DATA LRW/500/, LIW/30/
C NEQ = 12
C DO 10 I = 1,NEQ
C 10 Y(I) = 0.0D0
C Y(1) = 1.0D0
C T = 0.0D0
C TOUT = 0.1D0
C ITOL = 1
C RTOL = 1.0D-4
C ATOL = 1.0D-6
C ITASK = 1
C ISTATE = 1
C IOPT = 0
C MF = 121
C DO 40 IOUT = 1,5
C CALL DLSODES (FEX, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL,
C 1 ITASK, ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JEX, MF)
C WRITE(6,30)T,IWORK(11),RWORK(11),(Y(I),I=1,NEQ)
C 30 FORMAT(//' At t =',D11.3,4X,
C 1 ' No. steps =',I5,4X,' Last step =',D11.3/
C 2 ' Y array = ',4D14.5/13X,4D14.5/13X,4D14.5)
C IF (ISTATE .LT. 0) GO TO 80
C TOUT = TOUT*10.0D0
C 40 CONTINUE
C LENRW = IWORK(17)
C LENIW = IWORK(18)
C NST = IWORK(11)
C NFE = IWORK(12)
C NJE = IWORK(13)
C NLU = IWORK(21)
C NNZ = IWORK(19)
C NNZLU = IWORK(25) + IWORK(26) + NEQ
C WRITE (6,70) LENRW,LENIW,NST,NFE,NJE,NLU,NNZ,NNZLU
C 70 FORMAT(//' Required RWORK size =',I4,' IWORK size =',I4/
C 1 ' No. steps =',I4,' No. f-s =',I4,' No. J-s =',I4,
C 2 ' No. LU-s =',I4/' No. of nonzeros in J =',I5,
C 3 ' No. of nonzeros in LU =',I5)
C STOP
C 80 WRITE(6,90)ISTATE
C 90 FORMAT(///' Error halt.. ISTATE =',I3)
C STOP
C END
C
C SUBROUTINE FEX (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y, YDOT
C DOUBLE PRECISION RK1, RK2, RK3, RK4, RK5, RK6, RK7, RK8, RK9,
C 1 RK10, RK11, RK12, RK13, RK14, RK15, RK16, RK17
C DIMENSION Y(12), YDOT(12)
C DATA RK1/0.1D0/, RK2/10.0D0/, RK3/50.0D0/, RK4/2.5D0/, RK5/0.1D0/,
C 1 RK6/10.0D0/, RK7/50.0D0/, RK8/2.5D0/, RK9/50.0D0/, RK10/5.0D0/,
C 2 RK11/50.0D0/, RK12/50.0D0/, RK13/50.0D0/, RK14/30.0D0/,
C 3 RK15/100.0D0/, RK16/2.5D0/, RK17/100.0D0/, RK18/2.5D0/,
C 4 RK19/50.0D0/, RK20/50.0D0/
C YDOT(1) = -RK1*Y(1)
C YDOT(2) = RK1*Y(1) + RK11*RK14*Y(4) + RK19*RK14*Y(5)
C 1 - RK3*Y(2)*Y(3) - RK15*Y(2)*Y(12) - RK2*Y(2)
C YDOT(3) = RK2*Y(2) - RK5*Y(3) - RK3*Y(2)*Y(3) - RK7*Y(10)*Y(3)
C 1 + RK11*RK14*Y(4) + RK12*RK14*Y(6)
C YDOT(4) = RK3*Y(2)*Y(3) - RK11*RK14*Y(4) - RK4*Y(4)
C YDOT(5) = RK15*Y(2)*Y(12) - RK19*RK14*Y(5) - RK16*Y(5)
C YDOT(6) = RK7*Y(10)*Y(3) - RK12*RK14*Y(6) - RK8*Y(6)
C YDOT(7) = RK17*Y(10)*Y(12) - RK20*RK14*Y(7) - RK18*Y(7)
C YDOT(8) = RK9*Y(10) - RK13*RK14*Y(8) - RK10*Y(8)
C YDOT(9) = RK4*Y(4) + RK16*Y(5) + RK8*Y(6) + RK18*Y(7)
C YDOT(10) = RK5*Y(3) + RK12*RK14*Y(6) + RK20*RK14*Y(7)
C 1 + RK13*RK14*Y(8) - RK7*Y(10)*Y(3) - RK17*Y(10)*Y(12)
C 2 - RK6*Y(10) - RK9*Y(10)
C YDOT(11) = RK10*Y(8)
C YDOT(12) = RK6*Y(10) + RK19*RK14*Y(5) + RK20*RK14*Y(7)
C 1 - RK15*Y(2)*Y(12) - RK17*Y(10)*Y(12)
C RETURN
C END
C
C SUBROUTINE JEX (NEQ, T, Y, J, IA, JA, PDJ)
C DOUBLE PRECISION T, Y, PDJ
C DOUBLE PRECISION RK1, RK2, RK3, RK4, RK5, RK6, RK7, RK8, RK9,
C 1 RK10, RK11, RK12, RK13, RK14, RK15, RK16, RK17
C DIMENSION Y(12), IA(*), JA(*), PDJ(12)
C DATA RK1/0.1D0/, RK2/10.0D0/, RK3/50.0D0/, RK4/2.5D0/, RK5/0.1D0/,
C 1 RK6/10.0D0/, RK7/50.0D0/, RK8/2.5D0/, RK9/50.0D0/, RK10/5.0D0/,
C 2 RK11/50.0D0/, RK12/50.0D0/, RK13/50.0D0/, RK14/30.0D0/,
C 3 RK15/100.0D0/, RK16/2.5D0/, RK17/100.0D0/, RK18/2.5D0/,
C 4 RK19/50.0D0/, RK20/50.0D0/
C GO TO (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), J
C 1 PDJ(1) = -RK1
C PDJ(2) = RK1
C RETURN
C 2 PDJ(2) = -RK3*Y(3) - RK15*Y(12) - RK2
C PDJ(3) = RK2 - RK3*Y(3)
C PDJ(4) = RK3*Y(3)
C PDJ(5) = RK15*Y(12)
C PDJ(12) = -RK15*Y(12)
C RETURN
C 3 PDJ(2) = -RK3*Y(2)
C PDJ(3) = -RK5 - RK3*Y(2) - RK7*Y(10)
C PDJ(4) = RK3*Y(2)
C PDJ(6) = RK7*Y(10)
C PDJ(10) = RK5 - RK7*Y(10)
C RETURN
C 4 PDJ(2) = RK11*RK14
C PDJ(3) = RK11*RK14
C PDJ(4) = -RK11*RK14 - RK4
C PDJ(9) = RK4
C RETURN
C 5 PDJ(2) = RK19*RK14
C PDJ(5) = -RK19*RK14 - RK16
C PDJ(9) = RK16
C PDJ(12) = RK19*RK14
C RETURN
C 6 PDJ(3) = RK12*RK14
C PDJ(6) = -RK12*RK14 - RK8
C PDJ(9) = RK8
C PDJ(10) = RK12*RK14
C RETURN
C 7 PDJ(7) = -RK20*RK14 - RK18
C PDJ(9) = RK18
C PDJ(10) = RK20*RK14
C PDJ(12) = RK20*RK14
C RETURN
C 8 PDJ(8) = -RK13*RK14 - RK10
C PDJ(10) = RK13*RK14
C PDJ(11) = RK10
C 9 RETURN
C 10 PDJ(3) = -RK7*Y(3)
C PDJ(6) = RK7*Y(3)
C PDJ(7) = RK17*Y(12)
C PDJ(8) = RK9
C PDJ(10) = -RK7*Y(3) - RK17*Y(12) - RK6 - RK9
C PDJ(12) = RK6 - RK17*Y(12)
C 11 RETURN
C 12 PDJ(2) = -RK15*Y(2)
C PDJ(5) = RK15*Y(2)
C PDJ(7) = RK17*Y(10)
C PDJ(10) = -RK17*Y(10)
C PDJ(12) = -RK15*Y(2) - RK17*Y(10)
C RETURN
C END
C
C The output of this program (on a Cray-1 in single precision)
C is as follows:
C
C
C At t = 1.000e-01 No. steps = 12 Last step = 1.515e-02
C Y array = 9.90050e-01 6.28228e-03 3.65313e-03 7.51934e-07
C 1.12167e-09 1.18458e-09 1.77291e-12 3.26476e-07
C 5.46720e-08 9.99500e-06 4.48483e-08 2.76398e-06
C
C
C At t = 1.000e+00 No. steps = 33 Last step = 7.880e-02
C Y array = 9.04837e-01 9.13105e-03 8.20622e-02 2.49177e-05
C 1.85055e-06 1.96797e-06 1.46157e-07 2.39557e-05
C 3.26306e-05 7.21621e-04 5.06433e-05 3.05010e-03
C
C
C At t = 1.000e+01 No. steps = 48 Last step = 1.239e+00
C Y array = 3.67876e-01 3.68958e-03 3.65133e-01 4.48325e-05
C 6.10798e-05 4.33148e-05 5.90211e-05 1.18449e-04
C 3.15235e-03 3.56531e-03 4.15520e-03 2.48741e-01
C
C
C At t = 1.000e+02 No. steps = 91 Last step = 3.764e+00
C Y array = 4.44981e-05 4.42666e-07 4.47273e-04 -3.53257e-11
C 2.81577e-08 -9.67741e-11 2.77615e-07 1.45322e-07
C 1.56230e-02 4.37394e-06 1.60104e-02 9.52246e-01
C
C
C At t = 1.000e+03 No. steps = 111 Last step = 4.156e+02
C Y array = -2.65492e-13 2.60539e-14 -8.59563e-12 6.29355e-14
C -1.78066e-13 5.71471e-13 -1.47561e-12 4.58078e-15
C 1.56314e-02 1.37878e-13 1.60184e-02 9.52719e-01
C
C
C Required RWORK size = 442 IWORK size = 30
C No. steps = 111 No. f-s = 142 No. J-s = 2 No. LU-s = 20
C No. of nonzeros in J = 44 No. of nonzeros in LU = 50
C
C-----------------------------------------------------------------------
C Full Description of User Interface to DLSODES.
C
C The user interface to DLSODES consists of the following parts.
C
C 1. The call sequence to Subroutine DLSODES, which is a driver
C routine for the solver. This includes descriptions of both
C the call sequence arguments and of user-supplied routines.
C Following these descriptions is a description of
C optional inputs available through the call sequence, and then
C a description of optional outputs (in the work arrays).
C
C 2. Descriptions of other routines in the DLSODES package that may be
C (optionally) called by the user. These provide the ability to
C alter error message handling, save and restore the internal
C Common, and obtain specified derivatives of the solution y(t).
C
C 3. Descriptions of Common blocks to be declared in overlay
C or similar environments, or to be saved when doing an interrupt
C of the problem and continued solution later.
C
C 4. Description of two routines in the DLSODES package, either of
C which the user may replace with his/her own version, if desired.
C These relate to the measurement of errors.
C
C-----------------------------------------------------------------------
C Part 1. Call Sequence.
C
C The call sequence parameters used for input only are
C F, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK, IOPT, LRW, LIW, JAC, MF,
C and those used for both input and output are
C Y, T, ISTATE.
C The work arrays RWORK and IWORK are also used for conditional and
C optional inputs and optional outputs. (The term output here refers
C to the return from Subroutine DLSODES to the user's calling program.)
C
C The legality of input parameters will be thoroughly checked on the
C initial call for the problem, but not checked thereafter unless a
C change in input parameters is flagged by ISTATE = 3 on input.
C
C The descriptions of the call arguments are as follows.
C
C F = the name of the user-supplied subroutine defining the
C ODE system. The system must be put in the first-order
C form dy/dt = f(t,y), where f is a vector-valued function
C of the scalar t and the vector y. Subroutine F is to
C compute the function f. It is to have the form
C SUBROUTINE F (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y(*), YDOT(*)
C where NEQ, T, and Y are input, and the array YDOT = f(t,y)
C is output. Y and YDOT are arrays of length NEQ.
C Subroutine F should not alter y(1),...,y(NEQ).
C F must be declared External in the calling program.
C
C Subroutine F may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in F) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y below.
C
C If quantities computed in the F routine are needed
C externally to DLSODES, an extra call to F should be made
C for this purpose, for consistent and accurate results.
C If only the derivative dy/dt is needed, use DINTDY instead.
C
C NEQ = the size of the ODE system (number of first order
C ordinary differential equations). Used only for input.
C NEQ may be decreased, but not increased, during the problem.
C If NEQ is decreased (with ISTATE = 3 on input), the
C remaining components of Y should be left undisturbed, if
C these are to be accessed in F and/or JAC.
C
C Normally, NEQ is a scalar, and it is generally referred to
C as a scalar in this user interface description. However,
C NEQ may be an array, with NEQ(1) set to the system size.
C (The DLSODES package accesses only NEQ(1).) In either case,
C this parameter is passed as the NEQ argument in all calls
C to F and JAC. Hence, if it is an array, locations
C NEQ(2),... may be used to store other integer data and pass
C it to F and/or JAC. Subroutines F and/or JAC must include
C NEQ in a Dimension statement in that case.
C
C Y = a real array for the vector of dependent variables, of
C length NEQ or more. Used for both input and output on the
C first call (ISTATE = 1), and only for output on other calls.
C on the first call, Y must contain the vector of initial
C values. On output, Y contains the computed solution vector,
C evaluated at T. If desired, the Y array may be used
C for other purposes between calls to the solver.
C
C This array is passed as the Y argument in all calls to
C F and JAC. Hence its length may exceed NEQ, and locations
C Y(NEQ+1),... may be used to store other real data and
C pass it to F and/or JAC. (The DLSODES package accesses only
C Y(1),...,Y(NEQ).)
C
C T = the independent variable. On input, T is used only on the
C first call, as the initial point of the integration.
C on output, after each call, T is the value at which a
C computed solution Y is evaluated (usually the same as TOUT).
C On an error return, T is the farthest point reached.
C
C TOUT = the next value of t at which a computed solution is desired.
C Used only for input.
C
C When starting the problem (ISTATE = 1), TOUT may be equal
C to T for one call, then should .ne. T for the next call.
C For the initial T, an input value of TOUT .ne. T is used
C in order to determine the direction of the integration
C (i.e. the algebraic sign of the step sizes) and the rough
C scale of the problem. Integration in either direction
C (forward or backward in t) is permitted.
C
C If ITASK = 2 or 5 (one-step modes), TOUT is ignored after
C the first call (i.e. the first call with TOUT .ne. T).
C Otherwise, TOUT is required on every call.
C
C If ITASK = 1, 3, or 4, the values of TOUT need not be
C monotone, but a value of TOUT which backs up is limited
C to the current internal T interval, whose endpoints are
C TCUR - HU and TCUR (see optional outputs, below, for
C TCUR and HU).
C
C ITOL = an indicator for the type of error control. See
C description below under ATOL. Used only for input.
C
C RTOL = a relative error tolerance parameter, either a scalar or
C an array of length NEQ. See description below under ATOL.
C Input only.
C
C ATOL = an absolute error tolerance parameter, either a scalar or
C an array of length NEQ. Input only.
C
C The input parameters ITOL, RTOL, and ATOL determine
C the error control performed by the solver. The solver will
C control the vector E = (E(i)) of estimated local errors
C in y, according to an inequality of the form
C RMS-norm of ( E(i)/EWT(i) ) .le. 1,
C where EWT(i) = RTOL(i)*ABS(Y(i)) + ATOL(i),
C and the RMS-norm (root-mean-square norm) here is
C RMS-norm(v) = SQRT(sum v(i)**2 / NEQ). Here EWT = (EWT(i))
C is a vector of weights which must always be positive, and
C the values of RTOL and ATOL should all be non-negative.
C The following table gives the types (scalar/array) of
C RTOL and ATOL, and the corresponding form of EWT(i).
C
C ITOL RTOL ATOL EWT(i)
C 1 scalar scalar RTOL*ABS(Y(i)) + ATOL
C 2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
C 3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
C 4 array array RTOL(i)*ABS(Y(i)) + ATOL(i)
C
C When either of these parameters is a scalar, it need not
C be dimensioned in the user's calling program.
C
C If none of the above choices (with ITOL, RTOL, and ATOL
C fixed throughout the problem) is suitable, more general
C error controls can be obtained by substituting
C user-supplied routines for the setting of EWT and/or for
C the norm calculation. See Part 4 below.
C
C If global errors are to be estimated by making a repeated
C run on the same problem with smaller tolerances, then all
C components of RTOL and ATOL (i.e. of EWT) should be scaled
C down uniformly.
C
C ITASK = an index specifying the task to be performed.
C Input only. ITASK has the following values and meanings.
C 1 means normal computation of output values of y(t) at
C t = TOUT (by overshooting and interpolating).
C 2 means take one step only and return.
C 3 means stop at the first internal mesh point at or
C beyond t = TOUT and return.
C 4 means normal computation of output values of y(t) at
C t = TOUT but without overshooting t = TCRIT.
C TCRIT must be input as RWORK(1). TCRIT may be equal to
C or beyond TOUT, but not behind it in the direction of
C integration. This option is useful if the problem
C has a singularity at or beyond t = TCRIT.
C 5 means take one step, without passing TCRIT, and return.
C TCRIT must be input as RWORK(1).
C
C Note: If ITASK = 4 or 5 and the solver reaches TCRIT
C (within roundoff), it will return T = TCRIT (exactly) to
C indicate this (unless ITASK = 4 and TOUT comes before TCRIT,
C in which case answers at t = TOUT are returned first).
C
C ISTATE = an index used for input and output to specify the
C the state of the calculation.
C
C On input, the values of ISTATE are as follows.
C 1 means this is the first call for the problem
C (initializations will be done). See note below.
C 2 means this is not the first call, and the calculation
C is to continue normally, with no change in any input
C parameters except possibly TOUT and ITASK.
C (If ITOL, RTOL, and/or ATOL are changed between calls
C with ISTATE = 2, the new values will be used but not
C tested for legality.)
C 3 means this is not the first call, and the
C calculation is to continue normally, but with
C a change in input parameters other than
C TOUT and ITASK. Changes are allowed in
C NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, MF,
C the conditional inputs IA and JA,
C and any of the optional inputs except H0.
C In particular, if MITER = 1 or 2, a call with ISTATE = 3
C will cause the sparsity structure of the problem to be
C recomputed (or reread from IA and JA if MOSS = 0).
C Note: a preliminary call with TOUT = T is not counted
C as a first call here, as no initialization or checking of
C input is done. (Such a call is sometimes useful for the
C purpose of outputting the initial conditions.)
C Thus the first call for which TOUT .ne. T requires
C ISTATE = 1 on input.
C
C On output, ISTATE has the following values and meanings.
C 1 means nothing was done; TOUT = T and ISTATE = 1 on input.
C 2 means the integration was performed successfully.
C -1 means an excessive amount of work (more than MXSTEP
C steps) was done on this call, before completing the
C requested task, but the integration was otherwise
C successful as far as T. (MXSTEP is an optional input
C and is normally 500.) To continue, the user may
C simply reset ISTATE to a value .gt. 1 and call again
C (the excess work step counter will be reset to 0).
C In addition, the user may increase MXSTEP to avoid
C this error return (see below on optional inputs).
C -2 means too much accuracy was requested for the precision
C of the machine being used. This was detected before
C completing the requested task, but the integration
C was successful as far as T. To continue, the tolerance
C parameters must be reset, and ISTATE must be set
C to 3. The optional output TOLSF may be used for this
C purpose. (Note: If this condition is detected before
C taking any steps, then an illegal input return
C (ISTATE = -3) occurs instead.)
C -3 means illegal input was detected, before taking any
C integration steps. See written message for details.
C Note: If the solver detects an infinite loop of calls
C to the solver with illegal input, it will cause
C the run to stop.
C -4 means there were repeated error test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C The problem may have a singularity, or the input
C may be inappropriate.
C -5 means there were repeated convergence test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C This may be caused by an inaccurate Jacobian matrix,
C if one is being used.
C -6 means EWT(i) became zero for some i during the
C integration. Pure relative error control (ATOL(i)=0.0)
C was requested on a variable which has now vanished.
C The integration was successful as far as T.
C -7 means a fatal error return flag came from the sparse
C solver CDRV by way of DPRJS or DSOLSS (numerical
C factorization or backsolve). This should never happen.
C The integration was successful as far as T.
C
C Note: an error return with ISTATE = -1, -4, or -5 and with
C MITER = 1 or 2 may mean that the sparsity structure of the
C problem has changed significantly since it was last
C determined (or input). In that case, one can attempt to
C complete the integration by setting ISTATE = 3 on the next
C call, so that a new structure determination is done.
C
C Note: since the normal output value of ISTATE is 2,
C it does not need to be reset for normal continuation.
C Also, since a negative input value of ISTATE will be
C regarded as illegal, a negative output value requires the
C user to change it, and possibly other inputs, before
C calling the solver again.
C
C IOPT = an integer flag to specify whether or not any optional
C inputs are being used on this call. Input only.
C The optional inputs are listed separately below.
C IOPT = 0 means no optional inputs are being used.
C Default values will be used in all cases.
C IOPT = 1 means one or more optional inputs are being used.
C
C RWORK = a work array used for a mixture of real (double precision)
C and integer work space.
C The length of RWORK (in real words) must be at least
C 20 + NYH*(MAXORD + 1) + 3*NEQ + LWM where
C NYH = the initial value of NEQ,
C MAXORD = 12 (if METH = 1) or 5 (if METH = 2) (unless a
C smaller value is given as an optional input),
C LWM = 0 if MITER = 0,
C LWM = 2*NNZ + 2*NEQ + (NNZ+9*NEQ)/LENRAT if MITER = 1,
C LWM = 2*NNZ + 2*NEQ + (NNZ+10*NEQ)/LENRAT if MITER = 2,
C LWM = NEQ + 2 if MITER = 3.
C In the above formulas,
C NNZ = number of nonzero elements in the Jacobian matrix.
C LENRAT = the real to integer wordlength ratio (usually 1 in
C single precision and 2 in double precision).
C (See the MF description for METH and MITER.)
C Thus if MAXORD has its default value and NEQ is constant,
C the minimum length of RWORK is:
C 20 + 16*NEQ for MF = 10,
C 20 + 16*NEQ + LWM for MF = 11, 111, 211, 12, 112, 212,
C 22 + 17*NEQ for MF = 13,
C 20 + 9*NEQ for MF = 20,
C 20 + 9*NEQ + LWM for MF = 21, 121, 221, 22, 122, 222,
C 22 + 10*NEQ for MF = 23.
C If MITER = 1 or 2, the above formula for LWM is only a
C crude lower bound. The required length of RWORK cannot
C be readily predicted in general, as it depends on the
C sparsity structure of the problem. Some experimentation
C may be necessary.
C
C The first 20 words of RWORK are reserved for conditional
C and optional inputs and optional outputs.
C
C The following word in RWORK is a conditional input:
C RWORK(1) = TCRIT = critical value of t which the solver
C is not to overshoot. Required if ITASK is
C 4 or 5, and ignored otherwise. (See ITASK.)
C
C LRW = the length of the array RWORK, as declared by the user.
C (This will be checked by the solver.)
C
C IWORK = an integer work array. The length of IWORK must be at least
C 31 + NEQ + NNZ if MOSS = 0 and MITER = 1 or 2, or
C 30 otherwise.
C (NNZ is the number of nonzero elements in df/dy.)
C
C In DLSODES, IWORK is used only for conditional and
C optional inputs and optional outputs.
C
C The following two blocks of words in IWORK are conditional
C inputs, required if MOSS = 0 and MITER = 1 or 2, but not
C otherwise (see the description of MF for MOSS).
C IWORK(30+j) = IA(j) (j=1,...,NEQ+1)
C IWORK(31+NEQ+k) = JA(k) (k=1,...,NNZ)
C The two arrays IA and JA describe the sparsity structure
C to be assumed for the Jacobian matrix. JA contains the row
C indices where nonzero elements occur, reading in columnwise
C order, and IA contains the starting locations in JA of the
C descriptions of columns 1,...,NEQ, in that order, with
C IA(1) = 1. Thus, for each column index j = 1,...,NEQ, the
C values of the row index i in column j where a nonzero
C element may occur are given by
C i = JA(k), where IA(j) .le. k .lt. IA(j+1).
C If NNZ is the total number of nonzero locations assumed,
C then the length of the JA array is NNZ, and IA(NEQ+1) must
C be NNZ + 1. Duplicate entries are not allowed.
C
C LIW = the length of the array IWORK, as declared by the user.
C (This will be checked by the solver.)
C
C Note: The work arrays must not be altered between calls to DLSODES
C for the same problem, except possibly for the conditional and
C optional inputs, and except for the last 3*NEQ words of RWORK.
C The latter space is used for internal scratch space, and so is
C available for use by the user outside DLSODES between calls, if
C desired (but not for use by F or JAC).
C
C JAC = name of user-supplied routine (MITER = 1 or MOSS = 1) to
C compute the Jacobian matrix, df/dy, as a function of
C the scalar t and the vector y. It is to have the form
C SUBROUTINE JAC (NEQ, T, Y, J, IAN, JAN, PDJ)
C DOUBLE PRECISION T, Y(*), IAN(*), JAN(*), PDJ(*)
C where NEQ, T, Y, J, IAN, and JAN are input, and the array
C PDJ, of length NEQ, is to be loaded with column J
C of the Jacobian on output. Thus df(i)/dy(J) is to be
C loaded into PDJ(i) for all relevant values of i.
C Here T and Y have the same meaning as in Subroutine F,
C and J is a column index (1 to NEQ). IAN and JAN are
C undefined in calls to JAC for structure determination
C (MOSS = 1). otherwise, IAN and JAN are structure
C descriptors, as defined under optional outputs below, and
C so can be used to determine the relevant row indices i, if
C desired.
C JAC need not provide df/dy exactly. A crude
C approximation (possibly with greater sparsity) will do.
C In any case, PDJ is preset to zero by the solver,
C so that only the nonzero elements need be loaded by JAC.
C Calls to JAC are made with J = 1,...,NEQ, in that order, and
C each such set of calls is preceded by a call to F with the
C same arguments NEQ, T, and Y. Thus to gain some efficiency,
C intermediate quantities shared by both calculations may be
C saved in a user Common block by F and not recomputed by JAC,
C if desired. JAC must not alter its input arguments.
C JAC must be declared External in the calling program.
C Subroutine JAC may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in JAC) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y above.
C
C MF = the method flag. Used only for input.
C MF has three decimal digits-- MOSS, METH, MITER--
C MF = 100*MOSS + 10*METH + MITER.
C MOSS indicates the method to be used to obtain the sparsity
C structure of the Jacobian matrix if MITER = 1 or 2:
C MOSS = 0 means the user has supplied IA and JA
C (see descriptions under IWORK above).
C MOSS = 1 means the user has supplied JAC (see below)
C and the structure will be obtained from NEQ
C initial calls to JAC.
C MOSS = 2 means the structure will be obtained from NEQ+1
C initial calls to F.
C METH indicates the basic linear multistep method:
C METH = 1 means the implicit Adams method.
C METH = 2 means the method based on Backward
C Differentiation Formulas (BDFs).
C MITER indicates the corrector iteration method:
C MITER = 0 means functional iteration (no Jacobian matrix
C is involved).
C MITER = 1 means chord iteration with a user-supplied
C sparse Jacobian, given by Subroutine JAC.
C MITER = 2 means chord iteration with an internally
C generated (difference quotient) sparse Jacobian
C (using NGP extra calls to F per df/dy value,
C where NGP is an optional output described below.)
C MITER = 3 means chord iteration with an internally
C generated diagonal Jacobian approximation
C (using 1 extra call to F per df/dy evaluation).
C If MITER = 1 or MOSS = 1, the user must supply a Subroutine
C JAC (the name is arbitrary) as described above under JAC.
C Otherwise, a dummy argument can be used.
C
C The standard choices for MF are:
C MF = 10 for a nonstiff problem,
C MF = 21 or 22 for a stiff problem with IA/JA supplied
C (21 if JAC is supplied, 22 if not),
C MF = 121 for a stiff problem with JAC supplied,
C but not IA/JA,
C MF = 222 for a stiff problem with neither IA/JA nor
C JAC supplied.
C The sparseness structure can be changed during the
C problem by making a call to DLSODES with ISTATE = 3.
C-----------------------------------------------------------------------
C Optional Inputs.
C
C The following is a list of the optional inputs provided for in the
C call sequence. (See also Part 2.) For each such input variable,
C this table lists its name as used in this documentation, its
C location in the call sequence, its meaning, and the default value.
C The use of any of these inputs requires IOPT = 1, and in that
C case all of these inputs are examined. A value of zero for any
C of these optional inputs will cause the default value to be used.
C Thus to use a subset of the optional inputs, simply preload
C locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively, and
C then set those of interest to nonzero values.
C
C Name Location Meaning and Default Value
C
C H0 RWORK(5) the step size to be attempted on the first step.
C The default value is determined by the solver.
C
C HMAX RWORK(6) the maximum absolute step size allowed.
C The default value is infinite.
C
C HMIN RWORK(7) the minimum absolute step size allowed.
C The default value is 0. (This lower bound is not
C enforced on the final step before reaching TCRIT
C when ITASK = 4 or 5.)
C
C SETH RWORK(8) the element threshhold for sparsity determination
C when MOSS = 1 or 2. If the absolute value of
C an estimated Jacobian element is .le. SETH, it
C will be assumed to be absent in the structure.
C The default value of SETH is 0.
C
C MAXORD IWORK(5) the maximum order to be allowed. The default
C value is 12 if METH = 1, and 5 if METH = 2.
C If MAXORD exceeds the default value, it will
C be reduced to the default value.
C If MAXORD is changed during the problem, it may
C cause the current order to be reduced.
C
C MXSTEP IWORK(6) maximum number of (internally defined) steps
C allowed during one call to the solver.
C The default value is 500.
C
C MXHNIL IWORK(7) maximum number of messages printed (per problem)
C warning that T + H = T on a step (H = step size).
C This must be positive to result in a non-default
C value. The default value is 10.
C-----------------------------------------------------------------------
C Optional Outputs.
C
C As optional additional output from DLSODES, the variables listed
C below are quantities related to the performance of DLSODES
C which are available to the user. These are communicated by way of
C the work arrays, but also have internal mnemonic names as shown.
C Except where stated otherwise, all of these outputs are defined
C on any successful return from DLSODES, and on any return with
C ISTATE = -1, -2, -4, -5, or -6. On an illegal input return
C (ISTATE = -3), they will be unchanged from their existing values
C (if any), except possibly for TOLSF, LENRW, and LENIW.
C On any error return, outputs relevant to the error will be defined,
C as noted below.
C
C Name Location Meaning
C
C HU RWORK(11) the step size in t last used (successfully).
C
C HCUR RWORK(12) the step size to be attempted on the next step.
C
C TCUR RWORK(13) the current value of the independent variable
C which the solver has actually reached, i.e. the
C current internal mesh point in t. On output, TCUR
C will always be at least as far as the argument
C T, but may be farther (if interpolation was done).
C
C TOLSF RWORK(14) a tolerance scale factor, greater than 1.0,
C computed when a request for too much accuracy was
C detected (ISTATE = -3 if detected at the start of
C the problem, ISTATE = -2 otherwise). If ITOL is
C left unaltered but RTOL and ATOL are uniformly
C scaled up by a factor of TOLSF for the next call,
C then the solver is deemed likely to succeed.
C (The user may also ignore TOLSF and alter the
C tolerance parameters in any other way appropriate.)
C
C NST IWORK(11) the number of steps taken for the problem so far.
C
C NFE IWORK(12) the number of f evaluations for the problem so far,
C excluding those for structure determination
C (MOSS = 2).
C
C NJE IWORK(13) the number of Jacobian evaluations for the problem
C so far, excluding those for structure determination
C (MOSS = 1).
C
C NQU IWORK(14) the method order last used (successfully).
C
C NQCUR IWORK(15) the order to be attempted on the next step.
C
C IMXER IWORK(16) the index of the component of largest magnitude in
C the weighted local error vector ( E(i)/EWT(i) ),
C on an error return with ISTATE = -4 or -5.
C
C LENRW IWORK(17) the length of RWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C LENIW IWORK(18) the length of IWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C NNZ IWORK(19) the number of nonzero elements in the Jacobian
C matrix, including the diagonal (MITER = 1 or 2).
C (This may differ from that given by IA(NEQ+1)-1
C if MOSS = 0, because of added diagonal entries.)
C
C NGP IWORK(20) the number of groups of column indices, used in
C difference quotient Jacobian aproximations if
C MITER = 2. This is also the number of extra f
C evaluations needed for each Jacobian evaluation.
C
C NLU IWORK(21) the number of sparse LU decompositions for the
C problem so far.
C
C LYH IWORK(22) the base address in RWORK of the history array YH,
C described below in this list.
C
C IPIAN IWORK(23) the base address of the structure descriptor array
C IAN, described below in this list.
C
C IPJAN IWORK(24) the base address of the structure descriptor array
C JAN, described below in this list.
C
C NZL IWORK(25) the number of nonzero elements in the strict lower
C triangle of the LU factorization used in the chord
C iteration (MITER = 1 or 2).
C
C NZU IWORK(26) the number of nonzero elements in the strict upper
C triangle of the LU factorization used in the chord
C iteration (MITER = 1 or 2).
C The total number of nonzeros in the factorization
C is therefore NZL + NZU + NEQ.
C
C The following four arrays are segments of the RWORK array which
C may also be of interest to the user as optional outputs.
C For each array, the table below gives its internal name,
C its base address, and its description.
C For YH and ACOR, the base addresses are in RWORK (a real array).
C The integer arrays IAN and JAN are to be obtained by declaring an
C integer array IWK and identifying IWK(1) with RWORK(21), using either
C an equivalence statement or a subroutine call. Then the base
C addresses IPIAN (of IAN) and IPJAN (of JAN) in IWK are to be obtained
C as optional outputs IWORK(23) and IWORK(24), respectively.
C Thus IAN(1) is IWK(IPIAN), etc.
C
C Name Base Address Description
C
C IAN IPIAN (in IWK) structure descriptor array of size NEQ + 1.
C JAN IPJAN (in IWK) structure descriptor array of size NNZ.
C (see above) IAN and JAN together describe the sparsity
C structure of the Jacobian matrix, as used by
C DLSODES when MITER = 1 or 2.
C JAN contains the row indices of the nonzero
C locations, reading in columnwise order, and
C IAN contains the starting locations in JAN of
C the descriptions of columns 1,...,NEQ, in
C that order, with IAN(1) = 1. Thus for each
C j = 1,...,NEQ, the row indices i of the
C nonzero locations in column j are
C i = JAN(k), IAN(j) .le. k .lt. IAN(j+1).
C Note that IAN(NEQ+1) = NNZ + 1.
C (If MOSS = 0, IAN/JAN may differ from the
C input IA/JA because of a different ordering
C in each column, and added diagonal entries.)
C
C YH LYH the Nordsieck history array, of size NYH by
C (optional (NQCUR + 1), where NYH is the initial value
C output) of NEQ. For j = 0,1,...,NQCUR, column j+1
C of YH contains HCUR**j/factorial(j) times
C the j-th derivative of the interpolating
C polynomial currently representing the solution,
C evaluated at t = TCUR. The base address LYH
C is another optional output, listed above.
C
C ACOR LENRW-NEQ+1 array of size NEQ used for the accumulated
C corrections on each step, scaled on output
C to represent the estimated local error in y
C on the last step. This is the vector E in
C the description of the error control. It is
C defined only on a successful return from
C DLSODES.
C
C-----------------------------------------------------------------------
C Part 2. Other Routines Callable.
C
C The following are optional calls which the user may make to
C gain additional capabilities in conjunction with DLSODES.
C (The routines XSETUN and XSETF are designed to conform to the
C SLATEC error handling package.)
C
C Form of Call Function
C CALL XSETUN(LUN) Set the logical unit number, LUN, for
C output of messages from DLSODES, if
C the default is not desired.
C The default value of LUN is 6.
C
C CALL XSETF(MFLAG) Set a flag to control the printing of
C messages by DLSODES.
C MFLAG = 0 means do not print. (Danger:
C This risks losing valuable information.)
C MFLAG = 1 means print (the default).
C
C Either of the above calls may be made at
C any time and will take effect immediately.
C
C CALL DSRCMS(RSAV,ISAV,JOB) saves and restores the contents of
C the internal Common blocks used by
C DLSODES (see Part 3 below).
C RSAV must be a real array of length 224
C or more, and ISAV must be an integer
C array of length 71 or more.
C JOB=1 means save Common into RSAV/ISAV.
C JOB=2 means restore Common from RSAV/ISAV.
C DSRCMS is useful if one is
C interrupting a run and restarting
C later, or alternating between two or
C more problems solved with DLSODES.
C
C CALL DINTDY(,,,,,) Provide derivatives of y, of various
C (see below) orders, at a specified point t, if
C desired. It may be called only after
C a successful return from DLSODES.
C
C The detailed instructions for using DINTDY are as follows.
C The form of the call is:
C
C LYH = IWORK(22)
C CALL DINTDY (T, K, RWORK(LYH), NYH, DKY, IFLAG)
C
C The input parameters are:
C
C T = value of independent variable where answers are desired
C (normally the same as the T last returned by DLSODES).
C For valid results, T must lie between TCUR - HU and TCUR.
C (See optional outputs for TCUR and HU.)
C K = integer order of the derivative desired. K must satisfy
C 0 .le. K .le. NQCUR, where NQCUR is the current order
C (See optional outputs). The capability corresponding
C to K = 0, i.e. computing y(T), is already provided
C by DLSODES directly. Since NQCUR .ge. 1, the first
C derivative dy/dt is always available with DINTDY.
C LYH = the base address of the history array YH, obtained
C as an optional output as shown above.
C NYH = column length of YH, equal to the initial value of NEQ.
C
C The output parameters are:
C
C DKY = a real array of length NEQ containing the computed value
C of the K-th derivative of y(t).
C IFLAG = integer flag, returned as 0 if K and T were legal,
C -1 if K was illegal, and -2 if T was illegal.
C On an error return, a message is also written.
C-----------------------------------------------------------------------
C Part 3. Common Blocks.
C
C If DLSODES is to be used in an overlay situation, the user
C must declare, in the primary overlay, the variables in:
C (1) the call sequence to DLSODES, and
C (2) the two internal Common blocks
C /DLS001/ of length 255 (218 double precision words
C followed by 37 integer words),
C /DLSS01/ of length 40 (6 double precision words
C followed by 34 integer words),
C
C If DLSODES is used on a system in which the contents of internal
C Common blocks are not preserved between calls, the user should
C declare the above Common blocks in the calling program to insure
C that their contents are preserved.
C
C If the solution of a given problem by DLSODES is to be interrupted
C and then later continued, such as when restarting an interrupted run
C or alternating between two or more problems, the user should save,
C following the return from the last DLSODES call prior to the
C interruption, the contents of the call sequence variables and the
C internal Common blocks, and later restore these values before the
C next DLSODES call for that problem. To save and restore the Common
C blocks, use Subroutine DSRCMS (see Part 2 above).
C
C-----------------------------------------------------------------------
C Part 4. Optionally Replaceable Solver Routines.
C
C Below are descriptions of two routines in the DLSODES package which
C relate to the measurement of errors. Either routine can be
C replaced by a user-supplied version, if desired. However, since such
C a replacement may have a major impact on performance, it should be
C done only when absolutely necessary, and only with great caution.
C (Note: The means by which the package version of a routine is
C superseded by the user's version may be system-dependent.)
C
C (a) DEWSET.
C The following subroutine is called just before each internal
C integration step, and sets the array of error weights, EWT, as
C described under ITOL/RTOL/ATOL above:
C Subroutine DEWSET (NEQ, ITOL, RTOL, ATOL, YCUR, EWT)
C where NEQ, ITOL, RTOL, and ATOL are as in the DLSODES call sequence,
C YCUR contains the current dependent variable vector, and
C EWT is the array of weights set by DEWSET.
C
C If the user supplies this subroutine, it must return in EWT(i)
C (i = 1,...,NEQ) a positive quantity suitable for comparing errors
C in y(i) to. The EWT array returned by DEWSET is passed to the DVNORM
C routine (see below), and also used by DLSODES in the computation
C of the optional output IMXER, the diagonal Jacobian approximation,
C and the increments for difference quotient Jacobians.
C
C In the user-supplied version of DEWSET, it may be desirable to use
C the current values of derivatives of y. Derivatives up to order NQ
C are available from the history array YH, described above under
C optional outputs. In DEWSET, YH is identical to the YCUR array,
C extended to NQ + 1 columns with a column length of NYH and scale
C factors of H**j/factorial(j). On the first call for the problem,
C given by NST = 0, NQ is 1 and H is temporarily set to 1.0.
C NYH is the initial value of NEQ. The quantities NQ, H, and NST
C can be obtained by including in DEWSET the statements:
C DOUBLE PRECISION RLS
C COMMON /DLS001/ RLS(218),ILS(37)
C NQ = ILS(33)
C NST = ILS(34)
C H = RLS(212)
C Thus, for example, the current value of dy/dt can be obtained as
C YCUR(NYH+i)/H (i=1,...,NEQ) (and the division by H is
C unnecessary when NST = 0).
C
C (b) DVNORM.
C The following is a real function routine which computes the weighted
C root-mean-square norm of a vector v:
C D = DVNORM (N, V, W)
C where
C N = the length of the vector,
C V = real array of length N containing the vector,
C W = real array of length N containing weights,
C D = SQRT( (1/N) * sum(V(i)*W(i))**2 ).
C DVNORM is called with N = NEQ and with W(i) = 1.0/EWT(i), where
C EWT is as set by Subroutine DEWSET.
C
C If the user supplies this function, it should return a non-negative
C value of DVNORM suitable for use in the error control in DLSODES.
C None of the arguments should be altered by DVNORM.
C For example, a user-supplied DVNORM routine might:
C -substitute a max-norm of (V(i)*W(i)) for the RMS-norm, or
C -ignore some components of V in the norm, with the effect of
C suppressing the error control on those components of y.
C-----------------------------------------------------------------------
C
C***REVISION HISTORY (YYYYMMDD)
C 19810120 DATE WRITTEN
C 19820315 Upgraded MDI in ODRV package: operates on M + M-transpose.
C 19820426 Numerous revisions in use of work arrays;
C use wordlength ratio LENRAT; added IPISP & LRAT to Common;
C added optional outputs IPIAN/IPJAN;
C numerous corrections to comments.
C 19830503 Added routine CNTNZU; added NZL and NZU to /LSS001/;
C changed ADJLR call logic; added optional outputs NZL & NZU;
C revised counter initializations; revised PREP stmt. numbers;
C corrections to comments throughout.
C 19870320 Corrected jump on test of umax in CDRV routine;
C added ISTATE = -7 return.
C 19870330 Major update: corrected comments throughout;
C removed TRET from Common; rewrote EWSET with 4 loops;
C fixed t test in INTDY; added Cray directives in STODE;
C in STODE, fixed DELP init. and logic around PJAC call;
C combined routines to save/restore Common;
C passed LEVEL = 0 in error message calls (except run abort).
C 20010425 Major update: convert source lines to upper case;
C added *DECK lines; changed from 1 to * in dummy dimensions;
C changed names R1MACH/D1MACH to RUMACH/DUMACH;
C renamed routines for uniqueness across single/double prec.;
C converted intrinsic names to generic form;
C removed ILLIN and NTREP (data loaded) from Common;
C removed all 'own' variables from Common;
C changed error messages to quoted strings;
C replaced XERRWV/XERRWD with 1993 revised version;
C converted prologues, comments, error messages to mixed case;
C converted arithmetic IF statements to logical IF statements;
C numerous corrections to prologues and internal comments.
C 20010507 Converted single precision source to double precision.
C 20020502 Corrected declarations in descriptions of user routines.
C 20031105 Restored 'own' variables to Common blocks, to enable
C interrupt/restart feature.
C 20031112 Added SAVE statements for data-loaded constants.
C
C-----------------------------------------------------------------------
C Other routines in the DLSODES package.
C
C In addition to Subroutine DLSODES, the DLSODES package includes the
C following subroutines and function routines:
C DIPREP acts as an iterface between DLSODES and DPREP, and also does
C adjusting of work space pointers and work arrays.
C DPREP is called by DIPREP to compute sparsity and do sparse matrix
C preprocessing if MITER = 1 or 2.
C JGROUP is called by DPREP to compute groups of Jacobian column
C indices for use when MITER = 2.
C ADJLR adjusts the length of required sparse matrix work space.
C It is called by DPREP.
C CNTNZU is called by DPREP and counts the nonzero elements in the
C strict upper triangle of J + J-transpose, where J = df/dy.
C DINTDY computes an interpolated value of the y vector at t = TOUT.
C DSTODE is the core integrator, which does one step of the
C integration and the associated error control.
C DCFODE sets all method coefficients and test constants.
C DPRJS computes and preprocesses the Jacobian matrix J = df/dy
C and the Newton iteration matrix P = I - h*l0*J.
C DSOLSS manages solution of linear system in chord iteration.
C DEWSET sets the error weight vector EWT before each step.
C DVNORM computes the weighted RMS-norm of a vector.
C DSRCMS is a user-callable routine to save and restore
C the contents of the internal Common blocks.
C ODRV constructs a reordering of the rows and columns of
C a matrix by the minimum degree algorithm. ODRV is a
C driver routine which calls Subroutines MD, MDI, MDM,
C MDP, MDU, and SRO. See Ref. 2 for details. (The ODRV
C module has been modified since Ref. 2, however.)
C CDRV performs reordering, symbolic factorization, numerical
C factorization, or linear system solution operations,
C depending on a path argument ipath. CDRV is a
C driver routine which calls Subroutines NROC, NSFC,
C NNFC, NNSC, and NNTC. See Ref. 3 for details.
C DLSODES uses CDRV to solve linear systems in which the
C coefficient matrix is P = I - con*J, where I is the
C identity, con is a scalar, and J is an approximation to
C the Jacobian df/dy. Because CDRV deals with rowwise
C sparsity descriptions, CDRV works with P-transpose, not P.
C DUMACH computes the unit roundoff in a machine-independent manner.
C XERRWD, XSETUN, XSETF, IXSAV, and IUMACH handle the printing of all
C error messages and warnings. XERRWD is machine-dependent.
C Note: DVNORM, DUMACH, IXSAV, and IUMACH are function routines.
C All the others are subroutines.
C
C-----------------------------------------------------------------------
EXTERNAL DPRJS, DSOLSS
DOUBLE PRECISION DUMACH, DVNORM
INTEGER INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP,
1 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA,
2 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ,
3 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU
INTEGER I, I1, I2, IFLAG, IMAX, IMUL, IMXER, IPFLAG, IPGO, IREM,
1 J, KGO, LENRAT, LENYHT, LENIW, LENRW, LF0, LIA, LJA,
2 LRTEM, LWTEM, LYHD, LYHN, MF1, MORD, MXHNL0, MXSTP0, NCOLM
DOUBLE PRECISION ROWNS,
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
DOUBLE PRECISION CON0, CONMIN, CCMXJ, PSMALL, RBIG, SETH
DOUBLE PRECISION ATOLI, AYI, BIG, EWTI, H0, HMAX, HMX, RH, RTOLI,
1 TCRIT, TDIST, TNEXT, TOL, TOLSF, TP, SIZE, SUM, W0
DIMENSION MORD(2)
LOGICAL IHIT
CHARACTER*60 MSG
SAVE LENRAT, MORD, MXSTP0, MXHNL0
C-----------------------------------------------------------------------
C The following two internal Common blocks contain
C (a) variables which are local to any subroutine but whose values must
C be preserved between calls to the routine ("own" variables), and
C (b) variables which are communicated between subroutines.
C The block DLS001 is declared in subroutines DLSODES, DIPREP, DPREP,
C DINTDY, DSTODE, DPRJS, and DSOLSS.
C The block DLSS01 is declared in subroutines DLSODES, DIPREP, DPREP,
C DPRJS, and DSOLSS.
C Groups of variables are replaced by dummy arrays in the Common
C declarations in routines where those variables are not used.
C-----------------------------------------------------------------------
COMMON /DLS001/ ROWNS(209),
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
2 INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS(6),
3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
COMMON /DLSS01/ CON0, CONMIN, CCMXJ, PSMALL, RBIG, SETH,
1 IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP,
2 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA,
3 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ,
4 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU
C
DATA MORD(1),MORD(2)/12,5/, MXSTP0/500/, MXHNL0/10/
C-----------------------------------------------------------------------
C In the Data statement below, set LENRAT equal to the ratio of
C the wordlength for a real number to that for an integer. Usually,
C LENRAT = 1 for single precision and 2 for double precision. If the
C true ratio is not an integer, use the next smaller integer (.ge. 1).
C-----------------------------------------------------------------------
DATA LENRAT/2/
C-----------------------------------------------------------------------
C Block A.
C This code block is executed on every call.
C It tests ISTATE and ITASK for legality and branches appropriately.
C If ISTATE .gt. 1 but the flag INIT shows that initialization has
C not yet been done, an error return occurs.
C If ISTATE = 1 and TOUT = T, return immediately.
C-----------------------------------------------------------------------
IF (ISTATE .LT. 1 .OR. ISTATE .GT. 3) GO TO 601
IF (ITASK .LT. 1 .OR. ITASK .GT. 5) GO TO 602
IF (ISTATE .EQ. 1) GO TO 10
IF (INIT .EQ. 0) GO TO 603
IF (ISTATE .EQ. 2) GO TO 200
GO TO 20
10 INIT = 0
IF (TOUT .EQ. T) RETURN
C-----------------------------------------------------------------------
C Block B.
C The next code block is executed for the initial call (ISTATE = 1),
C or for a continuation call with parameter changes (ISTATE = 3).
C It contains checking of all inputs and various initializations.
C If ISTATE = 1, the final setting of work space pointers, the matrix
C preprocessing, and other initializations are done in Block C.
C
C First check legality of the non-optional inputs NEQ, ITOL, IOPT,
C MF, ML, and MU.
C-----------------------------------------------------------------------
20 IF (NEQ(1) .LE. 0) GO TO 604
IF (ISTATE .EQ. 1) GO TO 25
IF (NEQ(1) .GT. N) GO TO 605
25 N = NEQ(1)
IF (ITOL .LT. 1 .OR. ITOL .GT. 4) GO TO 606
IF (IOPT .LT. 0 .OR. IOPT .GT. 1) GO TO 607
MOSS = MF/100
MF1 = MF - 100*MOSS
METH = MF1/10
MITER = MF1 - 10*METH
IF (MOSS .LT. 0 .OR. MOSS .GT. 2) GO TO 608
IF (METH .LT. 1 .OR. METH .GT. 2) GO TO 608
IF (MITER .LT. 0 .OR. MITER .GT. 3) GO TO 608
IF (MITER .EQ. 0 .OR. MITER .EQ. 3) MOSS = 0
C Next process and check the optional inputs. --------------------------
IF (IOPT .EQ. 1) GO TO 40
MAXORD = MORD(METH)
MXSTEP = MXSTP0
MXHNIL = MXHNL0
IF (ISTATE .EQ. 1) H0 = 0.0D0
HMXI = 0.0D0
HMIN = 0.0D0
SETH = 0.0D0
GO TO 60
40 MAXORD = IWORK(5)
IF (MAXORD .LT. 0) GO TO 611
IF (MAXORD .EQ. 0) MAXORD = 100
MAXORD = MIN(MAXORD,MORD(METH))
MXSTEP = IWORK(6)
IF (MXSTEP .LT. 0) GO TO 612
IF (MXSTEP .EQ. 0) MXSTEP = MXSTP0
MXHNIL = IWORK(7)
IF (MXHNIL .LT. 0) GO TO 613
IF (MXHNIL .EQ. 0) MXHNIL = MXHNL0
IF (ISTATE .NE. 1) GO TO 50
H0 = RWORK(5)
IF ((TOUT - T)*H0 .LT. 0.0D0) GO TO 614
50 HMAX = RWORK(6)
IF (HMAX .LT. 0.0D0) GO TO 615
HMXI = 0.0D0
IF (HMAX .GT. 0.0D0) HMXI = 1.0D0/HMAX
HMIN = RWORK(7)
IF (HMIN .LT. 0.0D0) GO TO 616
SETH = RWORK(8)
IF (SETH .LT. 0.0D0) GO TO 609
C Check RTOL and ATOL for legality. ------------------------------------
60 RTOLI = RTOL(1)
ATOLI = ATOL(1)
DO 65 I = 1,N
IF (ITOL .GE. 3) RTOLI = RTOL(I)
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
IF (RTOLI .LT. 0.0D0) GO TO 619
IF (ATOLI .LT. 0.0D0) GO TO 620
65 CONTINUE
C-----------------------------------------------------------------------
C Compute required work array lengths, as far as possible, and test
C these against LRW and LIW. Then set tentative pointers for work
C arrays. Pointers to RWORK/IWORK segments are named by prefixing L to
C the name of the segment. E.g., the segment YH starts at RWORK(LYH).
C Segments of RWORK (in order) are denoted WM, YH, SAVF, EWT, ACOR.
C If MITER = 1 or 2, the required length of the matrix work space WM
C is not yet known, and so a crude minimum value is used for the
C initial tests of LRW and LIW, and YH is temporarily stored as far
C to the right in RWORK as possible, to leave the maximum amount
C of space for WM for matrix preprocessing. Thus if MITER = 1 or 2
C and MOSS .ne. 2, some of the segments of RWORK are temporarily
C omitted, as they are not needed in the preprocessing. These
C omitted segments are: ACOR if ISTATE = 1, EWT and ACOR if ISTATE = 3
C and MOSS = 1, and SAVF, EWT, and ACOR if ISTATE = 3 and MOSS = 0.
C-----------------------------------------------------------------------
LRAT = LENRAT
IF (ISTATE .EQ. 1) NYH = N
LWMIN = 0
IF (MITER .EQ. 1) LWMIN = 4*N + 10*N/LRAT
IF (MITER .EQ. 2) LWMIN = 4*N + 11*N/LRAT
IF (MITER .EQ. 3) LWMIN = N + 2
LENYH = (MAXORD+1)*NYH
LREST = LENYH + 3*N
LENRW = 20 + LWMIN + LREST
IWORK(17) = LENRW
LENIW = 30
IF (MOSS .EQ. 0 .AND. MITER .NE. 0 .AND. MITER .NE. 3)
1 LENIW = LENIW + N + 1
IWORK(18) = LENIW
IF (LENRW .GT. LRW) GO TO 617
IF (LENIW .GT. LIW) GO TO 618
LIA = 31
IF (MOSS .EQ. 0 .AND. MITER .NE. 0 .AND. MITER .NE. 3)
1 LENIW = LENIW + IWORK(LIA+N) - 1
IWORK(18) = LENIW
IF (LENIW .GT. LIW) GO TO 618
LJA = LIA + N + 1
LIA = MIN(LIA,LIW)
LJA = MIN(LJA,LIW)
LWM = 21
IF (ISTATE .EQ. 1) NQ = 1
NCOLM = MIN(NQ+1,MAXORD+2)
LENYHM = NCOLM*NYH
LENYHT = LENYH
IF (MITER .EQ. 1 .OR. MITER .EQ. 2) LENYHT = LENYHM
IMUL = 2
IF (ISTATE .EQ. 3) IMUL = MOSS
IF (MOSS .EQ. 2) IMUL = 3
LRTEM = LENYHT + IMUL*N
LWTEM = LWMIN
IF (MITER .EQ. 1 .OR. MITER .EQ. 2) LWTEM = LRW - 20 - LRTEM
LENWK = LWTEM
LYHN = LWM + LWTEM
LSAVF = LYHN + LENYHT
LEWT = LSAVF + N
LACOR = LEWT + N
ISTATC = ISTATE
IF (ISTATE .EQ. 1) GO TO 100
C-----------------------------------------------------------------------
C ISTATE = 3. Move YH to its new location.
C Note that only the part of YH needed for the next step, namely
C MIN(NQ+1,MAXORD+2) columns, is actually moved.
C A temporary error weight array EWT is loaded if MOSS = 2.
C Sparse matrix processing is done in DIPREP/DPREP if MITER = 1 or 2.
C If MAXORD was reduced below NQ, then the pointers are finally set
C so that SAVF is identical to YH(*,MAXORD+2).
C-----------------------------------------------------------------------
LYHD = LYH - LYHN
IMAX = LYHN - 1 + LENYHM
C Move YH. Move right if LYHD < 0; move left if LYHD > 0. -------------
IF (LYHD .LT. 0) THEN
DO 72 I = LYHN,IMAX
J = IMAX + LYHN - I
72 RWORK(J) = RWORK(J+LYHD)
ENDIF
IF (LYHD .GT. 0) THEN
DO 76 I = LYHN,IMAX
76 RWORK(I) = RWORK(I+LYHD)
ENDIF
80 LYH = LYHN
IWORK(22) = LYH
IF (MITER .EQ. 0 .OR. MITER .EQ. 3) GO TO 92
IF (MOSS .NE. 2) GO TO 85
C Temporarily load EWT if MITER = 1 or 2 and MOSS = 2. -----------------
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 82 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 621
82 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
85 CONTINUE
C DIPREP and DPREP do sparse matrix preprocessing if MITER = 1 or 2. ---
LSAVF = MIN(LSAVF,LRW)
LEWT = MIN(LEWT,LRW)
LACOR = MIN(LACOR,LRW)
CALL DIPREP (NEQ, Y, RWORK, IWORK(LIA),IWORK(LJA), IPFLAG, F, JAC)
LENRW = LWM - 1 + LENWK + LREST
IWORK(17) = LENRW
IF (IPFLAG .NE. -1) IWORK(23) = IPIAN
IF (IPFLAG .NE. -1) IWORK(24) = IPJAN
IPGO = -IPFLAG + 1
GO TO (90, 628, 629, 630, 631, 632, 633), IPGO
90 IWORK(22) = LYH
IF (LENRW .GT. LRW) GO TO 617
C Set flag to signal parameter changes to DSTODE. ----------------------
92 JSTART = -1
IF (N .EQ. NYH) GO TO 200
C NEQ was reduced. Zero part of YH to avoid undefined references. -----
I1 = LYH + L*NYH
I2 = LYH + (MAXORD + 1)*NYH - 1
IF (I1 .GT. I2) GO TO 200
DO 95 I = I1,I2
95 RWORK(I) = 0.0D0
GO TO 200
C-----------------------------------------------------------------------
C Block C.
C The next block is for the initial call only (ISTATE = 1).
C It contains all remaining initializations, the initial call to F,
C the sparse matrix preprocessing (MITER = 1 or 2), and the
C calculation of the initial step size.
C The error weights in EWT are inverted after being loaded.
C-----------------------------------------------------------------------
100 CONTINUE
LYH = LYHN
IWORK(22) = LYH
TN = T
NST = 0
H = 1.0D0
NNZ = 0
NGP = 0
NZL = 0
NZU = 0
C Load the initial value vector in YH. ---------------------------------
DO 105 I = 1,N
105 RWORK(I+LYH-1) = Y(I)
C Initial call to F. (LF0 points to YH(*,2).) -------------------------
LF0 = LYH + NYH
CALL F (NEQ, T, Y, RWORK(LF0))
NFE = 1
C Load and invert the EWT array. (H is temporarily set to 1.0.) -------
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 110 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 621
110 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
IF (MITER .EQ. 0 .OR. MITER .EQ. 3) GO TO 120
C DIPREP and DPREP do sparse matrix preprocessing if MITER = 1 or 2. ---
LACOR = MIN(LACOR,LRW)
CALL DIPREP (NEQ, Y, RWORK, IWORK(LIA),IWORK(LJA), IPFLAG, F, JAC)
LENRW = LWM - 1 + LENWK + LREST
IWORK(17) = LENRW
IF (IPFLAG .NE. -1) IWORK(23) = IPIAN
IF (IPFLAG .NE. -1) IWORK(24) = IPJAN
IPGO = -IPFLAG + 1
GO TO (115, 628, 629, 630, 631, 632, 633), IPGO
115 IWORK(22) = LYH
IF (LENRW .GT. LRW) GO TO 617
C Check TCRIT for legality (ITASK = 4 or 5). ---------------------------
120 CONTINUE
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 125
TCRIT = RWORK(1)
IF ((TCRIT - TOUT)*(TOUT - T) .LT. 0.0D0) GO TO 625
IF (H0 .NE. 0.0D0 .AND. (T + H0 - TCRIT)*H0 .GT. 0.0D0)
1 H0 = TCRIT - T
C Initialize all remaining parameters. ---------------------------------
125 UROUND = DUMACH()
JSTART = 0
IF (MITER .NE. 0) RWORK(LWM) = SQRT(UROUND)
MSBJ = 50
NSLJ = 0
CCMXJ = 0.2D0
PSMALL = 1000.0D0*UROUND
RBIG = 0.01D0/PSMALL
NHNIL = 0
NJE = 0
NLU = 0
NSLAST = 0
HU = 0.0D0
NQU = 0
CCMAX = 0.3D0
MAXCOR = 3
MSBP = 20
MXNCF = 10
C-----------------------------------------------------------------------
C The coding below computes the step size, H0, to be attempted on the
C first step, unless the user has supplied a value for this.
C First check that TOUT - T differs significantly from zero.
C A scalar tolerance quantity TOL is computed, as MAX(RTOL(i))
C if this is positive, or MAX(ATOL(i)/ABS(Y(i))) otherwise, adjusted
C so as to be between 100*UROUND and 1.0E-3.
C Then the computed value H0 is given by..
C NEQ
C H0**2 = TOL / ( w0**-2 + (1/NEQ) * Sum ( f(i)/ywt(i) )**2 )
C 1
C where w0 = MAX ( ABS(T), ABS(TOUT) ),
C f(i) = i-th component of initial value of f,
C ywt(i) = EWT(i)/TOL (a weight for y(i)).
C The sign of H0 is inferred from the initial values of TOUT and T.
C ABS(H0) is made .le. ABS(TOUT-T) in any case.
C-----------------------------------------------------------------------
LF0 = LYH + NYH
IF (H0 .NE. 0.0D0) GO TO 180
TDIST = ABS(TOUT - T)
W0 = MAX(ABS(T),ABS(TOUT))
IF (TDIST .LT. 2.0D0*UROUND*W0) GO TO 622
TOL = RTOL(1)
IF (ITOL .LE. 2) GO TO 140
DO 130 I = 1,N
130 TOL = MAX(TOL,RTOL(I))
140 IF (TOL .GT. 0.0D0) GO TO 160
ATOLI = ATOL(1)
DO 150 I = 1,N
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
AYI = ABS(Y(I))
IF (AYI .NE. 0.0D0) TOL = MAX(TOL,ATOLI/AYI)
150 CONTINUE
160 TOL = MAX(TOL,100.0D0*UROUND)
TOL = MIN(TOL,0.001D0)
SUM = DVNORM (N, RWORK(LF0), RWORK(LEWT))
SUM = 1.0D0/(TOL*W0*W0) + TOL*SUM**2
H0 = 1.0D0/SQRT(SUM)
H0 = MIN(H0,TDIST)
H0 = SIGN(H0,TOUT-T)
C Adjust H0 if necessary to meet HMAX bound. ---------------------------
180 RH = ABS(H0)*HMXI
IF (RH .GT. 1.0D0) H0 = H0/RH
C Load H with H0 and scale YH(*,2) by H0. ------------------------------
H = H0
DO 190 I = 1,N
190 RWORK(I+LF0-1) = H0*RWORK(I+LF0-1)
GO TO 270
C-----------------------------------------------------------------------
C Block D.
C The next code block is for continuation calls only (ISTATE = 2 or 3)
C and is to check stop conditions before taking a step.
C-----------------------------------------------------------------------
200 NSLAST = NST
GO TO (210, 250, 220, 230, 240), ITASK
210 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
220 TP = TN - HU*(1.0D0 + 100.0D0*UROUND)
IF ((TP - TOUT)*H .GT. 0.0D0) GO TO 623
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
GO TO 400
230 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
IF ((TCRIT - TOUT)*H .LT. 0.0D0) GO TO 625
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 245
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
240 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
245 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
IF (ISTATE .EQ. 2) JSTART = -2
C-----------------------------------------------------------------------
C Block E.
C The next block is normally executed for all calls and contains
C the call to the one-step core integrator DSTODE.
C
C This is a looping point for the integration steps.
C
C First check for too many steps being taken, update EWT (if not at
C start of problem), check for too much accuracy being requested, and
C check for H below the roundoff level in T.
C-----------------------------------------------------------------------
250 CONTINUE
IF ((NST-NSLAST) .GE. MXSTEP) GO TO 500
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 260 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 510
260 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
270 TOLSF = UROUND*DVNORM (N, RWORK(LYH), RWORK(LEWT))
IF (TOLSF .LE. 1.0D0) GO TO 280
TOLSF = TOLSF*2.0D0
IF (NST .EQ. 0) GO TO 626
GO TO 520
280 IF ((TN + H) .NE. TN) GO TO 290
NHNIL = NHNIL + 1
IF (NHNIL .GT. MXHNIL) GO TO 290
MSG = 'DLSODES- Warning..Internal T (=R1) and H (=R2) are'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' such that in the machine, T + H = T on the next step '
CALL XERRWD (MSG, 60, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' (H = step size). Solver will continue anyway.'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 2, TN, H)
IF (NHNIL .LT. MXHNIL) GO TO 290
MSG = 'DLSODES- Above warning has been issued I1 times. '
CALL XERRWD (MSG, 50, 102, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' It will not be issued again for this problem.'
CALL XERRWD (MSG, 50, 102, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
290 CONTINUE
C-----------------------------------------------------------------------
C CALL DSTODE(NEQ,Y,YH,NYH,YH,EWT,SAVF,ACOR,WM,WM,F,JAC,DPRJS,DSOLSS)
C-----------------------------------------------------------------------
CALL DSTODE (NEQ, Y, RWORK(LYH), NYH, RWORK(LYH), RWORK(LEWT),
1 RWORK(LSAVF), RWORK(LACOR), RWORK(LWM), RWORK(LWM),
2 F, JAC, DPRJS, DSOLSS)
KGO = 1 - KFLAG
GO TO (300, 530, 540, 550), KGO
C-----------------------------------------------------------------------
C Block F.
C The following block handles the case of a successful return from the
C core integrator (KFLAG = 0). Test for stop conditions.
C-----------------------------------------------------------------------
300 INIT = 1
GO TO (310, 400, 330, 340, 350), ITASK
C ITASK = 1. if TOUT has been reached, interpolate. -------------------
310 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
C ITASK = 3. Jump to exit if TOUT was reached. ------------------------
330 IF ((TN - TOUT)*H .GE. 0.0D0) GO TO 400
GO TO 250
C ITASK = 4. See if TOUT or TCRIT was reached. Adjust H if necessary.
340 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 345
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
345 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
JSTART = -2
GO TO 250
C ITASK = 5. See if TCRIT was reached and jump to exit. ---------------
350 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
C-----------------------------------------------------------------------
C Block G.
C The following block handles all successful returns from DLSODES.
C If ITASK .ne. 1, Y is loaded from YH and T is set accordingly.
C ISTATE is set to 2, and the optional outputs are loaded into the
C work arrays before returning.
C-----------------------------------------------------------------------
400 DO 410 I = 1,N
410 Y(I) = RWORK(I+LYH-1)
T = TN
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 420
IF (IHIT) T = TCRIT
420 ISTATE = 2
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(19) = NNZ
IWORK(20) = NGP
IWORK(21) = NLU
IWORK(25) = NZL
IWORK(26) = NZU
RETURN
C-----------------------------------------------------------------------
C Block H.
C The following block handles all unsuccessful returns other than
C those for illegal input. First the error message routine is called.
C If there was an error test or convergence test failure, IMXER is set.
C Then Y is loaded from YH and T is set to TN.
C The optional outputs are loaded into the work arrays before returning.
C-----------------------------------------------------------------------
C The maximum number of steps was taken before reaching TOUT. ----------
500 MSG = 'DLSODES- At current T (=R1), MXSTEP (=I1) steps '
CALL XERRWD (MSG, 50, 201, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' taken on this call before reaching TOUT '
CALL XERRWD (MSG, 50, 201, 0, 1, MXSTEP, 0, 1, TN, 0.0D0)
ISTATE = -1
GO TO 580
C EWT(i) .le. 0.0 for some i (not at start of problem). ----------------
510 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODES- At T (=R1), EWT(I1) has become R2 .le. 0.'
CALL XERRWD (MSG, 50, 202, 0, 1, I, 0, 2, TN, EWTI)
ISTATE = -6
GO TO 580
C Too much accuracy requested for machine precision. -------------------
520 MSG = 'DLSODES- At T (=R1), too much accuracy requested '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' for precision of machine.. See TOLSF (=R2) '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 2, TN, TOLSF)
RWORK(14) = TOLSF
ISTATE = -2
GO TO 580
C KFLAG = -1. Error test failed repeatedly or with ABS(H) = HMIN. -----
530 MSG = 'DLSODES- At T(=R1) and step size H(=R2), the error'
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' test failed repeatedly or with ABS(H) = HMIN'
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 2, TN, H)
ISTATE = -4
GO TO 560
C KFLAG = -2. Convergence failed repeatedly or with ABS(H) = HMIN. ----
540 MSG = 'DLSODES- At T (=R1) and step size H (=R2), the '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' corrector convergence failed repeatedly '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' or with ABS(H) = HMIN '
CALL XERRWD (MSG, 30, 205, 0, 0, 0, 0, 2, TN, H)
ISTATE = -5
GO TO 560
C KFLAG = -3. Fatal error flag returned by DPRJS or DSOLSS (CDRV). ----
550 MSG = 'DLSODES- At T (=R1) and step size H (=R2), a fatal'
CALL XERRWD (MSG, 50, 207, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' error flag was returned by CDRV (by way of '
CALL XERRWD (MSG, 50, 207, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' Subroutine DPRJS or DSOLSS) '
CALL XERRWD (MSG, 40, 207, 0, 0, 0, 0, 2, TN, H)
ISTATE = -7
GO TO 580
C Compute IMXER if relevant. -------------------------------------------
560 BIG = 0.0D0
IMXER = 1
DO 570 I = 1,N
SIZE = ABS(RWORK(I+LACOR-1)*RWORK(I+LEWT-1))
IF (BIG .GE. SIZE) GO TO 570
BIG = SIZE
IMXER = I
570 CONTINUE
IWORK(16) = IMXER
C Set Y vector, T, and optional outputs. -------------------------------
580 DO 590 I = 1,N
590 Y(I) = RWORK(I+LYH-1)
T = TN
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(19) = NNZ
IWORK(20) = NGP
IWORK(21) = NLU
IWORK(25) = NZL
IWORK(26) = NZU
RETURN
C-----------------------------------------------------------------------
C Block I.
C The following block handles all error returns due to illegal input
C (ISTATE = -3), as detected before calling the core integrator.
C First the error message routine is called. If the illegal input
C is a negative ISTATE, the run is aborted (apparent infinite loop).
C-----------------------------------------------------------------------
601 MSG = 'DLSODES- ISTATE (=I1) illegal.'
CALL XERRWD (MSG, 30, 1, 0, 1, ISTATE, 0, 0, 0.0D0, 0.0D0)
IF (ISTATE .LT. 0) GO TO 800
GO TO 700
602 MSG = 'DLSODES- ITASK (=I1) illegal. '
CALL XERRWD (MSG, 30, 2, 0, 1, ITASK, 0, 0, 0.0D0, 0.0D0)
GO TO 700
603 MSG = 'DLSODES- ISTATE.gt.1 but DLSODES not initialized. '
CALL XERRWD (MSG, 50, 3, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
GO TO 700
604 MSG = 'DLSODES- NEQ (=I1) .lt. 1 '
CALL XERRWD (MSG, 30, 4, 0, 1, NEQ(1), 0, 0, 0.0D0, 0.0D0)
GO TO 700
605 MSG = 'DLSODES- ISTATE = 3 and NEQ increased (I1 to I2). '
CALL XERRWD (MSG, 50, 5, 0, 2, N, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
606 MSG = 'DLSODES- ITOL (=I1) illegal. '
CALL XERRWD (MSG, 30, 6, 0, 1, ITOL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
607 MSG = 'DLSODES- IOPT (=I1) illegal. '
CALL XERRWD (MSG, 30, 7, 0, 1, IOPT, 0, 0, 0.0D0, 0.0D0)
GO TO 700
608 MSG = 'DLSODES- MF (=I1) illegal. '
CALL XERRWD (MSG, 30, 8, 0, 1, MF, 0, 0, 0.0D0, 0.0D0)
GO TO 700
609 MSG = 'DLSODES- SETH (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 9, 0, 0, 0, 0, 1, SETH, 0.0D0)
GO TO 700
611 MSG = 'DLSODES- MAXORD (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 11, 0, 1, MAXORD, 0, 0, 0.0D0, 0.0D0)
GO TO 700
612 MSG = 'DLSODES- MXSTEP (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 12, 0, 1, MXSTEP, 0, 0, 0.0D0, 0.0D0)
GO TO 700
613 MSG = 'DLSODES- MXHNIL (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 13, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
614 MSG = 'DLSODES- TOUT (=R1) behind T (=R2) '
CALL XERRWD (MSG, 40, 14, 0, 0, 0, 0, 2, TOUT, T)
MSG = ' Integration direction is given by H0 (=R1) '
CALL XERRWD (MSG, 50, 14, 0, 0, 0, 0, 1, H0, 0.0D0)
GO TO 700
615 MSG = 'DLSODES- HMAX (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 15, 0, 0, 0, 0, 1, HMAX, 0.0D0)
GO TO 700
616 MSG = 'DLSODES- HMIN (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 16, 0, 0, 0, 0, 1, HMIN, 0.0D0)
GO TO 700
617 MSG = 'DLSODES- RWORK length is insufficient to proceed. '
CALL XERRWD (MSG, 50, 17, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' Length needed is .ge. LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 17, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
618 MSG = 'DLSODES- IWORK length is insufficient to proceed. '
CALL XERRWD (MSG, 50, 18, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' Length needed is .ge. LENIW (=I1), exceeds LIW (=I2)'
CALL XERRWD (MSG, 60, 18, 0, 2, LENIW, LIW, 0, 0.0D0, 0.0D0)
GO TO 700
619 MSG = 'DLSODES- RTOL(I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 19, 0, 1, I, 0, 1, RTOLI, 0.0D0)
GO TO 700
620 MSG = 'DLSODES- ATOL(I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 20, 0, 1, I, 0, 1, ATOLI, 0.0D0)
GO TO 700
621 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODES- EWT(I1) is R1 .le. 0.0 '
CALL XERRWD (MSG, 40, 21, 0, 1, I, 0, 1, EWTI, 0.0D0)
GO TO 700
622 MSG='DLSODES- TOUT(=R1) too close to T(=R2) to start integration.'
CALL XERRWD (MSG, 60, 22, 0, 0, 0, 0, 2, TOUT, T)
GO TO 700
623 MSG='DLSODES- ITASK = I1 and TOUT (=R1) behind TCUR - HU (= R2) '
CALL XERRWD (MSG, 60, 23, 0, 1, ITASK, 0, 2, TOUT, TP)
GO TO 700
624 MSG='DLSODES- ITASK = 4 or 5 and TCRIT (=R1) behind TCUR (=R2) '
CALL XERRWD (MSG, 60, 24, 0, 0, 0, 0, 2, TCRIT, TN)
GO TO 700
625 MSG='DLSODES- ITASK = 4 or 5 and TCRIT (=R1) behind TOUT (=R2) '
CALL XERRWD (MSG, 60, 25, 0, 0, 0, 0, 2, TCRIT, TOUT)
GO TO 700
626 MSG = 'DLSODES- At start of problem, too much accuracy '
CALL XERRWD (MSG, 50, 26, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' requested for precision of machine.. See TOLSF (=R1) '
CALL XERRWD (MSG, 60, 26, 0, 0, 0, 0, 1, TOLSF, 0.0D0)
RWORK(14) = TOLSF
GO TO 700
627 MSG = 'DLSODES- Trouble in DINTDY. ITASK = I1, TOUT = R1'
CALL XERRWD (MSG, 50, 27, 0, 1, ITASK, 0, 1, TOUT, 0.0D0)
GO TO 700
628 MSG='DLSODES- RWORK length insufficient (for Subroutine DPREP). '
CALL XERRWD (MSG, 60, 28, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' Length needed is .ge. LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 28, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
629 MSG='DLSODES- RWORK length insufficient (for Subroutine JGROUP). '
CALL XERRWD (MSG, 60, 29, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' Length needed is .ge. LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 29, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
630 MSG='DLSODES- RWORK length insufficient (for Subroutine ODRV). '
CALL XERRWD (MSG, 60, 30, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' Length needed is .ge. LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 30, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
631 MSG='DLSODES- Error from ODRV in Yale Sparse Matrix Package. '
CALL XERRWD (MSG, 60, 31, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
IMUL = (IYS - 1)/N
IREM = IYS - IMUL*N
MSG=' At T (=R1), ODRV returned error flag = I1*NEQ + I2. '
CALL XERRWD (MSG, 60, 31, 0, 2, IMUL, IREM, 1, TN, 0.0D0)
GO TO 700
632 MSG='DLSODES- RWORK length insufficient (for Subroutine CDRV). '
CALL XERRWD (MSG, 60, 32, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' Length needed is .ge. LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 32, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
633 MSG='DLSODES- Error from CDRV in Yale Sparse Matrix Package. '
CALL XERRWD (MSG, 60, 33, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
IMUL = (IYS - 1)/N
IREM = IYS - IMUL*N
MSG=' At T (=R1), CDRV returned error flag = I1*NEQ + I2. '
CALL XERRWD (MSG, 60, 33, 0, 2, IMUL, IREM, 1, TN, 0.0D0)
IF (IMUL .EQ. 2) THEN
MSG=' Duplicate entry in sparsity structure descriptors. '
CALL XERRWD (MSG, 60, 33, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
IF (IMUL .EQ. 3 .OR. IMUL .EQ. 6) THEN
MSG=' Insufficient storage for NSFC (called by CDRV). '
CALL XERRWD (MSG, 60, 33, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
C
700 ISTATE = -3
RETURN
C
800 MSG = 'DLSODES- Run aborted.. apparent infinite loop. '
CALL XERRWD (MSG, 50, 303, 2, 0, 0, 0, 0, 0.0D0, 0.0D0)
RETURN
C----------------------- End of Subroutine DLSODES ---------------------
END
*DECK DLSODA
SUBROUTINE DLSODA (F, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
1 ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JAC, JT)
EXTERNAL F, JAC
INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK, LIW, JT
DOUBLE PRECISION Y, T, TOUT, RTOL, ATOL, RWORK
DIMENSION NEQ(*), Y(*), RTOL(*), ATOL(*), RWORK(LRW), IWORK(LIW)
C-----------------------------------------------------------------------
C This is the 12 November 2003 version of
C DLSODA: Livermore Solver for Ordinary Differential Equations, with
C Automatic method switching for stiff and nonstiff problems.
C
C This version is in double precision.
C
C DLSODA solves the initial value problem for stiff or nonstiff
C systems of first order ODEs,
C dy/dt = f(t,y) , or, in component form,
C dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(NEQ)) (i = 1,...,NEQ).
C
C This a variant version of the DLSODE package.
C It switches automatically between stiff and nonstiff methods.
C This means that the user does not have to determine whether the
C problem is stiff or not, and the solver will automatically choose the
C appropriate method. It always starts with the nonstiff method.
C
C Authors: Alan C. Hindmarsh
C Center for Applied Scientific Computing, L-561
C Lawrence Livermore National Laboratory
C Livermore, CA 94551
C and
C Linda R. Petzold
C Univ. of California at Santa Barbara
C Dept. of Computer Science
C Santa Barbara, CA 93106
C
C References:
C 1. Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE
C Solvers, in Scientific Computing, R. S. Stepleman et al. (Eds.),
C North-Holland, Amsterdam, 1983, pp. 55-64.
C 2. Linda R. Petzold, Automatic Selection of Methods for Solving
C Stiff and Nonstiff Systems of Ordinary Differential Equations,
C Siam J. Sci. Stat. Comput. 4 (1983), pp. 136-148.
C-----------------------------------------------------------------------
C Summary of Usage.
C
C Communication between the user and the DLSODA package, for normal
C situations, is summarized here. This summary describes only a subset
C of the full set of options available. See the full description for
C details, including alternative treatment of the Jacobian matrix,
C optional inputs and outputs, nonstandard options, and
C instructions for special situations. See also the example
C problem (with program and output) following this summary.
C
C A. First provide a subroutine of the form:
C SUBROUTINE F (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y(*), YDOT(*)
C which supplies the vector function f by loading YDOT(i) with f(i).
C
C B. Write a main program which calls Subroutine DLSODA once for
C each point at which answers are desired. This should also provide
C for possible use of logical unit 6 for output of error messages
C by DLSODA. On the first call to DLSODA, supply arguments as follows:
C F = name of subroutine for right-hand side vector f.
C This name must be declared External in calling program.
C NEQ = number of first order ODEs.
C Y = array of initial values, of length NEQ.
C T = the initial value of the independent variable.
C TOUT = first point where output is desired (.ne. T).
C ITOL = 1 or 2 according as ATOL (below) is a scalar or array.
C RTOL = relative tolerance parameter (scalar).
C ATOL = absolute tolerance parameter (scalar or array).
C the estimated local error in y(i) will be controlled so as
C to be less than
C EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
C EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
C Thus the local error test passes if, in each component,
C either the absolute error is less than ATOL (or ATOL(i)),
C or the relative error is less than RTOL.
C Use RTOL = 0.0 for pure absolute error control, and
C use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative error
C control. Caution: actual (global) errors may exceed these
C local tolerances, so choose them conservatively.
C ITASK = 1 for normal computation of output values of y at t = TOUT.
C ISTATE = integer flag (input and output). Set ISTATE = 1.
C IOPT = 0 to indicate no optional inputs used.
C RWORK = real work array of length at least:
C 22 + NEQ * MAX(16, NEQ + 9).
C See also Paragraph E below.
C LRW = declared length of RWORK (in user's dimension).
C IWORK = integer work array of length at least 20 + NEQ.
C LIW = declared length of IWORK (in user's dimension).
C JAC = name of subroutine for Jacobian matrix.
C Use a dummy name. See also Paragraph E below.
C JT = Jacobian type indicator. Set JT = 2.
C See also Paragraph E below.
C Note that the main program must declare arrays Y, RWORK, IWORK,
C and possibly ATOL.
C
C C. The output from the first call (or any call) is:
C Y = array of computed values of y(t) vector.
C T = corresponding value of independent variable (normally TOUT).
C ISTATE = 2 if DLSODA was successful, negative otherwise.
C -1 means excess work done on this call (perhaps wrong JT).
C -2 means excess accuracy requested (tolerances too small).
C -3 means illegal input detected (see printed message).
C -4 means repeated error test failures (check all inputs).
C -5 means repeated convergence failures (perhaps bad Jacobian
C supplied or wrong choice of JT or tolerances).
C -6 means error weight became zero during problem. (Solution
C component i vanished, and ATOL or ATOL(i) = 0.)
C -7 means work space insufficient to finish (see messages).
C
C D. To continue the integration after a successful return, simply
C reset TOUT and call DLSODA again. No other parameters need be reset.
C
C E. Note: If and when DLSODA regards the problem as stiff, and
C switches methods accordingly, it must make use of the NEQ by NEQ
C Jacobian matrix, J = df/dy. For the sake of simplicity, the
C inputs to DLSODA recommended in Paragraph B above cause DLSODA to
C treat J as a full matrix, and to approximate it internally by
C difference quotients. Alternatively, J can be treated as a band
C matrix (with great potential reduction in the size of the RWORK
C array). Also, in either the full or banded case, the user can supply
C J in closed form, with a routine whose name is passed as the JAC
C argument. These alternatives are described in the paragraphs on
C RWORK, JAC, and JT in the full description of the call sequence below.
C
C-----------------------------------------------------------------------
C Example Problem.
C
C The following is a simple example problem, with the coding
C needed for its solution by DLSODA. The problem is from chemical
C kinetics, and consists of the following three rate equations:
C dy1/dt = -.04*y1 + 1.e4*y2*y3
C dy2/dt = .04*y1 - 1.e4*y2*y3 - 3.e7*y2**2
C dy3/dt = 3.e7*y2**2
C on the interval from t = 0.0 to t = 4.e10, with initial conditions
C y1 = 1.0, y2 = y3 = 0. The problem is stiff.
C
C The following coding solves this problem with DLSODA,
C printing results at t = .4, 4., ..., 4.e10. It uses
C ITOL = 2 and ATOL much smaller for y2 than y1 or y3 because
C y2 has much smaller values.
C At the end of the run, statistical quantities of interest are
C printed (see optional outputs in the full description below).
C
C EXTERNAL FEX
C DOUBLE PRECISION ATOL, RTOL, RWORK, T, TOUT, Y
C DIMENSION Y(3), ATOL(3), RWORK(70), IWORK(23)
C NEQ = 3
C Y(1) = 1.
C Y(2) = 0.
C Y(3) = 0.
C T = 0.
C TOUT = .4
C ITOL = 2
C RTOL = 1.D-4
C ATOL(1) = 1.D-6
C ATOL(2) = 1.D-10
C ATOL(3) = 1.D-6
C ITASK = 1
C ISTATE = 1
C IOPT = 0
C LRW = 70
C LIW = 23
C JT = 2
C DO 40 IOUT = 1,12
C CALL DLSODA(FEX,NEQ,Y,T,TOUT,ITOL,RTOL,ATOL,ITASK,ISTATE,
C 1 IOPT,RWORK,LRW,IWORK,LIW,JDUM,JT)
C WRITE(6,20)T,Y(1),Y(2),Y(3)
C 20 FORMAT(' At t =',D12.4,' Y =',3D14.6)
C IF (ISTATE .LT. 0) GO TO 80
C 40 TOUT = TOUT*10.
C WRITE(6,60)IWORK(11),IWORK(12),IWORK(13),IWORK(19),RWORK(15)
C 60 FORMAT(/' No. steps =',I4,' No. f-s =',I4,' No. J-s =',I4/
C 1 ' Method last used =',I2,' Last switch was at t =',D12.4)
C STOP
C 80 WRITE(6,90)ISTATE
C 90 FORMAT(///' Error halt.. ISTATE =',I3)
C STOP
C END
C
C SUBROUTINE FEX (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y, YDOT
C DIMENSION Y(3), YDOT(3)
C YDOT(1) = -.04*Y(1) + 1.D4*Y(2)*Y(3)
C YDOT(3) = 3.D7*Y(2)*Y(2)
C YDOT(2) = -YDOT(1) - YDOT(3)
C RETURN
C END
C
C The output of this program (on a CDC-7600 in single precision)
C is as follows:
C
C At t = 4.0000e-01 y = 9.851712e-01 3.386380e-05 1.479493e-02
C At t = 4.0000e+00 Y = 9.055333e-01 2.240655e-05 9.444430e-02
C At t = 4.0000e+01 Y = 7.158403e-01 9.186334e-06 2.841505e-01
C At t = 4.0000e+02 Y = 4.505250e-01 3.222964e-06 5.494717e-01
C At t = 4.0000e+03 Y = 1.831975e-01 8.941774e-07 8.168016e-01
C At t = 4.0000e+04 Y = 3.898730e-02 1.621940e-07 9.610125e-01
C At t = 4.0000e+05 Y = 4.936363e-03 1.984221e-08 9.950636e-01
C At t = 4.0000e+06 Y = 5.161831e-04 2.065786e-09 9.994838e-01
C At t = 4.0000e+07 Y = 5.179817e-05 2.072032e-10 9.999482e-01
C At t = 4.0000e+08 Y = 5.283401e-06 2.113371e-11 9.999947e-01
C At t = 4.0000e+09 Y = 4.659031e-07 1.863613e-12 9.999995e-01
C At t = 4.0000e+10 Y = 1.404280e-08 5.617126e-14 1.000000e+00
C
C No. steps = 361 No. f-s = 693 No. J-s = 64
C Method last used = 2 Last switch was at t = 6.0092e-03
C-----------------------------------------------------------------------
C Full description of user interface to DLSODA.
C
C The user interface to DLSODA consists of the following parts.
C
C 1. The call sequence to Subroutine DLSODA, which is a driver
C routine for the solver. This includes descriptions of both
C the call sequence arguments and of user-supplied routines.
C following these descriptions is a description of
C optional inputs available through the call sequence, and then
C a description of optional outputs (in the work arrays).
C
C 2. Descriptions of other routines in the DLSODA package that may be
C (optionally) called by the user. These provide the ability to
C alter error message handling, save and restore the internal
C Common, and obtain specified derivatives of the solution y(t).
C
C 3. Descriptions of Common blocks to be declared in overlay
C or similar environments, or to be saved when doing an interrupt
C of the problem and continued solution later.
C
C 4. Description of a subroutine in the DLSODA package,
C which the user may replace with his/her own version, if desired.
C this relates to the measurement of errors.
C
C-----------------------------------------------------------------------
C Part 1. Call Sequence.
C
C The call sequence parameters used for input only are
C F, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK, IOPT, LRW, LIW, JAC, JT,
C and those used for both input and output are
C Y, T, ISTATE.
C The work arrays RWORK and IWORK are also used for conditional and
C optional inputs and optional outputs. (The term output here refers
C to the return from Subroutine DLSODA to the user's calling program.)
C
C The legality of input parameters will be thoroughly checked on the
C initial call for the problem, but not checked thereafter unless a
C change in input parameters is flagged by ISTATE = 3 on input.
C
C The descriptions of the call arguments are as follows.
C
C F = the name of the user-supplied subroutine defining the
C ODE system. The system must be put in the first-order
C form dy/dt = f(t,y), where f is a vector-valued function
C of the scalar t and the vector y. Subroutine F is to
C compute the function f. It is to have the form
C SUBROUTINE F (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y(*), YDOT(*)
C where NEQ, T, and Y are input, and the array YDOT = f(t,y)
C is output. Y and YDOT are arrays of length NEQ.
C Subroutine F should not alter Y(1),...,Y(NEQ).
C F must be declared External in the calling program.
C
C Subroutine F may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in F) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y below.
C
C If quantities computed in the F routine are needed
C externally to DLSODA, an extra call to F should be made
C for this purpose, for consistent and accurate results.
C If only the derivative dy/dt is needed, use DINTDY instead.
C
C NEQ = the size of the ODE system (number of first order
C ordinary differential equations). Used only for input.
C NEQ may be decreased, but not increased, during the problem.
C If NEQ is decreased (with ISTATE = 3 on input), the
C remaining components of Y should be left undisturbed, if
C these are to be accessed in F and/or JAC.
C
C Normally, NEQ is a scalar, and it is generally referred to
C as a scalar in this user interface description. However,
C NEQ may be an array, with NEQ(1) set to the system size.
C (The DLSODA package accesses only NEQ(1).) In either case,
C this parameter is passed as the NEQ argument in all calls
C to F and JAC. Hence, if it is an array, locations
C NEQ(2),... may be used to store other integer data and pass
C it to F and/or JAC. Subroutines F and/or JAC must include
C NEQ in a Dimension statement in that case.
C
C Y = a real array for the vector of dependent variables, of
C length NEQ or more. Used for both input and output on the
C first call (ISTATE = 1), and only for output on other calls.
C On the first call, Y must contain the vector of initial
C values. On output, Y contains the computed solution vector,
C evaluated at T. If desired, the Y array may be used
C for other purposes between calls to the solver.
C
C This array is passed as the Y argument in all calls to
C F and JAC. Hence its length may exceed NEQ, and locations
C Y(NEQ+1),... may be used to store other real data and
C pass it to F and/or JAC. (The DLSODA package accesses only
C Y(1),...,Y(NEQ).)
C
C T = the independent variable. On input, T is used only on the
C first call, as the initial point of the integration.
C on output, after each call, T is the value at which a
C computed solution Y is evaluated (usually the same as TOUT).
C on an error return, T is the farthest point reached.
C
C TOUT = the next value of t at which a computed solution is desired.
C Used only for input.
C
C When starting the problem (ISTATE = 1), TOUT may be equal
C to T for one call, then should .ne. T for the next call.
C For the initial t, an input value of TOUT .ne. T is used
C in order to determine the direction of the integration
C (i.e. the algebraic sign of the step sizes) and the rough
C scale of the problem. Integration in either direction
C (forward or backward in t) is permitted.
C
C If ITASK = 2 or 5 (one-step modes), TOUT is ignored after
C the first call (i.e. the first call with TOUT .ne. T).
C Otherwise, TOUT is required on every call.
C
C If ITASK = 1, 3, or 4, the values of TOUT need not be
C monotone, but a value of TOUT which backs up is limited
C to the current internal T interval, whose endpoints are
C TCUR - HU and TCUR (see optional outputs, below, for
C TCUR and HU).
C
C ITOL = an indicator for the type of error control. See
C description below under ATOL. Used only for input.
C
C RTOL = a relative error tolerance parameter, either a scalar or
C an array of length NEQ. See description below under ATOL.
C Input only.
C
C ATOL = an absolute error tolerance parameter, either a scalar or
C an array of length NEQ. Input only.
C
C The input parameters ITOL, RTOL, and ATOL determine
C the error control performed by the solver. The solver will
C control the vector E = (E(i)) of estimated local errors
C in y, according to an inequality of the form
C max-norm of ( E(i)/EWT(i) ) .le. 1,
C where EWT = (EWT(i)) is a vector of positive error weights.
C The values of RTOL and ATOL should all be non-negative.
C The following table gives the types (scalar/array) of
C RTOL and ATOL, and the corresponding form of EWT(i).
C
C ITOL RTOL ATOL EWT(i)
C 1 scalar scalar RTOL*ABS(Y(i)) + ATOL
C 2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
C 3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
C 4 array array RTOL(i)*ABS(Y(i)) + ATOL(i)
C
C When either of these parameters is a scalar, it need not
C be dimensioned in the user's calling program.
C
C If none of the above choices (with ITOL, RTOL, and ATOL
C fixed throughout the problem) is suitable, more general
C error controls can be obtained by substituting a
C user-supplied routine for the setting of EWT.
C See Part 4 below.
C
C If global errors are to be estimated by making a repeated
C run on the same problem with smaller tolerances, then all
C components of RTOL and ATOL (i.e. of EWT) should be scaled
C down uniformly.
C
C ITASK = an index specifying the task to be performed.
C Input only. ITASK has the following values and meanings.
C 1 means normal computation of output values of y(t) at
C t = TOUT (by overshooting and interpolating).
C 2 means take one step only and return.
C 3 means stop at the first internal mesh point at or
C beyond t = TOUT and return.
C 4 means normal computation of output values of y(t) at
C t = TOUT but without overshooting t = TCRIT.
C TCRIT must be input as RWORK(1). TCRIT may be equal to
C or beyond TOUT, but not behind it in the direction of
C integration. This option is useful if the problem
C has a singularity at or beyond t = TCRIT.
C 5 means take one step, without passing TCRIT, and return.
C TCRIT must be input as RWORK(1).
C
C Note: If ITASK = 4 or 5 and the solver reaches TCRIT
C (within roundoff), it will return T = TCRIT (exactly) to
C indicate this (unless ITASK = 4 and TOUT comes before TCRIT,
C in which case answers at t = TOUT are returned first).
C
C ISTATE = an index used for input and output to specify the
C the state of the calculation.
C
C On input, the values of ISTATE are as follows.
C 1 means this is the first call for the problem
C (initializations will be done). See note below.
C 2 means this is not the first call, and the calculation
C is to continue normally, with no change in any input
C parameters except possibly TOUT and ITASK.
C (If ITOL, RTOL, and/or ATOL are changed between calls
C with ISTATE = 2, the new values will be used but not
C tested for legality.)
C 3 means this is not the first call, and the
C calculation is to continue normally, but with
C a change in input parameters other than
C TOUT and ITASK. Changes are allowed in
C NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, JT, ML, MU,
C and any optional inputs except H0, MXORDN, and MXORDS.
C (See IWORK description for ML and MU.)
C Note: A preliminary call with TOUT = T is not counted
C as a first call here, as no initialization or checking of
C input is done. (Such a call is sometimes useful for the
C purpose of outputting the initial conditions.)
C Thus the first call for which TOUT .ne. T requires
C ISTATE = 1 on input.
C
C On output, ISTATE has the following values and meanings.
C 1 means nothing was done; TOUT = T and ISTATE = 1 on input.
C 2 means the integration was performed successfully.
C -1 means an excessive amount of work (more than MXSTEP
C steps) was done on this call, before completing the
C requested task, but the integration was otherwise
C successful as far as T. (MXSTEP is an optional input
C and is normally 500.) To continue, the user may
C simply reset ISTATE to a value .gt. 1 and call again
C (the excess work step counter will be reset to 0).
C In addition, the user may increase MXSTEP to avoid
C this error return (see below on optional inputs).
C -2 means too much accuracy was requested for the precision
C of the machine being used. This was detected before
C completing the requested task, but the integration
C was successful as far as T. To continue, the tolerance
C parameters must be reset, and ISTATE must be set
C to 3. The optional output TOLSF may be used for this
C purpose. (Note: If this condition is detected before
C taking any steps, then an illegal input return
C (ISTATE = -3) occurs instead.)
C -3 means illegal input was detected, before taking any
C integration steps. See written message for details.
C Note: If the solver detects an infinite loop of calls
C to the solver with illegal input, it will cause
C the run to stop.
C -4 means there were repeated error test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C The problem may have a singularity, or the input
C may be inappropriate.
C -5 means there were repeated convergence test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C This may be caused by an inaccurate Jacobian matrix,
C if one is being used.
C -6 means EWT(i) became zero for some i during the
C integration. Pure relative error control (ATOL(i)=0.0)
C was requested on a variable which has now vanished.
C The integration was successful as far as T.
C -7 means the length of RWORK and/or IWORK was too small to
C proceed, but the integration was successful as far as T.
C This happens when DLSODA chooses to switch methods
C but LRW and/or LIW is too small for the new method.
C
C Note: Since the normal output value of ISTATE is 2,
C it does not need to be reset for normal continuation.
C Also, since a negative input value of ISTATE will be
C regarded as illegal, a negative output value requires the
C user to change it, and possibly other inputs, before
C calling the solver again.
C
C IOPT = an integer flag to specify whether or not any optional
C inputs are being used on this call. Input only.
C The optional inputs are listed separately below.
C IOPT = 0 means no optional inputs are being used.
C default values will be used in all cases.
C IOPT = 1 means one or more optional inputs are being used.
C
C RWORK = a real array (double precision) for work space, and (in the
C first 20 words) for conditional and optional inputs and
C optional outputs.
C As DLSODA switches automatically between stiff and nonstiff
C methods, the required length of RWORK can change during the
C problem. Thus the RWORK array passed to DLSODA can either
C have a static (fixed) length large enough for both methods,
C or have a dynamic (changing) length altered by the calling
C program in response to output from DLSODA.
C
C --- Fixed Length Case ---
C If the RWORK length is to be fixed, it should be at least
C MAX (LRN, LRS),
C where LRN and LRS are the RWORK lengths required when the
C current method is nonstiff or stiff, respectively.
C
C The separate RWORK length requirements LRN and LRS are
C as follows:
C IF NEQ is constant and the maximum method orders have
C their default values, then
C LRN = 20 + 16*NEQ,
C LRS = 22 + 9*NEQ + NEQ**2 if JT = 1 or 2,
C LRS = 22 + 10*NEQ + (2*ML+MU)*NEQ if JT = 4 or 5.
C Under any other conditions, LRN and LRS are given by:
C LRN = 20 + NYH*(MXORDN+1) + 3*NEQ,
C LRS = 20 + NYH*(MXORDS+1) + 3*NEQ + LMAT,
C where
C NYH = the initial value of NEQ,
C MXORDN = 12, unless a smaller value is given as an
C optional input,
C MXORDS = 5, unless a smaller value is given as an
C optional input,
C LMAT = length of matrix work space:
C LMAT = NEQ**2 + 2 if JT = 1 or 2,
C LMAT = (2*ML + MU + 1)*NEQ + 2 if JT = 4 or 5.
C
C --- Dynamic Length Case ---
C If the length of RWORK is to be dynamic, then it should
C be at least LRN or LRS, as defined above, depending on the
C current method. Initially, it must be at least LRN (since
C DLSODA starts with the nonstiff method). On any return
C from DLSODA, the optional output MCUR indicates the current
C method. If MCUR differs from the value it had on the
C previous return, or if there has only been one call to
C DLSODA and MCUR is now 2, then DLSODA has switched
C methods during the last call, and the length of RWORK
C should be reset (to LRN if MCUR = 1, or to LRS if
C MCUR = 2). (An increase in the RWORK length is required
C if DLSODA returned ISTATE = -7, but not otherwise.)
C After resetting the length, call DLSODA with ISTATE = 3
C to signal that change.
C
C LRW = the length of the array RWORK, as declared by the user.
C (This will be checked by the solver.)
C
C IWORK = an integer array for work space.
C As DLSODA switches automatically between stiff and nonstiff
C methods, the required length of IWORK can change during
C problem, between
C LIS = 20 + NEQ and LIN = 20,
C respectively. Thus the IWORK array passed to DLSODA can
C either have a fixed length of at least 20 + NEQ, or have a
C dynamic length of at least LIN or LIS, depending on the
C current method. The comments on dynamic length under
C RWORK above apply here. Initially, this length need
C only be at least LIN = 20.
C
C The first few words of IWORK are used for conditional and
C optional inputs and optional outputs.
C
C The following 2 words in IWORK are conditional inputs:
C IWORK(1) = ML these are the lower and upper
C IWORK(2) = MU half-bandwidths, respectively, of the
C banded Jacobian, excluding the main diagonal.
C The band is defined by the matrix locations
C (i,j) with i-ML .le. j .le. i+MU. ML and MU
C must satisfy 0 .le. ML,MU .le. NEQ-1.
C These are required if JT is 4 or 5, and
C ignored otherwise. ML and MU may in fact be
C the band parameters for a matrix to which
C df/dy is only approximately equal.
C
C LIW = the length of the array IWORK, as declared by the user.
C (This will be checked by the solver.)
C
C Note: The base addresses of the work arrays must not be
C altered between calls to DLSODA for the same problem.
C The contents of the work arrays must not be altered
C between calls, except possibly for the conditional and
C optional inputs, and except for the last 3*NEQ words of RWORK.
C The latter space is used for internal scratch space, and so is
C available for use by the user outside DLSODA between calls, if
C desired (but not for use by F or JAC).
C
C JAC = the name of the user-supplied routine to compute the
C Jacobian matrix, df/dy, if JT = 1 or 4. The JAC routine
C is optional, but if the problem is expected to be stiff much
C of the time, you are encouraged to supply JAC, for the sake
C of efficiency. (Alternatively, set JT = 2 or 5 to have
C DLSODA compute df/dy internally by difference quotients.)
C If and when DLSODA uses df/dy, it treats this NEQ by NEQ
C matrix either as full (JT = 1 or 2), or as banded (JT =
C 4 or 5) with half-bandwidths ML and MU (discussed under
C IWORK above). In either case, if JT = 1 or 4, the JAC
C routine must compute df/dy as a function of the scalar t
C and the vector y. It is to have the form
C SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD)
C DOUBLE PRECISION T, Y(*), PD(NROWPD,*)
C where NEQ, T, Y, ML, MU, and NROWPD are input and the array
C PD is to be loaded with partial derivatives (elements of
C the Jacobian matrix) on output. PD must be given a first
C dimension of NROWPD. T and Y have the same meaning as in
C Subroutine F.
C In the full matrix case (JT = 1), ML and MU are
C ignored, and the Jacobian is to be loaded into PD in
C columnwise manner, with df(i)/dy(j) loaded into PD(i,j).
C In the band matrix case (JT = 4), the elements
C within the band are to be loaded into PD in columnwise
C manner, with diagonal lines of df/dy loaded into the rows
C of PD. Thus df(i)/dy(j) is to be loaded into PD(i-j+MU+1,j).
C ML and MU are the half-bandwidth parameters (see IWORK).
C The locations in PD in the two triangular areas which
C correspond to nonexistent matrix elements can be ignored
C or loaded arbitrarily, as they are overwritten by DLSODA.
C JAC need not provide df/dy exactly. A crude
C approximation (possibly with a smaller bandwidth) will do.
C In either case, PD is preset to zero by the solver,
C so that only the nonzero elements need be loaded by JAC.
C Each call to JAC is preceded by a call to F with the same
C arguments NEQ, T, and Y. Thus to gain some efficiency,
C intermediate quantities shared by both calculations may be
C saved in a user Common block by F and not recomputed by JAC,
C if desired. Also, JAC may alter the Y array, if desired.
C JAC must be declared External in the calling program.
C Subroutine JAC may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in JAC) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y above.
C
C JT = Jacobian type indicator. Used only for input.
C JT specifies how the Jacobian matrix df/dy will be
C treated, if and when DLSODA requires this matrix.
C JT has the following values and meanings:
C 1 means a user-supplied full (NEQ by NEQ) Jacobian.
C 2 means an internally generated (difference quotient) full
C Jacobian (using NEQ extra calls to F per df/dy value).
C 4 means a user-supplied banded Jacobian.
C 5 means an internally generated banded Jacobian (using
C ML+MU+1 extra calls to F per df/dy evaluation).
C If JT = 1 or 4, the user must supply a Subroutine JAC
C (the name is arbitrary) as described above under JAC.
C If JT = 2 or 5, a dummy argument can be used.
C-----------------------------------------------------------------------
C Optional Inputs.
C
C The following is a list of the optional inputs provided for in the
C call sequence. (See also Part 2.) For each such input variable,
C this table lists its name as used in this documentation, its
C location in the call sequence, its meaning, and the default value.
C The use of any of these inputs requires IOPT = 1, and in that
C case all of these inputs are examined. A value of zero for any
C of these optional inputs will cause the default value to be used.
C Thus to use a subset of the optional inputs, simply preload
C locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively, and
C then set those of interest to nonzero values.
C
C Name Location Meaning and Default Value
C
C H0 RWORK(5) the step size to be attempted on the first step.
C The default value is determined by the solver.
C
C HMAX RWORK(6) the maximum absolute step size allowed.
C The default value is infinite.
C
C HMIN RWORK(7) the minimum absolute step size allowed.
C The default value is 0. (This lower bound is not
C enforced on the final step before reaching TCRIT
C when ITASK = 4 or 5.)
C
C IXPR IWORK(5) flag to generate extra printing at method switches.
C IXPR = 0 means no extra printing (the default).
C IXPR = 1 means print data on each switch.
C T, H, and NST will be printed on the same logical
C unit as used for error messages.
C
C MXSTEP IWORK(6) maximum number of (internally defined) steps
C allowed during one call to the solver.
C The default value is 500.
C
C MXHNIL IWORK(7) maximum number of messages printed (per problem)
C warning that T + H = T on a step (H = step size).
C This must be positive to result in a non-default
C value. The default value is 10.
C
C MXORDN IWORK(8) the maximum order to be allowed for the nonstiff
C (Adams) method. the default value is 12.
C if MXORDN exceeds the default value, it will
C be reduced to the default value.
C MXORDN is held constant during the problem.
C
C MXORDS IWORK(9) the maximum order to be allowed for the stiff
C (BDF) method. The default value is 5.
C If MXORDS exceeds the default value, it will
C be reduced to the default value.
C MXORDS is held constant during the problem.
C-----------------------------------------------------------------------
C Optional Outputs.
C
C As optional additional output from DLSODA, the variables listed
C below are quantities related to the performance of DLSODA
C which are available to the user. These are communicated by way of
C the work arrays, but also have internal mnemonic names as shown.
C except where stated otherwise, all of these outputs are defined
C on any successful return from DLSODA, and on any return with
C ISTATE = -1, -2, -4, -5, or -6. On an illegal input return
C (ISTATE = -3), they will be unchanged from their existing values
C (if any), except possibly for TOLSF, LENRW, and LENIW.
C On any error return, outputs relevant to the error will be defined,
C as noted below.
C
C Name Location Meaning
C
C HU RWORK(11) the step size in t last used (successfully).
C
C HCUR RWORK(12) the step size to be attempted on the next step.
C
C TCUR RWORK(13) the current value of the independent variable
C which the solver has actually reached, i.e. the
C current internal mesh point in t. On output, TCUR
C will always be at least as far as the argument
C T, but may be farther (if interpolation was done).
C
C TOLSF RWORK(14) a tolerance scale factor, greater than 1.0,
C computed when a request for too much accuracy was
C detected (ISTATE = -3 if detected at the start of
C the problem, ISTATE = -2 otherwise). If ITOL is
C left unaltered but RTOL and ATOL are uniformly
C scaled up by a factor of TOLSF for the next call,
C then the solver is deemed likely to succeed.
C (The user may also ignore TOLSF and alter the
C tolerance parameters in any other way appropriate.)
C
C TSW RWORK(15) the value of t at the time of the last method
C switch, if any.
C
C NST IWORK(11) the number of steps taken for the problem so far.
C
C NFE IWORK(12) the number of f evaluations for the problem so far.
C
C NJE IWORK(13) the number of Jacobian evaluations (and of matrix
C LU decompositions) for the problem so far.
C
C NQU IWORK(14) the method order last used (successfully).
C
C NQCUR IWORK(15) the order to be attempted on the next step.
C
C IMXER IWORK(16) the index of the component of largest magnitude in
C the weighted local error vector ( E(i)/EWT(i) ),
C on an error return with ISTATE = -4 or -5.
C
C LENRW IWORK(17) the length of RWORK actually required, assuming
C that the length of RWORK is to be fixed for the
C rest of the problem, and that switching may occur.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C LENIW IWORK(18) the length of IWORK actually required, assuming
C that the length of IWORK is to be fixed for the
C rest of the problem, and that switching may occur.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C MUSED IWORK(19) the method indicator for the last successful step:
C 1 means Adams (nonstiff), 2 means BDF (stiff).
C
C MCUR IWORK(20) the current method indicator:
C 1 means Adams (nonstiff), 2 means BDF (stiff).
C This is the method to be attempted
C on the next step. Thus it differs from MUSED
C only if a method switch has just been made.
C
C The following two arrays are segments of the RWORK array which
C may also be of interest to the user as optional outputs.
C For each array, the table below gives its internal name,
C its base address in RWORK, and its description.
C
C Name Base Address Description
C
C YH 21 the Nordsieck history array, of size NYH by
C (NQCUR + 1), where NYH is the initial value
C of NEQ. For j = 0,1,...,NQCUR, column j+1
C of YH contains HCUR**j/factorial(j) times
C the j-th derivative of the interpolating
C polynomial currently representing the solution,
C evaluated at T = TCUR.
C
C ACOR LACOR array of size NEQ used for the accumulated
C (from Common corrections on each step, scaled on output
C as noted) to represent the estimated local error in y
C on the last step. This is the vector E in
C the description of the error control. It is
C defined only on a successful return from
C DLSODA. The base address LACOR is obtained by
C including in the user's program the
C following 2 lines:
C COMMON /DLS001/ RLS(218), ILS(37)
C LACOR = ILS(22)
C
C-----------------------------------------------------------------------
C Part 2. Other Routines Callable.
C
C The following are optional calls which the user may make to
C gain additional capabilities in conjunction with DLSODA.
C (The routines XSETUN and XSETF are designed to conform to the
C SLATEC error handling package.)
C
C Form of Call Function
C CALL XSETUN(LUN) set the logical unit number, LUN, for
C output of messages from DLSODA, if
C the default is not desired.
C The default value of LUN is 6.
C
C CALL XSETF(MFLAG) set a flag to control the printing of
C messages by DLSODA.
C MFLAG = 0 means do not print. (Danger:
C This risks losing valuable information.)
C MFLAG = 1 means print (the default).
C
C Either of the above calls may be made at
C any time and will take effect immediately.
C
C CALL DSRCMA(RSAV,ISAV,JOB) saves and restores the contents of
C the internal Common blocks used by
C DLSODA (see Part 3 below).
C RSAV must be a real array of length 240
C or more, and ISAV must be an integer
C array of length 46 or more.
C JOB=1 means save Common into RSAV/ISAV.
C JOB=2 means restore Common from RSAV/ISAV.
C DSRCMA is useful if one is
C interrupting a run and restarting
C later, or alternating between two or
C more problems solved with DLSODA.
C
C CALL DINTDY(,,,,,) provide derivatives of y, of various
C (see below) orders, at a specified point t, if
C desired. It may be called only after
C a successful return from DLSODA.
C
C The detailed instructions for using DINTDY are as follows.
C The form of the call is:
C
C CALL DINTDY (T, K, RWORK(21), NYH, DKY, IFLAG)
C
C The input parameters are:
C
C T = value of independent variable where answers are desired
C (normally the same as the T last returned by DLSODA).
C For valid results, T must lie between TCUR - HU and TCUR.
C (See optional outputs for TCUR and HU.)
C K = integer order of the derivative desired. K must satisfy
C 0 .le. K .le. NQCUR, where NQCUR is the current order
C (see optional outputs). The capability corresponding
C to K = 0, i.e. computing y(T), is already provided
C by DLSODA directly. Since NQCUR .ge. 1, the first
C derivative dy/dt is always available with DINTDY.
C RWORK(21) = the base address of the history array YH.
C NYH = column length of YH, equal to the initial value of NEQ.
C
C The output parameters are:
C
C DKY = a real array of length NEQ containing the computed value
C of the K-th derivative of y(t).
C IFLAG = integer flag, returned as 0 if K and T were legal,
C -1 if K was illegal, and -2 if T was illegal.
C On an error return, a message is also written.
C-----------------------------------------------------------------------
C Part 3. Common Blocks.
C
C If DLSODA is to be used in an overlay situation, the user
C must declare, in the primary overlay, the variables in:
C (1) the call sequence to DLSODA, and
C (2) the two internal Common blocks
C /DLS001/ of length 255 (218 double precision words
C followed by 37 integer words),
C /DLSA01/ of length 31 (22 double precision words
C followed by 9 integer words).
C
C If DLSODA is used on a system in which the contents of internal
C Common blocks are not preserved between calls, the user should
C declare the above Common blocks in the calling program to insure
C that their contents are preserved.
C
C If the solution of a given problem by DLSODA is to be interrupted
C and then later continued, such as when restarting an interrupted run
C or alternating between two or more problems, the user should save,
C following the return from the last DLSODA call prior to the
C interruption, the contents of the call sequence variables and the
C internal Common blocks, and later restore these values before the
C next DLSODA call for that problem. To save and restore the Common
C blocks, use Subroutine DSRCMA (see Part 2 above).
C
C-----------------------------------------------------------------------
C Part 4. Optionally Replaceable Solver Routines.
C
C Below is a description of a routine in the DLSODA package which
C relates to the measurement of errors, and can be
C replaced by a user-supplied version, if desired. However, since such
C a replacement may have a major impact on performance, it should be
C done only when absolutely necessary, and only with great caution.
C (Note: The means by which the package version of a routine is
C superseded by the user's version may be system-dependent.)
C
C (a) DEWSET.
C The following subroutine is called just before each internal
C integration step, and sets the array of error weights, EWT, as
C described under ITOL/RTOL/ATOL above:
C Subroutine DEWSET (NEQ, ITOL, RTOL, ATOL, YCUR, EWT)
C where NEQ, ITOL, RTOL, and ATOL are as in the DLSODA call sequence,
C YCUR contains the current dependent variable vector, and
C EWT is the array of weights set by DEWSET.
C
C If the user supplies this subroutine, it must return in EWT(i)
C (i = 1,...,NEQ) a positive quantity suitable for comparing errors
C in y(i) to. The EWT array returned by DEWSET is passed to the
C DMNORM routine, and also used by DLSODA in the computation
C of the optional output IMXER, and the increments for difference
C quotient Jacobians.
C
C In the user-supplied version of DEWSET, it may be desirable to use
C the current values of derivatives of y. Derivatives up to order NQ
C are available from the history array YH, described above under
C optional outputs. In DEWSET, YH is identical to the YCUR array,
C extended to NQ + 1 columns with a column length of NYH and scale
C factors of H**j/factorial(j). On the first call for the problem,
C given by NST = 0, NQ is 1 and H is temporarily set to 1.0.
C NYH is the initial value of NEQ. The quantities NQ, H, and NST
C can be obtained by including in DEWSET the statements:
C DOUBLE PRECISION RLS
C COMMON /DLS001/ RLS(218),ILS(37)
C NQ = ILS(33)
C NST = ILS(34)
C H = RLS(212)
C Thus, for example, the current value of dy/dt can be obtained as
C YCUR(NYH+i)/H (i=1,...,NEQ) (and the division by H is
C unnecessary when NST = 0).
C-----------------------------------------------------------------------
C
C***REVISION HISTORY (YYYYMMDD)
C 19811102 DATE WRITTEN
C 19820126 Fixed bug in tests of work space lengths;
C minor corrections in main prologue and comments.
C 19870330 Major update: corrected comments throughout;
C removed TRET from Common; rewrote EWSET with 4 loops;
C fixed t test in INTDY; added Cray directives in STODA;
C in STODA, fixed DELP init. and logic around PJAC call;
C combined routines to save/restore Common;
C passed LEVEL = 0 in error message calls (except run abort).
C 19970225 Fixed lines setting JSTART = -2 in Subroutine LSODA.
C 20010425 Major update: convert source lines to upper case;
C added *DECK lines; changed from 1 to * in dummy dimensions;
C changed names R1MACH/D1MACH to RUMACH/DUMACH;
C renamed routines for uniqueness across single/double prec.;
C converted intrinsic names to generic form;
C removed ILLIN and NTREP (data loaded) from Common;
C removed all 'own' variables from Common;
C changed error messages to quoted strings;
C replaced XERRWV/XERRWD with 1993 revised version;
C converted prologues, comments, error messages to mixed case;
C numerous corrections to prologues and internal comments.
C 20010507 Converted single precision source to double precision.
C 20010613 Revised excess accuracy test (to match rest of ODEPACK).
C 20010808 Fixed bug in DPRJA (matrix in DBNORM call).
C 20020502 Corrected declarations in descriptions of user routines.
C 20031105 Restored 'own' variables to Common blocks, to enable
C interrupt/restart feature.
C 20031112 Added SAVE statements for data-loaded constants.
C
C-----------------------------------------------------------------------
C Other routines in the DLSODA package.
C
C In addition to Subroutine DLSODA, the DLSODA package includes the
C following subroutines and function routines:
C DINTDY computes an interpolated value of the y vector at t = TOUT.
C DSTODA is the core integrator, which does one step of the
C integration and the associated error control.
C DCFODE sets all method coefficients and test constants.
C DPRJA computes and preprocesses the Jacobian matrix J = df/dy
C and the Newton iteration matrix P = I - h*l0*J.
C DSOLSY manages solution of linear system in chord iteration.
C DEWSET sets the error weight vector EWT before each step.
C DMNORM computes the weighted max-norm of a vector.
C DFNORM computes the norm of a full matrix consistent with the
C weighted max-norm on vectors.
C DBNORM computes the norm of a band matrix consistent with the
C weighted max-norm on vectors.
C DSRCMA is a user-callable routine to save and restore
C the contents of the internal Common blocks.
C DGEFA and DGESL are routines from LINPACK for solving full
C systems of linear algebraic equations.
C DGBFA and DGBSL are routines from LINPACK for solving banded
C linear systems.
C DUMACH computes the unit roundoff in a machine-independent manner.
C XERRWD, XSETUN, XSETF, IXSAV, and IUMACH handle the printing of all
C error messages and warnings. XERRWD is machine-dependent.
C Note: DMNORM, DFNORM, DBNORM, DUMACH, IXSAV, and IUMACH are
C function routines. All the others are subroutines.
C
C-----------------------------------------------------------------------
EXTERNAL DPRJA, DSOLSY
DOUBLE PRECISION DUMACH, DMNORM
INTEGER INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER INSUFR, INSUFI, IXPR, IOWNS2, JTYP, MUSED, MXORDN, MXORDS
INTEGER I, I1, I2, IFLAG, IMXER, KGO, LF0,
1 LENIW, LENRW, LENWM, ML, MORD, MU, MXHNL0, MXSTP0
INTEGER LEN1, LEN1C, LEN1N, LEN1S, LEN2, LENIWC, LENRWC
DOUBLE PRECISION ROWNS,
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
DOUBLE PRECISION TSW, ROWNS2, PDNORM
DOUBLE PRECISION ATOLI, AYI, BIG, EWTI, H0, HMAX, HMX, RH, RTOLI,
1 TCRIT, TDIST, TNEXT, TOL, TOLSF, TP, SIZE, SUM, W0
DIMENSION MORD(2)
LOGICAL IHIT
CHARACTER*60 MSG
SAVE MORD, MXSTP0, MXHNL0
C-----------------------------------------------------------------------
C The following two internal Common blocks contain
C (a) variables which are local to any subroutine but whose values must
C be preserved between calls to the routine ("own" variables), and
C (b) variables which are communicated between subroutines.
C The block DLS001 is declared in subroutines DLSODA, DINTDY, DSTODA,
C DPRJA, and DSOLSY.
C The block DLSA01 is declared in subroutines DLSODA, DSTODA, and DPRJA.
C Groups of variables are replaced by dummy arrays in the Common
C declarations in routines where those variables are not used.
C-----------------------------------------------------------------------
COMMON /DLS001/ ROWNS(209),
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
2 INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS(6),
3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
COMMON /DLSA01/ TSW, ROWNS2(20), PDNORM,
1 INSUFR, INSUFI, IXPR, IOWNS2(2), JTYP, MUSED, MXORDN, MXORDS
C
DATA MORD(1),MORD(2)/12,5/, MXSTP0/500/, MXHNL0/10/
C-----------------------------------------------------------------------
C Block A.
C This code block is executed on every call.
C It tests ISTATE and ITASK for legality and branches appropriately.
C If ISTATE .gt. 1 but the flag INIT shows that initialization has
C not yet been done, an error return occurs.
C If ISTATE = 1 and TOUT = T, return immediately.
C-----------------------------------------------------------------------
IF (ISTATE .LT. 1 .OR. ISTATE .GT. 3) GO TO 601
IF (ITASK .LT. 1 .OR. ITASK .GT. 5) GO TO 602
IF (ISTATE .EQ. 1) GO TO 10
IF (INIT .EQ. 0) GO TO 603
IF (ISTATE .EQ. 2) GO TO 200
GO TO 20
10 INIT = 0
IF (TOUT .EQ. T) RETURN
C-----------------------------------------------------------------------
C Block B.
C The next code block is executed for the initial call (ISTATE = 1),
C or for a continuation call with parameter changes (ISTATE = 3).
C It contains checking of all inputs and various initializations.
C
C First check legality of the non-optional inputs NEQ, ITOL, IOPT,
C JT, ML, and MU.
C-----------------------------------------------------------------------
20 IF (NEQ(1) .LE. 0) GO TO 604
IF (ISTATE .EQ. 1) GO TO 25
IF (NEQ(1) .GT. N) GO TO 605
25 N = NEQ(1)
IF (ITOL .LT. 1 .OR. ITOL .GT. 4) GO TO 606
IF (IOPT .LT. 0 .OR. IOPT .GT. 1) GO TO 607
IF (JT .EQ. 3 .OR. JT .LT. 1 .OR. JT .GT. 5) GO TO 608
JTYP = JT
IF (JT .LE. 2) GO TO 30
ML = IWORK(1)
MU = IWORK(2)
IF (ML .LT. 0 .OR. ML .GE. N) GO TO 609
IF (MU .LT. 0 .OR. MU .GE. N) GO TO 610
30 CONTINUE
C Next process and check the optional inputs. --------------------------
IF (IOPT .EQ. 1) GO TO 40
IXPR = 0
MXSTEP = MXSTP0
MXHNIL = MXHNL0
HMXI = 0.0D0
HMIN = 0.0D0
IF (ISTATE .NE. 1) GO TO 60
H0 = 0.0D0
MXORDN = MORD(1)
MXORDS = MORD(2)
GO TO 60
40 IXPR = IWORK(5)
IF (IXPR .LT. 0 .OR. IXPR .GT. 1) GO TO 611
MXSTEP = IWORK(6)
IF (MXSTEP .LT. 0) GO TO 612
IF (MXSTEP .EQ. 0) MXSTEP = MXSTP0
MXHNIL = IWORK(7)
IF (MXHNIL .LT. 0) GO TO 613
IF (MXHNIL .EQ. 0) MXHNIL = MXHNL0
IF (ISTATE .NE. 1) GO TO 50
H0 = RWORK(5)
MXORDN = IWORK(8)
IF (MXORDN .LT. 0) GO TO 628
IF (MXORDN .EQ. 0) MXORDN = 100
MXORDN = MIN(MXORDN,MORD(1))
MXORDS = IWORK(9)
IF (MXORDS .LT. 0) GO TO 629
IF (MXORDS .EQ. 0) MXORDS = 100
MXORDS = MIN(MXORDS,MORD(2))
IF ((TOUT - T)*H0 .LT. 0.0D0) GO TO 614
50 HMAX = RWORK(6)
IF (HMAX .LT. 0.0D0) GO TO 615
HMXI = 0.0D0
IF (HMAX .GT. 0.0D0) HMXI = 1.0D0/HMAX
HMIN = RWORK(7)
IF (HMIN .LT. 0.0D0) GO TO 616
C-----------------------------------------------------------------------
C Set work array pointers and check lengths LRW and LIW.
C If ISTATE = 1, METH is initialized to 1 here to facilitate the
C checking of work space lengths.
C Pointers to segments of RWORK and IWORK are named by prefixing L to
C the name of the segment. E.g., the segment YH starts at RWORK(LYH).
C Segments of RWORK (in order) are denoted YH, WM, EWT, SAVF, ACOR.
C If the lengths provided are insufficient for the current method,
C an error return occurs. This is treated as illegal input on the
C first call, but as a problem interruption with ISTATE = -7 on a
C continuation call. If the lengths are sufficient for the current
C method but not for both methods, a warning message is sent.
C-----------------------------------------------------------------------
60 IF (ISTATE .EQ. 1) METH = 1
IF (ISTATE .EQ. 1) NYH = N
LYH = 21
LEN1N = 20 + (MXORDN + 1)*NYH
LEN1S = 20 + (MXORDS + 1)*NYH
LWM = LEN1S + 1
IF (JT .LE. 2) LENWM = N*N + 2
IF (JT .GE. 4) LENWM = (2*ML + MU + 1)*N + 2
LEN1S = LEN1S + LENWM
LEN1C = LEN1N
IF (METH .EQ. 2) LEN1C = LEN1S
LEN1 = MAX(LEN1N,LEN1S)
LEN2 = 3*N
LENRW = LEN1 + LEN2
LENRWC = LEN1C + LEN2
IWORK(17) = LENRW
LIWM = 1
LENIW = 20 + N
LENIWC = 20
IF (METH .EQ. 2) LENIWC = LENIW
IWORK(18) = LENIW
IF (ISTATE .EQ. 1 .AND. LRW .LT. LENRWC) GO TO 617
IF (ISTATE .EQ. 1 .AND. LIW .LT. LENIWC) GO TO 618
IF (ISTATE .EQ. 3 .AND. LRW .LT. LENRWC) GO TO 550
IF (ISTATE .EQ. 3 .AND. LIW .LT. LENIWC) GO TO 555
LEWT = LEN1 + 1
INSUFR = 0
IF (LRW .GE. LENRW) GO TO 65
INSUFR = 2
LEWT = LEN1C + 1
MSG='DLSODA- Warning.. RWORK length is sufficient for now, but '
CALL XERRWD (MSG, 60, 103, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' may not be later. Integration will proceed anyway. '
CALL XERRWD (MSG, 60, 103, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' Length needed is LENRW = I1, while LRW = I2.'
CALL XERRWD (MSG, 50, 103, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
65 LSAVF = LEWT + N
LACOR = LSAVF + N
INSUFI = 0
IF (LIW .GE. LENIW) GO TO 70
INSUFI = 2
MSG='DLSODA- Warning.. IWORK length is sufficient for now, but '
CALL XERRWD (MSG, 60, 104, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' may not be later. Integration will proceed anyway. '
CALL XERRWD (MSG, 60, 104, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' Length needed is LENIW = I1, while LIW = I2.'
CALL XERRWD (MSG, 50, 104, 0, 2, LENIW, LIW, 0, 0.0D0, 0.0D0)
70 CONTINUE
C Check RTOL and ATOL for legality. ------------------------------------
RTOLI = RTOL(1)
ATOLI = ATOL(1)
DO 75 I = 1,N
IF (ITOL .GE. 3) RTOLI = RTOL(I)
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
IF (RTOLI .LT. 0.0D0) GO TO 619
IF (ATOLI .LT. 0.0D0) GO TO 620
75 CONTINUE
IF (ISTATE .EQ. 1) GO TO 100
C If ISTATE = 3, set flag to signal parameter changes to DSTODA. -------
JSTART = -1
IF (N .EQ. NYH) GO TO 200
C NEQ was reduced. Zero part of YH to avoid undefined references. -----
I1 = LYH + L*NYH
I2 = LYH + (MAXORD + 1)*NYH - 1
IF (I1 .GT. I2) GO TO 200
DO 95 I = I1,I2
95 RWORK(I) = 0.0D0
GO TO 200
C-----------------------------------------------------------------------
C Block C.
C The next block is for the initial call only (ISTATE = 1).
C It contains all remaining initializations, the initial call to F,
C and the calculation of the initial step size.
C The error weights in EWT are inverted after being loaded.
C-----------------------------------------------------------------------
100 UROUND = DUMACH()
TN = T
TSW = T
MAXORD = MXORDN
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 110
TCRIT = RWORK(1)
IF ((TCRIT - TOUT)*(TOUT - T) .LT. 0.0D0) GO TO 625
IF (H0 .NE. 0.0D0 .AND. (T + H0 - TCRIT)*H0 .GT. 0.0D0)
1 H0 = TCRIT - T
110 JSTART = 0
NHNIL = 0
NST = 0
NJE = 0
NSLAST = 0
HU = 0.0D0
NQU = 0
MUSED = 0
MITER = 0
CCMAX = 0.3D0
MAXCOR = 3
MSBP = 20
MXNCF = 10
C Initial call to F. (LF0 points to YH(*,2).) -------------------------
LF0 = LYH + NYH
CALL F (NEQ, T, Y, RWORK(LF0))
NFE = 1
C Load the initial value vector in YH. ---------------------------------
DO 115 I = 1,N
115 RWORK(I+LYH-1) = Y(I)
C Load and invert the EWT array. (H is temporarily set to 1.0.) -------
NQ = 1
H = 1.0D0
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 120 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 621
120 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
C-----------------------------------------------------------------------
C The coding below computes the step size, H0, to be attempted on the
C first step, unless the user has supplied a value for this.
C First check that TOUT - T differs significantly from zero.
C A scalar tolerance quantity TOL is computed, as MAX(RTOL(i))
C if this is positive, or MAX(ATOL(i)/ABS(Y(i))) otherwise, adjusted
C so as to be between 100*UROUND and 1.0E-3.
C Then the computed value H0 is given by:
C
C H0**(-2) = 1./(TOL * w0**2) + TOL * (norm(F))**2
C
C where w0 = MAX ( ABS(T), ABS(TOUT) ),
C F = the initial value of the vector f(t,y), and
C norm() = the weighted vector norm used throughout, given by
C the DMNORM function routine, and weighted by the
C tolerances initially loaded into the EWT array.
C The sign of H0 is inferred from the initial values of TOUT and T.
C ABS(H0) is made .le. ABS(TOUT-T) in any case.
C-----------------------------------------------------------------------
IF (H0 .NE. 0.0D0) GO TO 180
TDIST = ABS(TOUT - T)
W0 = MAX(ABS(T),ABS(TOUT))
IF (TDIST .LT. 2.0D0*UROUND*W0) GO TO 622
TOL = RTOL(1)
IF (ITOL .LE. 2) GO TO 140
DO 130 I = 1,N
130 TOL = MAX(TOL,RTOL(I))
140 IF (TOL .GT. 0.0D0) GO TO 160
ATOLI = ATOL(1)
DO 150 I = 1,N
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
AYI = ABS(Y(I))
IF (AYI .NE. 0.0D0) TOL = MAX(TOL,ATOLI/AYI)
150 CONTINUE
160 TOL = MAX(TOL,100.0D0*UROUND)
TOL = MIN(TOL,0.001D0)
SUM = DMNORM (N, RWORK(LF0), RWORK(LEWT))
SUM = 1.0D0/(TOL*W0*W0) + TOL*SUM**2
H0 = 1.0D0/SQRT(SUM)
H0 = MIN(H0,TDIST)
H0 = SIGN(H0,TOUT-T)
C Adjust H0 if necessary to meet HMAX bound. ---------------------------
180 RH = ABS(H0)*HMXI
IF (RH .GT. 1.0D0) H0 = H0/RH
C Load H with H0 and scale YH(*,2) by H0. ------------------------------
H = H0
DO 190 I = 1,N
190 RWORK(I+LF0-1) = H0*RWORK(I+LF0-1)
GO TO 270
C-----------------------------------------------------------------------
C Block D.
C The next code block is for continuation calls only (ISTATE = 2 or 3)
C and is to check stop conditions before taking a step.
C-----------------------------------------------------------------------
200 NSLAST = NST
GO TO (210, 250, 220, 230, 240), ITASK
210 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
220 TP = TN - HU*(1.0D0 + 100.0D0*UROUND)
IF ((TP - TOUT)*H .GT. 0.0D0) GO TO 623
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
T = TN
GO TO 400
230 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
IF ((TCRIT - TOUT)*H .LT. 0.0D0) GO TO 625
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 245
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
240 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
245 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) T = TCRIT
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
IF (ISTATE .EQ. 2 .AND. JSTART .GE. 0) JSTART = -2
C-----------------------------------------------------------------------
C Block E.
C The next block is normally executed for all calls and contains
C the call to the one-step core integrator DSTODA.
C
C This is a looping point for the integration steps.
C
C First check for too many steps being taken, update EWT (if not at
C start of problem), check for too much accuracy being requested, and
C check for H below the roundoff level in T.
C-----------------------------------------------------------------------
250 CONTINUE
IF (METH .EQ. MUSED) GO TO 255
IF (INSUFR .EQ. 1) GO TO 550
IF (INSUFI .EQ. 1) GO TO 555
255 IF ((NST-NSLAST) .GE. MXSTEP) GO TO 500
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 260 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 510
260 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
270 TOLSF = UROUND*DMNORM (N, RWORK(LYH), RWORK(LEWT))
IF (TOLSF .LE. 1.0D0) GO TO 280
TOLSF = TOLSF*2.0D0
IF (NST .EQ. 0) GO TO 626
GO TO 520
280 IF ((TN + H) .NE. TN) GO TO 290
NHNIL = NHNIL + 1
IF (NHNIL .GT. MXHNIL) GO TO 290
MSG = 'DLSODA- Warning..Internal T (=R1) and H (=R2) are'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' such that in the machine, T + H = T on the next step '
CALL XERRWD (MSG, 60, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' (H = step size). Solver will continue anyway.'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 2, TN, H)
IF (NHNIL .LT. MXHNIL) GO TO 290
MSG = 'DLSODA- Above warning has been issued I1 times. '
CALL XERRWD (MSG, 50, 102, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' It will not be issued again for this problem.'
CALL XERRWD (MSG, 50, 102, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
290 CONTINUE
C-----------------------------------------------------------------------
C CALL DSTODA(NEQ,Y,YH,NYH,YH,EWT,SAVF,ACOR,WM,IWM,F,JAC,DPRJA,DSOLSY)
C-----------------------------------------------------------------------
CALL DSTODA (NEQ, Y, RWORK(LYH), NYH, RWORK(LYH), RWORK(LEWT),
1 RWORK(LSAVF), RWORK(LACOR), RWORK(LWM), IWORK(LIWM),
2 F, JAC, DPRJA, DSOLSY)
KGO = 1 - KFLAG
GO TO (300, 530, 540), KGO
C-----------------------------------------------------------------------
C Block F.
C The following block handles the case of a successful return from the
C core integrator (KFLAG = 0).
C If a method switch was just made, record TSW, reset MAXORD,
C set JSTART to -1 to signal DSTODA to complete the switch,
C and do extra printing of data if IXPR = 1.
C Then, in any case, check for stop conditions.
C-----------------------------------------------------------------------
300 INIT = 1
IF (METH .EQ. MUSED) GO TO 310
TSW = TN
MAXORD = MXORDN
IF (METH .EQ. 2) MAXORD = MXORDS
IF (METH .EQ. 2) RWORK(LWM) = SQRT(UROUND)
INSUFR = MIN(INSUFR,1)
INSUFI = MIN(INSUFI,1)
JSTART = -1
IF (IXPR .EQ. 0) GO TO 310
IF (METH .EQ. 2) THEN
MSG='DLSODA- A switch to the BDF (stiff) method has occurred '
CALL XERRWD (MSG, 60, 105, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
IF (METH .EQ. 1) THEN
MSG='DLSODA- A switch to the Adams (nonstiff) method has occurred'
CALL XERRWD (MSG, 60, 106, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
MSG=' at T = R1, tentative step size H = R2, step NST = I1 '
CALL XERRWD (MSG, 60, 107, 0, 1, NST, 0, 2, TN, H)
310 GO TO (320, 400, 330, 340, 350), ITASK
C ITASK = 1. If TOUT has been reached, interpolate. -------------------
320 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
C ITASK = 3. Jump to exit if TOUT was reached. ------------------------
330 IF ((TN - TOUT)*H .GE. 0.0D0) GO TO 400
GO TO 250
C ITASK = 4. See if TOUT or TCRIT was reached. Adjust H if necessary.
340 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 345
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
345 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
IF (JSTART .GE. 0) JSTART = -2
GO TO 250
C ITASK = 5. See if TCRIT was reached and jump to exit. ---------------
350 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
C-----------------------------------------------------------------------
C Block G.
C The following block handles all successful returns from DLSODA.
C If ITASK .ne. 1, Y is loaded from YH and T is set accordingly.
C ISTATE is set to 2, and the optional outputs are loaded into the
C work arrays before returning.
C-----------------------------------------------------------------------
400 DO 410 I = 1,N
410 Y(I) = RWORK(I+LYH-1)
T = TN
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 420
IF (IHIT) T = TCRIT
420 ISTATE = 2
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
RWORK(15) = TSW
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(19) = MUSED
IWORK(20) = METH
RETURN
C-----------------------------------------------------------------------
C Block H.
C The following block handles all unsuccessful returns other than
C those for illegal input. First the error message routine is called.
C If there was an error test or convergence test failure, IMXER is set.
C Then Y is loaded from YH and T is set to TN.
C The optional outputs are loaded into the work arrays before returning.
C-----------------------------------------------------------------------
C The maximum number of steps was taken before reaching TOUT. ----------
500 MSG = 'DLSODA- At current T (=R1), MXSTEP (=I1) steps '
CALL XERRWD (MSG, 50, 201, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' taken on this call before reaching TOUT '
CALL XERRWD (MSG, 50, 201, 0, 1, MXSTEP, 0, 1, TN, 0.0D0)
ISTATE = -1
GO TO 580
C EWT(i) .le. 0.0 for some i (not at start of problem). ----------------
510 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODA- At T (=R1), EWT(I1) has become R2 .le. 0.'
CALL XERRWD (MSG, 50, 202, 0, 1, I, 0, 2, TN, EWTI)
ISTATE = -6
GO TO 580
C Too much accuracy requested for machine precision. -------------------
520 MSG = 'DLSODA- At T (=R1), too much accuracy requested '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' for precision of machine.. See TOLSF (=R2) '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 2, TN, TOLSF)
RWORK(14) = TOLSF
ISTATE = -2
GO TO 580
C KFLAG = -1. Error test failed repeatedly or with ABS(H) = HMIN. -----
530 MSG = 'DLSODA- At T(=R1) and step size H(=R2), the error'
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' test failed repeatedly or with ABS(H) = HMIN'
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 2, TN, H)
ISTATE = -4
GO TO 560
C KFLAG = -2. Convergence failed repeatedly or with ABS(H) = HMIN. ----
540 MSG = 'DLSODA- At T (=R1) and step size H (=R2), the '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' corrector convergence failed repeatedly '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' or with ABS(H) = HMIN '
CALL XERRWD (MSG, 30, 205, 0, 0, 0, 0, 2, TN, H)
ISTATE = -5
GO TO 560
C RWORK length too small to proceed. -----------------------------------
550 MSG = 'DLSODA- At current T(=R1), RWORK length too small'
CALL XERRWD (MSG, 50, 206, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' to proceed. The integration was otherwise successful.'
CALL XERRWD (MSG, 60, 206, 0, 0, 0, 0, 1, TN, 0.0D0)
ISTATE = -7
GO TO 580
C IWORK length too small to proceed. -----------------------------------
555 MSG = 'DLSODA- At current T(=R1), IWORK length too small'
CALL XERRWD (MSG, 50, 207, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' to proceed. The integration was otherwise successful.'
CALL XERRWD (MSG, 60, 207, 0, 0, 0, 0, 1, TN, 0.0D0)
ISTATE = -7
GO TO 580
C Compute IMXER if relevant. -------------------------------------------
560 BIG = 0.0D0
IMXER = 1
DO 570 I = 1,N
SIZE = ABS(RWORK(I+LACOR-1)*RWORK(I+LEWT-1))
IF (BIG .GE. SIZE) GO TO 570
BIG = SIZE
IMXER = I
570 CONTINUE
IWORK(16) = IMXER
C Set Y vector, T, and optional outputs. -------------------------------
580 DO 590 I = 1,N
590 Y(I) = RWORK(I+LYH-1)
T = TN
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
RWORK(15) = TSW
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(19) = MUSED
IWORK(20) = METH
RETURN
C-----------------------------------------------------------------------
C Block I.
C The following block handles all error returns due to illegal input
C (ISTATE = -3), as detected before calling the core integrator.
C First the error message routine is called. If the illegal input
C is a negative ISTATE, the run is aborted (apparent infinite loop).
C-----------------------------------------------------------------------
601 MSG = 'DLSODA- ISTATE (=I1) illegal.'
CALL XERRWD (MSG, 30, 1, 0, 1, ISTATE, 0, 0, 0.0D0, 0.0D0)
IF (ISTATE .LT. 0) GO TO 800
GO TO 700
602 MSG = 'DLSODA- ITASK (=I1) illegal. '
CALL XERRWD (MSG, 30, 2, 0, 1, ITASK, 0, 0, 0.0D0, 0.0D0)
GO TO 700
603 MSG = 'DLSODA- ISTATE .gt. 1 but DLSODA not initialized.'
CALL XERRWD (MSG, 50, 3, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
GO TO 700
604 MSG = 'DLSODA- NEQ (=I1) .lt. 1 '
CALL XERRWD (MSG, 30, 4, 0, 1, NEQ(1), 0, 0, 0.0D0, 0.0D0)
GO TO 700
605 MSG = 'DLSODA- ISTATE = 3 and NEQ increased (I1 to I2). '
CALL XERRWD (MSG, 50, 5, 0, 2, N, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
606 MSG = 'DLSODA- ITOL (=I1) illegal. '
CALL XERRWD (MSG, 30, 6, 0, 1, ITOL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
607 MSG = 'DLSODA- IOPT (=I1) illegal. '
CALL XERRWD (MSG, 30, 7, 0, 1, IOPT, 0, 0, 0.0D0, 0.0D0)
GO TO 700
608 MSG = 'DLSODA- JT (=I1) illegal. '
CALL XERRWD (MSG, 30, 8, 0, 1, JT, 0, 0, 0.0D0, 0.0D0)
GO TO 700
609 MSG = 'DLSODA- ML (=I1) illegal: .lt.0 or .ge.NEQ (=I2) '
CALL XERRWD (MSG, 50, 9, 0, 2, ML, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
610 MSG = 'DLSODA- MU (=I1) illegal: .lt.0 or .ge.NEQ (=I2) '
CALL XERRWD (MSG, 50, 10, 0, 2, MU, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
611 MSG = 'DLSODA- IXPR (=I1) illegal. '
CALL XERRWD (MSG, 30, 11, 0, 1, IXPR, 0, 0, 0.0D0, 0.0D0)
GO TO 700
612 MSG = 'DLSODA- MXSTEP (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 12, 0, 1, MXSTEP, 0, 0, 0.0D0, 0.0D0)
GO TO 700
613 MSG = 'DLSODA- MXHNIL (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 13, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
614 MSG = 'DLSODA- TOUT (=R1) behind T (=R2) '
CALL XERRWD (MSG, 40, 14, 0, 0, 0, 0, 2, TOUT, T)
MSG = ' Integration direction is given by H0 (=R1) '
CALL XERRWD (MSG, 50, 14, 0, 0, 0, 0, 1, H0, 0.0D0)
GO TO 700
615 MSG = 'DLSODA- HMAX (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 15, 0, 0, 0, 0, 1, HMAX, 0.0D0)
GO TO 700
616 MSG = 'DLSODA- HMIN (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 16, 0, 0, 0, 0, 1, HMIN, 0.0D0)
GO TO 700
617 MSG='DLSODA- RWORK length needed, LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 17, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
618 MSG='DLSODA- IWORK length needed, LENIW (=I1), exceeds LIW (=I2)'
CALL XERRWD (MSG, 60, 18, 0, 2, LENIW, LIW, 0, 0.0D0, 0.0D0)
GO TO 700
619 MSG = 'DLSODA- RTOL(I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 19, 0, 1, I, 0, 1, RTOLI, 0.0D0)
GO TO 700
620 MSG = 'DLSODA- ATOL(I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 20, 0, 1, I, 0, 1, ATOLI, 0.0D0)
GO TO 700
621 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODA- EWT(I1) is R1 .le. 0.0 '
CALL XERRWD (MSG, 40, 21, 0, 1, I, 0, 1, EWTI, 0.0D0)
GO TO 700
622 MSG='DLSODA- TOUT(=R1) too close to T(=R2) to start integration.'
CALL XERRWD (MSG, 60, 22, 0, 0, 0, 0, 2, TOUT, T)
GO TO 700
623 MSG='DLSODA- ITASK = I1 and TOUT (=R1) behind TCUR - HU (= R2) '
CALL XERRWD (MSG, 60, 23, 0, 1, ITASK, 0, 2, TOUT, TP)
GO TO 700
624 MSG='DLSODA- ITASK = 4 or 5 and TCRIT (=R1) behind TCUR (=R2) '
CALL XERRWD (MSG, 60, 24, 0, 0, 0, 0, 2, TCRIT, TN)
GO TO 700
625 MSG='DLSODA- ITASK = 4 or 5 and TCRIT (=R1) behind TOUT (=R2) '
CALL XERRWD (MSG, 60, 25, 0, 0, 0, 0, 2, TCRIT, TOUT)
GO TO 700
626 MSG = 'DLSODA- At start of problem, too much accuracy '
CALL XERRWD (MSG, 50, 26, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' requested for precision of machine.. See TOLSF (=R1) '
CALL XERRWD (MSG, 60, 26, 0, 0, 0, 0, 1, TOLSF, 0.0D0)
RWORK(14) = TOLSF
GO TO 700
627 MSG = 'DLSODA- Trouble in DINTDY. ITASK = I1, TOUT = R1'
CALL XERRWD (MSG, 50, 27, 0, 1, ITASK, 0, 1, TOUT, 0.0D0)
GO TO 700
628 MSG = 'DLSODA- MXORDN (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 28, 0, 1, MXORDN, 0, 0, 0.0D0, 0.0D0)
GO TO 700
629 MSG = 'DLSODA- MXORDS (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 29, 0, 1, MXORDS, 0, 0, 0.0D0, 0.0D0)
C
700 ISTATE = -3
RETURN
C
800 MSG = 'DLSODA- Run aborted.. apparent infinite loop. '
CALL XERRWD (MSG, 50, 303, 2, 0, 0, 0, 0, 0.0D0, 0.0D0)
RETURN
C----------------------- End of Subroutine DLSODA ----------------------
END
*DECK DLSODAR
SUBROUTINE DLSODAR (F, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
1 ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JAC, JT,
2 G, NG, JROOT)
EXTERNAL F, JAC, G
INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK, LIW, JT,
1 NG, JROOT
DOUBLE PRECISION Y, T, TOUT, RTOL, ATOL, RWORK
DIMENSION NEQ(*), Y(*), RTOL(*), ATOL(*), RWORK(LRW), IWORK(LIW),
1 JROOT(NG)
C-----------------------------------------------------------------------
C This is the 12 November 2003 version of
C DLSODAR: Livermore Solver for Ordinary Differential Equations, with
C Automatic method switching for stiff and nonstiff problems,
C and with Root-finding.
C
C This version is in double precision.
C
C DLSODAR solves the initial value problem for stiff or nonstiff
C systems of first order ODEs,
C dy/dt = f(t,y) , or, in component form,
C dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(NEQ)) (i = 1,...,NEQ).
C At the same time, it locates the roots of any of a set of functions
C g(i) = g(i,t,y(1),...,y(NEQ)) (i = 1,...,ng).
C
C This a variant version of the DLSODE package. It differs from it
C in two ways:
C (a) It switches automatically between stiff and nonstiff methods.
C This means that the user does not have to determine whether the
C problem is stiff or not, and the solver will automatically choose the
C appropriate method. It always starts with the nonstiff method.
C (b) It finds the root of at least one of a set of constraint
C functions g(i) of the independent and dependent variables.
C It finds only those roots for which some g(i), as a function
C of t, changes sign in the interval of integration.
C It then returns the solution at the root, if that occurs
C sooner than the specified stop condition, and otherwise returns
C the solution according the specified stop condition.
C
C Authors: Alan C. Hindmarsh,
C Center for Applied Scientific Computing, L-561
C Lawrence Livermore National Laboratory
C Livermore, CA 94551
C and
C Linda R. Petzold
C Univ. of California at Santa Barbara
C Dept. of Computer Science
C Santa Barbara, CA 93106
C
C References:
C 1. Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE
C Solvers, in Scientific Computing, R. S. Stepleman et al. (Eds.),
C North-Holland, Amsterdam, 1983, pp. 55-64.
C 2. Linda R. Petzold, Automatic Selection of Methods for Solving
C Stiff and Nonstiff Systems of Ordinary Differential Equations,
C Siam J. Sci. Stat. Comput. 4 (1983), pp. 136-148.
C 3. Kathie L. Hiebert and Lawrence F. Shampine, Implicitly Defined
C Output Points for Solutions of ODEs, Sandia Report SAND80-0180,
C February 1980.
C-----------------------------------------------------------------------
C Summary of Usage.
C
C Communication between the user and the DLSODAR package, for normal
C situations, is summarized here. This summary describes only a subset
C of the full set of options available. See the full description for
C details, including alternative treatment of the Jacobian matrix,
C optional inputs and outputs, nonstandard options, and
C instructions for special situations. See also the example
C problem (with program and output) following this summary.
C
C A. First provide a subroutine of the form:
C SUBROUTINE F (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y(*), YDOT(*)
C which supplies the vector function f by loading YDOT(i) with f(i).
C
C B. Provide a subroutine of the form:
C SUBROUTINE G (NEQ, T, Y, NG, GOUT)
C DOUBLE PRECISION T, Y(*), GOUT(NG)
C which supplies the vector function g by loading GOUT(i) with
C g(i), the i-th constraint function whose root is sought.
C
C C. Write a main program which calls Subroutine DLSODAR once for
C each point at which answers are desired. This should also provide
C for possible use of logical unit 6 for output of error messages by
C DLSODAR. On the first call to DLSODAR, supply arguments as follows:
C F = name of subroutine for right-hand side vector f.
C This name must be declared External in calling program.
C NEQ = number of first order ODEs.
C Y = array of initial values, of length NEQ.
C T = the initial value of the independent variable.
C TOUT = first point where output is desired (.ne. T).
C ITOL = 1 or 2 according as ATOL (below) is a scalar or array.
C RTOL = relative tolerance parameter (scalar).
C ATOL = absolute tolerance parameter (scalar or array).
C the estimated local error in y(i) will be controlled so as
C to be less than
C EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
C EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
C Thus the local error test passes if, in each component,
C either the absolute error is less than ATOL (or ATOL(i)),
C or the relative error is less than RTOL.
C Use RTOL = 0.0 for pure absolute error control, and
C use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative error
C control. Caution: actual (global) errors may exceed these
C local tolerances, so choose them conservatively.
C ITASK = 1 for normal computation of output values of y at t = TOUT.
C ISTATE = integer flag (input and output). Set ISTATE = 1.
C IOPT = 0 to indicate no optional inputs used.
C RWORK = real work array of length at least:
C 22 + NEQ * MAX(16, NEQ + 9) + 3*NG.
C See also Paragraph F below.
C LRW = declared length of RWORK (in user's dimension).
C IWORK = integer work array of length at least 20 + NEQ.
C LIW = declared length of IWORK (in user's dimension).
C JAC = name of subroutine for Jacobian matrix.
C Use a dummy name. See also Paragraph F below.
C JT = Jacobian type indicator. Set JT = 2.
C See also Paragraph F below.
C G = name of subroutine for constraint functions, whose
C roots are desired during the integration.
C This name must be declared External in calling program.
C NG = number of constraint functions g(i). If there are none,
C set NG = 0, and pass a dummy name for G.
C JROOT = integer array of length NG for output of root information.
C See next paragraph.
C Note that the main program must declare arrays Y, RWORK, IWORK,
C JROOT, and possibly ATOL.
C
C D. The output from the first call (or any call) is:
C Y = array of computed values of y(t) vector.
C T = corresponding value of independent variable. This is
C TOUT if ISTATE = 2, or the root location if ISTATE = 3,
C or the farthest point reached if DLSODAR was unsuccessful.
C ISTATE = 2 or 3 if DLSODAR was successful, negative otherwise.
C 2 means no root was found, and TOUT was reached as desired.
C 3 means a root was found prior to reaching TOUT.
C -1 means excess work done on this call (perhaps wrong JT).
C -2 means excess accuracy requested (tolerances too small).
C -3 means illegal input detected (see printed message).
C -4 means repeated error test failures (check all inputs).
C -5 means repeated convergence failures (perhaps bad Jacobian
C supplied or wrong choice of JT or tolerances).
C -6 means error weight became zero during problem. (Solution
C component i vanished, and ATOL or ATOL(i) = 0.)
C -7 means work space insufficient to finish (see messages).
C JROOT = array showing roots found if ISTATE = 3 on return.
C JROOT(i) = 1 if g(i) has a root at t, or 0 otherwise.
C
C E. To continue the integration after a successful return, proceed
C as follows:
C (a) If ISTATE = 2 on return, reset TOUT and call DLSODAR again.
C (b) If ISTATE = 3 on return, reset ISTATE to 2, call DLSODAR again.
C In either case, no other parameters need be reset.
C
C F. Note: If and when DLSODAR regards the problem as stiff, and
C switches methods accordingly, it must make use of the NEQ by NEQ
C Jacobian matrix, J = df/dy. For the sake of simplicity, the
C inputs to DLSODAR recommended in Paragraph C above cause DLSODAR to
C treat J as a full matrix, and to approximate it internally by
C difference quotients. Alternatively, J can be treated as a band
C matrix (with great potential reduction in the size of the RWORK
C array). Also, in either the full or banded case, the user can supply
C J in closed form, with a routine whose name is passed as the JAC
C argument. These alternatives are described in the paragraphs on
C RWORK, JAC, and JT in the full description of the call sequence below.
C
C-----------------------------------------------------------------------
C Example Problem.
C
C The following is a simple example problem, with the coding
C needed for its solution by DLSODAR. The problem is from chemical
C kinetics, and consists of the following three rate equations:
C dy1/dt = -.04*y1 + 1.e4*y2*y3
C dy2/dt = .04*y1 - 1.e4*y2*y3 - 3.e7*y2**2
C dy3/dt = 3.e7*y2**2
C on the interval from t = 0.0 to t = 4.e10, with initial conditions
C y1 = 1.0, y2 = y3 = 0. The problem is stiff.
C In addition, we want to find the values of t, y1, y2, and y3 at which
C (1) y1 reaches the value 1.e-4, and
C (2) y3 reaches the value 1.e-2.
C
C The following coding solves this problem with DLSODAR,
C printing results at t = .4, 4., ..., 4.e10, and at the computed
C roots. It uses ITOL = 2 and ATOL much smaller for y2 than y1 or y3
C because y2 has much smaller values.
C At the end of the run, statistical quantities of interest are
C printed (see optional outputs in the full description below).
C
C EXTERNAL FEX, GEX
C DOUBLE PRECISION ATOL, RTOL, RWORK, T, TOUT, Y
C DIMENSION Y(3), ATOL(3), RWORK(76), IWORK(23), JROOT(2)
C NEQ = 3
C Y(1) = 1.
C Y(2) = 0.
C Y(3) = 0.
C T = 0.
C TOUT = .4
C ITOL = 2
C RTOL = 1.D-4
C ATOL(1) = 1.D-6
C ATOL(2) = 1.D-10
C ATOL(3) = 1.D-6
C ITASK = 1
C ISTATE = 1
C IOPT = 0
C LRW = 76
C LIW = 23
C JT = 2
C NG = 2
C DO 40 IOUT = 1,12
C 10 CALL DLSODAR(FEX,NEQ,Y,T,TOUT,ITOL,RTOL,ATOL,ITASK,ISTATE,
C 1 IOPT,RWORK,LRW,IWORK,LIW,JDUM,JT,GEX,NG,JROOT)
C WRITE(6,20)T,Y(1),Y(2),Y(3)
C 20 FORMAT(' At t =',D12.4,' Y =',3D14.6)
C IF (ISTATE .LT. 0) GO TO 80
C IF (ISTATE .EQ. 2) GO TO 40
C WRITE(6,30)JROOT(1),JROOT(2)
C 30 FORMAT(5X,' The above line is a root, JROOT =',2I5)
C ISTATE = 2
C GO TO 10
C 40 TOUT = TOUT*10.
C WRITE(6,60)IWORK(11),IWORK(12),IWORK(13),IWORK(10),
C 1 IWORK(19),RWORK(15)
C 60 FORMAT(/' No. steps =',I4,' No. f-s =',I4,' No. J-s =',I4,
C 1 ' No. g-s =',I4/
C 2 ' Method last used =',I2,' Last switch was at t =',D12.4)
C STOP
C 80 WRITE(6,90)ISTATE
C 90 FORMAT(///' Error halt.. ISTATE =',I3)
C STOP
C END
C
C SUBROUTINE FEX (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y, YDOT
C DIMENSION Y(3), YDOT(3)
C YDOT(1) = -.04*Y(1) + 1.D4*Y(2)*Y(3)
C YDOT(3) = 3.D7*Y(2)*Y(2)
C YDOT(2) = -YDOT(1) - YDOT(3)
C RETURN
C END
C
C SUBROUTINE GEX (NEQ, T, Y, NG, GOUT)
C DOUBLE PRECISION T, Y, GOUT
C DIMENSION Y(3), GOUT(2)
C GOUT(1) = Y(1) - 1.D-4
C GOUT(2) = Y(3) - 1.D-2
C RETURN
C END
C
C The output of this program (on a CDC-7600 in single precision)
C is as follows:
C
C At t = 2.6400e-01 y = 9.899653e-01 3.470563e-05 1.000000e-02
C The above line is a root, JROOT = 0 1
C At t = 4.0000e-01 Y = 9.851712e-01 3.386380e-05 1.479493e-02
C At t = 4.0000e+00 Y = 9.055333e-01 2.240655e-05 9.444430e-02
C At t = 4.0000e+01 Y = 7.158403e-01 9.186334e-06 2.841505e-01
C At t = 4.0000e+02 Y = 4.505250e-01 3.222964e-06 5.494717e-01
C At t = 4.0000e+03 Y = 1.831975e-01 8.941774e-07 8.168016e-01
C At t = 4.0000e+04 Y = 3.898730e-02 1.621940e-07 9.610125e-01
C At t = 4.0000e+05 Y = 4.936363e-03 1.984221e-08 9.950636e-01
C At t = 4.0000e+06 Y = 5.161831e-04 2.065786e-09 9.994838e-01
C At t = 2.0745e+07 Y = 1.000000e-04 4.000395e-10 9.999000e-01
C The above line is a root, JROOT = 1 0
C At t = 4.0000e+07 Y = 5.179817e-05 2.072032e-10 9.999482e-01
C At t = 4.0000e+08 Y = 5.283401e-06 2.113371e-11 9.999947e-01
C At t = 4.0000e+09 Y = 4.659031e-07 1.863613e-12 9.999995e-01
C At t = 4.0000e+10 Y = 1.404280e-08 5.617126e-14 1.000000e+00
C
C No. steps = 361 No. f-s = 693 No. J-s = 64 No. g-s = 390
C Method last used = 2 Last switch was at t = 6.0092e-03
C
C-----------------------------------------------------------------------
C Full Description of User Interface to DLSODAR.
C
C The user interface to DLSODAR consists of the following parts.
C
C 1. The call sequence to Subroutine DLSODAR, which is a driver
C routine for the solver. This includes descriptions of both
C the call sequence arguments and of user-supplied routines.
C Following these descriptions is a description of
C optional inputs available through the call sequence, and then
C a description of optional outputs (in the work arrays).
C
C 2. Descriptions of other routines in the DLSODAR package that may be
C (optionally) called by the user. These provide the ability to
C alter error message handling, save and restore the internal
C Common, and obtain specified derivatives of the solution y(t).
C
C 3. Descriptions of Common blocks to be declared in overlay
C or similar environments, or to be saved when doing an interrupt
C of the problem and continued solution later.
C
C 4. Description of a subroutine in the DLSODAR package,
C which the user may replace with his/her own version, if desired.
C this relates to the measurement of errors.
C
C-----------------------------------------------------------------------
C Part 1. Call Sequence.
C
C The call sequence parameters used for input only are
C F, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK, IOPT, LRW, LIW, JAC,
C JT, G, and NG,
C that used only for output is JROOT,
C and those used for both input and output are
C Y, T, ISTATE.
C The work arrays RWORK and IWORK are also used for conditional and
C optional inputs and optional outputs. (The term output here refers
C to the return from Subroutine DLSODAR to the user's calling program.)
C
C The legality of input parameters will be thoroughly checked on the
C initial call for the problem, but not checked thereafter unless a
C change in input parameters is flagged by ISTATE = 3 on input.
C
C The descriptions of the call arguments are as follows.
C
C F = the name of the user-supplied subroutine defining the
C ODE system. The system must be put in the first-order
C form dy/dt = f(t,y), where f is a vector-valued function
C of the scalar t and the vector y. Subroutine F is to
C compute the function f. It is to have the form
C SUBROUTINE F (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y(*), YDOT(*)
C where NEQ, T, and Y are input, and the array YDOT = f(t,y)
C is output. Y and YDOT are arrays of length NEQ.
C Subroutine F should not alter Y(1),...,Y(NEQ).
C F must be declared External in the calling program.
C
C Subroutine F may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in F) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y below.
C
C If quantities computed in the F routine are needed
C externally to DLSODAR, an extra call to F should be made
C for this purpose, for consistent and accurate results.
C If only the derivative dy/dt is needed, use DINTDY instead.
C
C NEQ = the size of the ODE system (number of first order
C ordinary differential equations). Used only for input.
C NEQ may be decreased, but not increased, during the problem.
C If NEQ is decreased (with ISTATE = 3 on input), the
C remaining components of Y should be left undisturbed, if
C these are to be accessed in F and/or JAC.
C
C Normally, NEQ is a scalar, and it is generally referred to
C as a scalar in this user interface description. However,
C NEQ may be an array, with NEQ(1) set to the system size.
C (The DLSODAR package accesses only NEQ(1).) In either case,
C this parameter is passed as the NEQ argument in all calls
C to F, JAC, and G. Hence, if it is an array, locations
C NEQ(2),... may be used to store other integer data and pass
C it to F, JAC, and G. Each such subroutine must include
C NEQ in a Dimension statement in that case.
C
C Y = a real array for the vector of dependent variables, of
C length NEQ or more. Used for both input and output on the
C first call (ISTATE = 1), and only for output on other calls.
C On the first call, Y must contain the vector of initial
C values. On output, Y contains the computed solution vector,
C evaluated at T. If desired, the Y array may be used
C for other purposes between calls to the solver.
C
C This array is passed as the Y argument in all calls to F,
C JAC, and G. Hence its length may exceed NEQ, and locations
C Y(NEQ+1),... may be used to store other real data and
C pass it to F, JAC, and G. (The DLSODAR package accesses only
C Y(1),...,Y(NEQ).)
C
C T = the independent variable. On input, T is used only on the
C first call, as the initial point of the integration.
C On output, after each call, T is the value at which a
C computed solution y is evaluated (usually the same as TOUT).
C If a root was found, T is the computed location of the
C root reached first, on output.
C On an error return, T is the farthest point reached.
C
C TOUT = the next value of t at which a computed solution is desired.
C Used only for input.
C
C When starting the problem (ISTATE = 1), TOUT may be equal
C to T for one call, then should .ne. T for the next call.
C For the initial T, an input value of TOUT .ne. T is used
C in order to determine the direction of the integration
C (i.e. the algebraic sign of the step sizes) and the rough
C scale of the problem. Integration in either direction
C (forward or backward in t) is permitted.
C
C If ITASK = 2 or 5 (one-step modes), TOUT is ignored after
C the first call (i.e. the first call with TOUT .ne. T).
C Otherwise, TOUT is required on every call.
C
C If ITASK = 1, 3, or 4, the values of TOUT need not be
C monotone, but a value of TOUT which backs up is limited
C to the current internal T interval, whose endpoints are
C TCUR - HU and TCUR (see optional outputs, below, for
C TCUR and HU).
C
C ITOL = an indicator for the type of error control. See
C description below under ATOL. Used only for input.
C
C RTOL = a relative error tolerance parameter, either a scalar or
C an array of length NEQ. See description below under ATOL.
C Input only.
C
C ATOL = an absolute error tolerance parameter, either a scalar or
C an array of length NEQ. Input only.
C
C The input parameters ITOL, RTOL, and ATOL determine
C the error control performed by the solver. The solver will
C control the vector E = (E(i)) of estimated local errors
C in y, according to an inequality of the form
C max-norm of ( E(i)/EWT(i) ) .le. 1,
C where EWT = (EWT(i)) is a vector of positive error weights.
C The values of RTOL and ATOL should all be non-negative.
C The following table gives the types (scalar/array) of
C RTOL and ATOL, and the corresponding form of EWT(i).
C
C ITOL RTOL ATOL EWT(i)
C 1 scalar scalar RTOL*ABS(Y(i)) + ATOL
C 2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
C 3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
C 4 array array RTOL(i)*ABS(Y(i)) + ATOL(i)
C
C When either of these parameters is a scalar, it need not
C be dimensioned in the user's calling program.
C
C If none of the above choices (with ITOL, RTOL, and ATOL
C fixed throughout the problem) is suitable, more general
C error controls can be obtained by substituting a
C user-supplied routine for the setting of EWT.
C See Part 4 below.
C
C If global errors are to be estimated by making a repeated
C run on the same problem with smaller tolerances, then all
C components of RTOL and ATOL (i.e. of EWT) should be scaled
C down uniformly.
C
C ITASK = an index specifying the task to be performed.
C input only. ITASK has the following values and meanings.
C 1 means normal computation of output values of y(t) at
C t = TOUT (by overshooting and interpolating).
C 2 means take one step only and return.
C 3 means stop at the first internal mesh point at or
C beyond t = TOUT and return.
C 4 means normal computation of output values of y(t) at
C t = TOUT but without overshooting t = TCRIT.
C TCRIT must be input as RWORK(1). TCRIT may be equal to
C or beyond TOUT, but not behind it in the direction of
C integration. This option is useful if the problem
C has a singularity at or beyond t = TCRIT.
C 5 means take one step, without passing TCRIT, and return.
C TCRIT must be input as RWORK(1).
C
C Note: If ITASK = 4 or 5 and the solver reaches TCRIT
C (within roundoff), it will return T = TCRIT (exactly) to
C indicate this (unless ITASK = 4 and TOUT comes before TCRIT,
C in which case answers at t = TOUT are returned first).
C
C ISTATE = an index used for input and output to specify the
C the state of the calculation.
C
C On input, the values of ISTATE are as follows.
C 1 means this is the first call for the problem
C (initializations will be done). See note below.
C 2 means this is not the first call, and the calculation
C is to continue normally, with no change in any input
C parameters except possibly TOUT and ITASK.
C (If ITOL, RTOL, and/or ATOL are changed between calls
C with ISTATE = 2, the new values will be used but not
C tested for legality.)
C 3 means this is not the first call, and the
C calculation is to continue normally, but with
C a change in input parameters other than
C TOUT and ITASK. Changes are allowed in
C NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, JT, ML, MU,
C and any optional inputs except H0, MXORDN, and MXORDS.
C (See IWORK description for ML and MU.)
C In addition, immediately following a return with
C ISTATE = 3 (root found), NG and G may be changed.
C (But changing NG from 0 to .gt. 0 is not allowed.)
C Note: A preliminary call with TOUT = T is not counted
C as a first call here, as no initialization or checking of
C input is done. (Such a call is sometimes useful for the
C purpose of outputting the initial conditions.)
C Thus the first call for which TOUT .ne. T requires
C ISTATE = 1 on input.
C
C On output, ISTATE has the following values and meanings.
C 1 means nothing was done; TOUT = t and ISTATE = 1 on input.
C 2 means the integration was performed successfully, and
C no roots were found.
C 3 means the integration was successful, and one or more
C roots were found before satisfying the stop condition
C specified by ITASK. See JROOT.
C -1 means an excessive amount of work (more than MXSTEP
C steps) was done on this call, before completing the
C requested task, but the integration was otherwise
C successful as far as T. (MXSTEP is an optional input
C and is normally 500.) To continue, the user may
C simply reset ISTATE to a value .gt. 1 and call again
C (the excess work step counter will be reset to 0).
C In addition, the user may increase MXSTEP to avoid
C this error return (see below on optional inputs).
C -2 means too much accuracy was requested for the precision
C of the machine being used. This was detected before
C completing the requested task, but the integration
C was successful as far as T. To continue, the tolerance
C parameters must be reset, and ISTATE must be set
C to 3. The optional output TOLSF may be used for this
C purpose. (Note: If this condition is detected before
C taking any steps, then an illegal input return
C (ISTATE = -3) occurs instead.)
C -3 means illegal input was detected, before taking any
C integration steps. See written message for details.
C Note: If the solver detects an infinite loop of calls
C to the solver with illegal input, it will cause
C the run to stop.
C -4 means there were repeated error test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C The problem may have a singularity, or the input
C may be inappropriate.
C -5 means there were repeated convergence test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C This may be caused by an inaccurate Jacobian matrix,
C if one is being used.
C -6 means EWT(i) became zero for some i during the
C integration. Pure relative error control (ATOL(i)=0.0)
C was requested on a variable which has now vanished.
C The integration was successful as far as T.
C -7 means the length of RWORK and/or IWORK was too small to
C proceed, but the integration was successful as far as T.
C This happens when DLSODAR chooses to switch methods
C but LRW and/or LIW is too small for the new method.
C
C Note: Since the normal output value of ISTATE is 2,
C it does not need to be reset for normal continuation.
C Also, since a negative input value of ISTATE will be
C regarded as illegal, a negative output value requires the
C user to change it, and possibly other inputs, before
C calling the solver again.
C
C IOPT = an integer flag to specify whether or not any optional
C inputs are being used on this call. Input only.
C The optional inputs are listed separately below.
C IOPT = 0 means no optional inputs are being used.
C Default values will be used in all cases.
C IOPT = 1 means one or more optional inputs are being used.
C
C RWORK = a real array (double precision) for work space, and (in the
C first 20 words) for conditional and optional inputs and
C optional outputs.
C As DLSODAR switches automatically between stiff and nonstiff
C methods, the required length of RWORK can change during the
C problem. Thus the RWORK array passed to DLSODAR can either
C have a static (fixed) length large enough for both methods,
C or have a dynamic (changing) length altered by the calling
C program in response to output from DLSODAR.
C
C --- Fixed Length Case ---
C If the RWORK length is to be fixed, it should be at least
C max (LRN, LRS),
C where LRN and LRS are the RWORK lengths required when the
C current method is nonstiff or stiff, respectively.
C
C The separate RWORK length requirements LRN and LRS are
C as follows:
C If NEQ is constant and the maximum method orders have
C their default values, then
C LRN = 20 + 16*NEQ + 3*NG,
C LRS = 22 + 9*NEQ + NEQ**2 + 3*NG (JT = 1 or 2),
C LRS = 22 + 10*NEQ + (2*ML+MU)*NEQ + 3*NG (JT = 4 or 5).
C Under any other conditions, LRN and LRS are given by:
C LRN = 20 + NYH*(MXORDN+1) + 3*NEQ + 3*NG,
C LRS = 20 + NYH*(MXORDS+1) + 3*NEQ + LMAT + 3*NG,
C where
C NYH = the initial value of NEQ,
C MXORDN = 12, unless a smaller value is given as an
C optional input,
C MXORDS = 5, unless a smaller value is given as an
C optional input,
C LMAT = length of matrix work space:
C LMAT = NEQ**2 + 2 if JT = 1 or 2,
C LMAT = (2*ML + MU + 1)*NEQ + 2 if JT = 4 or 5.
C
C --- Dynamic Length Case ---
C If the length of RWORK is to be dynamic, then it should
C be at least LRN or LRS, as defined above, depending on the
C current method. Initially, it must be at least LRN (since
C DLSODAR starts with the nonstiff method). On any return
C from DLSODAR, the optional output MCUR indicates the current
C method. If MCUR differs from the value it had on the
C previous return, or if there has only been one call to
C DLSODAR and MCUR is now 2, then DLSODAR has switched
C methods during the last call, and the length of RWORK
C should be reset (to LRN if MCUR = 1, or to LRS if
C MCUR = 2). (An increase in the RWORK length is required
C if DLSODAR returned ISTATE = -7, but not otherwise.)
C After resetting the length, call DLSODAR with ISTATE = 3
C to signal that change.
C
C LRW = the length of the array RWORK, as declared by the user.
C (This will be checked by the solver.)
C
C IWORK = an integer array for work space.
C As DLSODAR switches automatically between stiff and nonstiff
C methods, the required length of IWORK can change during
C problem, between
C LIS = 20 + NEQ and LIN = 20,
C respectively. Thus the IWORK array passed to DLSODAR can
C either have a fixed length of at least 20 + NEQ, or have a
C dynamic length of at least LIN or LIS, depending on the
C current method. The comments on dynamic length under
C RWORK above apply here. Initially, this length need
C only be at least LIN = 20.
C
C The first few words of IWORK are used for conditional and
C optional inputs and optional outputs.
C
C The following 2 words in IWORK are conditional inputs:
C IWORK(1) = ML These are the lower and upper
C IWORK(2) = MU half-bandwidths, respectively, of the
C banded Jacobian, excluding the main diagonal.
C The band is defined by the matrix locations
C (i,j) with i-ML .le. j .le. i+MU. ML and MU
C must satisfy 0 .le. ML,MU .le. NEQ-1.
C These are required if JT is 4 or 5, and
C ignored otherwise. ML and MU may in fact be
C the band parameters for a matrix to which
C df/dy is only approximately equal.
C
C LIW = the length of the array IWORK, as declared by the user.
C (This will be checked by the solver.)
C
C Note: The base addresses of the work arrays must not be
C altered between calls to DLSODAR for the same problem.
C The contents of the work arrays must not be altered
C between calls, except possibly for the conditional and
C optional inputs, and except for the last 3*NEQ words of RWORK.
C The latter space is used for internal scratch space, and so is
C available for use by the user outside DLSODAR between calls, if
C desired (but not for use by F, JAC, or G).
C
C JAC = the name of the user-supplied routine to compute the
C Jacobian matrix, df/dy, if JT = 1 or 4. The JAC routine
C is optional, but if the problem is expected to be stiff much
C of the time, you are encouraged to supply JAC, for the sake
C of efficiency. (Alternatively, set JT = 2 or 5 to have
C DLSODAR compute df/dy internally by difference quotients.)
C If and when DLSODAR uses df/dy, it treats this NEQ by NEQ
C matrix either as full (JT = 1 or 2), or as banded (JT =
C 4 or 5) with half-bandwidths ML and MU (discussed under
C IWORK above). In either case, if JT = 1 or 4, the JAC
C routine must compute df/dy as a function of the scalar t
C and the vector y. It is to have the form
C SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD)
C DOUBLE PRECISION T, Y(*), PD(NROWPD,*)
C where NEQ, T, Y, ML, MU, and NROWPD are input and the array
C PD is to be loaded with partial derivatives (elements of
C the Jacobian matrix) on output. PD must be given a first
C dimension of NROWPD. T and Y have the same meaning as in
C Subroutine F.
C In the full matrix case (JT = 1), ML and MU are
C ignored, and the Jacobian is to be loaded into PD in
C columnwise manner, with df(i)/dy(j) loaded into pd(i,j).
C In the band matrix case (JT = 4), the elements
C within the band are to be loaded into PD in columnwise
C manner, with diagonal lines of df/dy loaded into the rows
C of PD. Thus df(i)/dy(j) is to be loaded into PD(i-j+MU+1,j).
C ML and MU are the half-bandwidth parameters (see IWORK).
C The locations in PD in the two triangular areas which
C correspond to nonexistent matrix elements can be ignored
C or loaded arbitrarily, as they are overwritten by DLSODAR.
C JAC need not provide df/dy exactly. A crude
C approximation (possibly with a smaller bandwidth) will do.
C In either case, PD is preset to zero by the solver,
C so that only the nonzero elements need be loaded by JAC.
C Each call to JAC is preceded by a call to F with the same
C arguments NEQ, T, and Y. Thus to gain some efficiency,
C intermediate quantities shared by both calculations may be
C saved in a user Common block by F and not recomputed by JAC,
C if desired. Also, JAC may alter the Y array, if desired.
C JAC must be declared External in the calling program.
C Subroutine JAC may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in JAC) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y above.
C
C JT = Jacobian type indicator. Used only for input.
C JT specifies how the Jacobian matrix df/dy will be
C treated, if and when DLSODAR requires this matrix.
C JT has the following values and meanings:
C 1 means a user-supplied full (NEQ by NEQ) Jacobian.
C 2 means an internally generated (difference quotient) full
C Jacobian (using NEQ extra calls to F per df/dy value).
C 4 means a user-supplied banded Jacobian.
C 5 means an internally generated banded Jacobian (using
C ML+MU+1 extra calls to F per df/dy evaluation).
C If JT = 1 or 4, the user must supply a Subroutine JAC
C (the name is arbitrary) as described above under JAC.
C If JT = 2 or 5, a dummy argument can be used.
C
C G = the name of subroutine for constraint functions, whose
C roots are desired during the integration. It is to have
C the form
C SUBROUTINE G (NEQ, T, Y, NG, GOUT)
C DOUBLE PRECISION T, Y(*), GOUT(NG)
C where NEQ, T, Y, and NG are input, and the array GOUT
C is output. NEQ, T, and Y have the same meaning as in
C the F routine, and GOUT is an array of length NG.
C For i = 1,...,NG, this routine is to load into GOUT(i)
C the value at (T,Y) of the i-th constraint function g(i).
C DLSODAR will find roots of the g(i) of odd multiplicity
C (i.e. sign changes) as they occur during the integration.
C G must be declared External in the calling program.
C
C Caution: Because of numerical errors in the functions
C g(i) due to roundoff and integration error, DLSODAR may
C return false roots, or return the same root at two or more
C nearly equal values of t. If such false roots are
C suspected, the user should consider smaller error tolerances
C and/or higher precision in the evaluation of the g(i).
C
C If a root of some g(i) defines the end of the problem,
C the input to DLSODAR should nevertheless allow integration
C to a point slightly past that root, so that DLSODAR can
C locate the root by interpolation.
C
C Subroutine G may access user-defined quantities in
C NEQ(2),... and Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in G) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y above.
C
C NG = number of constraint functions g(i). If there are none,
C set NG = 0, and pass a dummy name for G.
C
C JROOT = integer array of length NG. Used only for output.
C On a return with ISTATE = 3 (one or more roots found),
C JROOT(i) = 1 if g(i) has a root at T, or JROOT(i) = 0 if not.
C-----------------------------------------------------------------------
C Optional Inputs.
C
C The following is a list of the optional inputs provided for in the
C call sequence. (See also Part 2.) For each such input variable,
C this table lists its name as used in this documentation, its
C location in the call sequence, its meaning, and the default value.
C The use of any of these inputs requires IOPT = 1, and in that
C case all of these inputs are examined. A value of zero for any
C of these optional inputs will cause the default value to be used.
C Thus to use a subset of the optional inputs, simply preload
C locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively, and
C then set those of interest to nonzero values.
C
C Name Location Meaning and Default Value
C
C H0 RWORK(5) the step size to be attempted on the first step.
C The default value is determined by the solver.
C
C HMAX RWORK(6) the maximum absolute step size allowed.
C The default value is infinite.
C
C HMIN RWORK(7) the minimum absolute step size allowed.
C The default value is 0. (This lower bound is not
C enforced on the final step before reaching TCRIT
C when ITASK = 4 or 5.)
C
C IXPR IWORK(5) flag to generate extra printing at method switches.
C IXPR = 0 means no extra printing (the default).
C IXPR = 1 means print data on each switch.
C T, H, and NST will be printed on the same logical
C unit as used for error messages.
C
C MXSTEP IWORK(6) maximum number of (internally defined) steps
C allowed during one call to the solver.
C The default value is 500.
C
C MXHNIL IWORK(7) maximum number of messages printed (per problem)
C warning that T + H = T on a step (H = step size).
C This must be positive to result in a non-default
C value. The default value is 10.
C
C MXORDN IWORK(8) the maximum order to be allowed for the nonstiff
C (Adams) method. The default value is 12.
C If MXORDN exceeds the default value, it will
C be reduced to the default value.
C MXORDN is held constant during the problem.
C
C MXORDS IWORK(9) the maximum order to be allowed for the stiff
C (BDF) method. The default value is 5.
C If MXORDS exceeds the default value, it will
C be reduced to the default value.
C MXORDS is held constant during the problem.
C-----------------------------------------------------------------------
C Optional Outputs.
C
C As optional additional output from DLSODAR, the variables listed
C below are quantities related to the performance of DLSODAR
C which are available to the user. These are communicated by way of
C the work arrays, but also have internal mnemonic names as shown.
C Except where stated otherwise, all of these outputs are defined
C on any successful return from DLSODAR, and on any return with
C ISTATE = -1, -2, -4, -5, or -6. On an illegal input return
C (ISTATE = -3), they will be unchanged from their existing values
C (if any), except possibly for TOLSF, LENRW, and LENIW.
C On any error return, outputs relevant to the error will be defined,
C as noted below.
C
C Name Location Meaning
C
C HU RWORK(11) the step size in t last used (successfully).
C
C HCUR RWORK(12) the step size to be attempted on the next step.
C
C TCUR RWORK(13) the current value of the independent variable
C which the solver has actually reached, i.e. the
C current internal mesh point in t. On output, TCUR
C will always be at least as far as the argument
C T, but may be farther (if interpolation was done).
C
C TOLSF RWORK(14) a tolerance scale factor, greater than 1.0,
C computed when a request for too much accuracy was
C detected (ISTATE = -3 if detected at the start of
C the problem, ISTATE = -2 otherwise). If ITOL is
C left unaltered but RTOL and ATOL are uniformly
C scaled up by a factor of TOLSF for the next call,
C then the solver is deemed likely to succeed.
C (The user may also ignore TOLSF and alter the
C tolerance parameters in any other way appropriate.)
C
C TSW RWORK(15) the value of t at the time of the last method
C switch, if any.
C
C NGE IWORK(10) the number of g evaluations for the problem so far.
C
C NST IWORK(11) the number of steps taken for the problem so far.
C
C NFE IWORK(12) the number of f evaluations for the problem so far.
C
C NJE IWORK(13) the number of Jacobian evaluations (and of matrix
C LU decompositions) for the problem so far.
C
C NQU IWORK(14) the method order last used (successfully).
C
C NQCUR IWORK(15) the order to be attempted on the next step.
C
C IMXER IWORK(16) the index of the component of largest magnitude in
C the weighted local error vector ( E(i)/EWT(i) ),
C on an error return with ISTATE = -4 or -5.
C
C LENRW IWORK(17) the length of RWORK actually required, assuming
C that the length of RWORK is to be fixed for the
C rest of the problem, and that switching may occur.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C LENIW IWORK(18) the length of IWORK actually required, assuming
C that the length of IWORK is to be fixed for the
C rest of the problem, and that switching may occur.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C MUSED IWORK(19) the method indicator for the last successful step:
C 1 means Adams (nonstiff), 2 means BDF (stiff).
C
C MCUR IWORK(20) the current method indicator:
C 1 means Adams (nonstiff), 2 means BDF (stiff).
C This is the method to be attempted
C on the next step. Thus it differs from MUSED
C only if a method switch has just been made.
C
C The following two arrays are segments of the RWORK array which
C may also be of interest to the user as optional outputs.
C For each array, the table below gives its internal name,
C its base address in RWORK, and its description.
C
C Name Base Address Description
C
C YH 21 + 3*NG the Nordsieck history array, of size NYH by
C (NQCUR + 1), where NYH is the initial value
C of NEQ. For j = 0,1,...,NQCUR, column j+1
C of YH contains HCUR**j/factorial(j) times
C the j-th derivative of the interpolating
C polynomial currently representing the solution,
C evaluated at t = TCUR.
C
C ACOR LACOR array of size NEQ used for the accumulated
C (from Common corrections on each step, scaled on output
C as noted) to represent the estimated local error in y
C on the last step. This is the vector E in
C the description of the error control. It is
C defined only on a successful return from
C DLSODAR. The base address LACOR is obtained by
C including in the user's program the
C following 2 lines:
C COMMON /DLS001/ RLS(218), ILS(37)
C LACOR = ILS(22)
C
C-----------------------------------------------------------------------
C Part 2. Other Routines Callable.
C
C The following are optional calls which the user may make to
C gain additional capabilities in conjunction with DLSODAR.
C (The routines XSETUN and XSETF are designed to conform to the
C SLATEC error handling package.)
C
C Form of Call Function
C CALL XSETUN(LUN) Set the logical unit number, LUN, for
C output of messages from DLSODAR, if
C the default is not desired.
C The default value of LUN is 6.
C
C CALL XSETF(MFLAG) Set a flag to control the printing of
C messages by DLSODAR.
C MFLAG = 0 means do not print. (Danger:
C This risks losing valuable information.)
C MFLAG = 1 means print (the default).
C
C Either of the above calls may be made at
C any time and will take effect immediately.
C
C CALL DSRCAR(RSAV,ISAV,JOB) saves and restores the contents of
C the internal Common blocks used by
C DLSODAR (see Part 3 below).
C RSAV must be a real array of length 245
C or more, and ISAV must be an integer
C array of length 55 or more.
C JOB=1 means save Common into RSAV/ISAV.
C JOB=2 means restore Common from RSAV/ISAV.
C DSRCAR is useful if one is
C interrupting a run and restarting
C later, or alternating between two or
C more problems solved with DLSODAR.
C
C CALL DINTDY(,,,,,) Provide derivatives of y, of various
C (see below) orders, at a specified point t, if
C desired. It may be called only after
C a successful return from DLSODAR.
C
C The detailed instructions for using DINTDY are as follows.
C The form of the call is:
C
C LYH = 21 + 3*NG
C CALL DINTDY (T, K, RWORK(LYH), NYH, DKY, IFLAG)
C
C The input parameters are:
C
C T = value of independent variable where answers are desired
C (normally the same as the T last returned by DLSODAR).
C For valid results, T must lie between TCUR - HU and TCUR.
C (See optional outputs for TCUR and HU.)
C K = integer order of the derivative desired. K must satisfy
C 0 .le. K .le. NQCUR, where NQCUR is the current order
C (see optional outputs). The capability corresponding
C to K = 0, i.e. computing y(t), is already provided
C by DLSODAR directly. Since NQCUR .ge. 1, the first
C derivative dy/dt is always available with DINTDY.
C LYH = 21 + 3*NG = base address in RWORK of the history array YH.
C NYH = column length of YH, equal to the initial value of NEQ.
C
C The output parameters are:
C
C DKY = a real array of length NEQ containing the computed value
C of the K-th derivative of y(t).
C IFLAG = integer flag, returned as 0 if K and T were legal,
C -1 if K was illegal, and -2 if T was illegal.
C On an error return, a message is also written.
C-----------------------------------------------------------------------
C Part 3. Common Blocks.
C
C If DLSODAR is to be used in an overlay situation, the user
C must declare, in the primary overlay, the variables in:
C (1) the call sequence to DLSODAR, and
C (2) the three internal Common blocks
C /DLS001/ of length 255 (218 double precision words
C followed by 37 integer words),
C /DLSA01/ of length 31 (22 double precision words
C followed by 9 integer words).
C /DLSR01/ of length 7 (3 double precision words
C followed by 4 integer words).
C
C If DLSODAR is used on a system in which the contents of internal
C Common blocks are not preserved between calls, the user should
C declare the above Common blocks in the calling program to insure
C that their contents are preserved.
C
C If the solution of a given problem by DLSODAR is to be interrupted
C and then later continued, such as when restarting an interrupted run
C or alternating between two or more problems, the user should save,
C following the return from the last DLSODAR call prior to the
C interruption, the contents of the call sequence variables and the
C internal Common blocks, and later restore these values before the
C next DLSODAR call for that problem. To save and restore the Common
C blocks, use Subroutine DSRCAR (see Part 2 above).
C
C-----------------------------------------------------------------------
C Part 4. Optionally Replaceable Solver Routines.
C
C Below is a description of a routine in the DLSODAR package which
C relates to the measurement of errors, and can be
C replaced by a user-supplied version, if desired. However, since such
C a replacement may have a major impact on performance, it should be
C done only when absolutely necessary, and only with great caution.
C (Note: The means by which the package version of a routine is
C superseded by the user's version may be system-dependent.)
C
C (a) DEWSET.
C The following subroutine is called just before each internal
C integration step, and sets the array of error weights, EWT, as
C described under ITOL/RTOL/ATOL above:
C Subroutine DEWSET (NEQ, ITOL, RTOL, ATOL, YCUR, EWT)
C where NEQ, ITOL, RTOL, and ATOL are as in the DLSODAR call sequence,
C YCUR contains the current dependent variable vector, and
C EWT is the array of weights set by DEWSET.
C
C If the user supplies this subroutine, it must return in EWT(i)
C (i = 1,...,NEQ) a positive quantity suitable for comparing errors
C in y(i) to. The EWT array returned by DEWSET is passed to the
C DMNORM routine, and also used by DLSODAR in the computation
C of the optional output IMXER, and the increments for difference
C quotient Jacobians.
C
C In the user-supplied version of DEWSET, it may be desirable to use
C the current values of derivatives of y. Derivatives up to order NQ
C are available from the history array YH, described above under
C optional outputs. In DEWSET, YH is identical to the YCUR array,
C extended to NQ + 1 columns with a column length of NYH and scale
C factors of H**j/factorial(j). On the first call for the problem,
C given by NST = 0, NQ is 1 and H is temporarily set to 1.0.
C NYH is the initial value of NEQ. The quantities NQ, H, and NST
C can be obtained by including in DEWSET the statements:
C DOUBLE PRECISION RLS
C COMMON /DLS001/ RLS(218),ILS(37)
C NQ = ILS(33)
C NST = ILS(34)
C H = RLS(212)
C Thus, for example, the current value of dy/dt can be obtained as
C YCUR(NYH+i)/H (i=1,...,NEQ) (and the division by H is
C unnecessary when NST = 0).
C-----------------------------------------------------------------------
C
C***REVISION HISTORY (YYYYMMDD)
C 19811102 DATE WRITTEN
C 19820126 Fixed bug in tests of work space lengths;
C minor corrections in main prologue and comments.
C 19820507 Fixed bug in RCHEK in setting HMING.
C 19870330 Major update: corrected comments throughout;
C removed TRET from Common; rewrote EWSET with 4 loops;
C fixed t test in INTDY; added Cray directives in STODA;
C in STODA, fixed DELP init. and logic around PJAC call;
C combined routines to save/restore Common;
C passed LEVEL = 0 in error message calls (except run abort).
C 19970225 Fixed lines setting JSTART = -2 in Subroutine LSODAR.
C 20010425 Major update: convert source lines to upper case;
C added *DECK lines; changed from 1 to * in dummy dimensions;
C changed names R1MACH/D1MACH to RUMACH/DUMACH;
C renamed routines for uniqueness across single/double prec.;
C converted intrinsic names to generic form;
C removed ILLIN and NTREP (data loaded) from Common;
C removed all 'own' variables from Common;
C changed error messages to quoted strings;
C replaced XERRWV/XERRWD with 1993 revised version;
C converted prologues, comments, error messages to mixed case;
C numerous corrections to prologues and internal comments.
C 20010507 Converted single precision source to double precision.
C 20010613 Revised excess accuracy test (to match rest of ODEPACK).
C 20010808 Fixed bug in DPRJA (matrix in DBNORM call).
C 20020502 Corrected declarations in descriptions of user routines.
C 20031105 Restored 'own' variables to Common blocks, to enable
C interrupt/restart feature.
C 20031112 Added SAVE statements for data-loaded constants.
C
C-----------------------------------------------------------------------
C Other routines in the DLSODAR package.
C
C In addition to Subroutine DLSODAR, the DLSODAR package includes the
C following subroutines and function routines:
C DRCHEK does preliminary checking for roots, and serves as an
C interface between Subroutine DLSODAR and Subroutine DROOTS.
C DROOTS finds the leftmost root of a set of functions.
C DINTDY computes an interpolated value of the y vector at t = TOUT.
C DSTODA is the core integrator, which does one step of the
C integration and the associated error control.
C DCFODE sets all method coefficients and test constants.
C DPRJA computes and preprocesses the Jacobian matrix J = df/dy
C and the Newton iteration matrix P = I - h*l0*J.
C DSOLSY manages solution of linear system in chord iteration.
C DEWSET sets the error weight vector EWT before each step.
C DMNORM computes the weighted max-norm of a vector.
C DFNORM computes the norm of a full matrix consistent with the
C weighted max-norm on vectors.
C DBNORM computes the norm of a band matrix consistent with the
C weighted max-norm on vectors.
C DSRCAR is a user-callable routine to save and restore
C the contents of the internal Common blocks.
C DGEFA and DGESL are routines from LINPACK for solving full
C systems of linear algebraic equations.
C DGBFA and DGBSL are routines from LINPACK for solving banded
C linear systems.
C DCOPY is one of the basic linear algebra modules (BLAS).
C DUMACH computes the unit roundoff in a machine-independent manner.
C XERRWD, XSETUN, XSETF, IXSAV, and IUMACH handle the printing of all
C error messages and warnings. XERRWD is machine-dependent.
C Note: DMNORM, DFNORM, DBNORM, DUMACH, IXSAV, and IUMACH are
C function routines. All the others are subroutines.
C
C-----------------------------------------------------------------------
EXTERNAL DPRJA, DSOLSY
DOUBLE PRECISION DUMACH, DMNORM
INTEGER INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER INSUFR, INSUFI, IXPR, IOWNS2, JTYP, MUSED, MXORDN, MXORDS
INTEGER LG0, LG1, LGX, IOWNR3, IRFND, ITASKC, NGC, NGE
INTEGER I, I1, I2, IFLAG, IMXER, KGO, LENIW,
1 LENRW, LENWM, LF0, ML, MORD, MU, MXHNL0, MXSTP0
INTEGER LEN1, LEN1C, LEN1N, LEN1S, LEN2, LENIWC, LENRWC
INTEGER IRFP, IRT, LENYH, LYHNEW
DOUBLE PRECISION ROWNS,
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
DOUBLE PRECISION TSW, ROWNS2, PDNORM
DOUBLE PRECISION ROWNR3, T0, TLAST, TOUTC
DOUBLE PRECISION ATOLI, AYI, BIG, EWTI, H0, HMAX, HMX, RH, RTOLI,
1 TCRIT, TDIST, TNEXT, TOL, TOLSF, TP, SIZE, SUM, W0
DIMENSION MORD(2)
LOGICAL IHIT
CHARACTER*60 MSG
SAVE MORD, MXSTP0, MXHNL0
C-----------------------------------------------------------------------
C The following three internal Common blocks contain
C (a) variables which are local to any subroutine but whose values must
C be preserved between calls to the routine ("own" variables), and
C (b) variables which are communicated between subroutines.
C The block DLS001 is declared in subroutines DLSODAR, DINTDY, DSTODA,
C DPRJA, and DSOLSY.
C The block DLSA01 is declared in subroutines DLSODAR, DSTODA, DPRJA.
C The block DLSR01 is declared in subroutines DLSODAR, DRCHEK, DROOTS.
C Groups of variables are replaced by dummy arrays in the Common
C declarations in routines where those variables are not used.
C-----------------------------------------------------------------------
COMMON /DLS001/ ROWNS(209),
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
2 INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS(6),
3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
COMMON /DLSA01/ TSW, ROWNS2(20), PDNORM,
1 INSUFR, INSUFI, IXPR, IOWNS2(2), JTYP, MUSED, MXORDN, MXORDS
C
COMMON /DLSR01/ ROWNR3(2), T0, TLAST, TOUTC,
1 LG0, LG1, LGX, IOWNR3(2), IRFND, ITASKC, NGC, NGE
C
DATA MORD(1),MORD(2)/12,5/, MXSTP0/500/, MXHNL0/10/
C-----------------------------------------------------------------------
C Block A.
C This code block is executed on every call.
C It tests ISTATE and ITASK for legality and branches appropriately.
C If ISTATE .gt. 1 but the flag INIT shows that initialization has
C not yet been done, an error return occurs.
C If ISTATE = 1 and TOUT = T, return immediately.
C-----------------------------------------------------------------------
IF (ISTATE .LT. 1 .OR. ISTATE .GT. 3) GO TO 601
IF (ITASK .LT. 1 .OR. ITASK .GT. 5) GO TO 602
ITASKC = ITASK
IF (ISTATE .EQ. 1) GO TO 10
IF (INIT .EQ. 0) GO TO 603
IF (ISTATE .EQ. 2) GO TO 200
GO TO 20
10 INIT = 0
IF (TOUT .EQ. T) RETURN
C-----------------------------------------------------------------------
C Block B.
C The next code block is executed for the initial call (ISTATE = 1),
C or for a continuation call with parameter changes (ISTATE = 3).
C It contains checking of all inputs and various initializations.
C
C First check legality of the non-optional inputs NEQ, ITOL, IOPT,
C JT, ML, MU, and NG.
C-----------------------------------------------------------------------
20 IF (NEQ(1) .LE. 0) GO TO 604
IF (ISTATE .EQ. 1) GO TO 25
IF (NEQ(1) .GT. N) GO TO 605
25 N = NEQ(1)
IF (ITOL .LT. 1 .OR. ITOL .GT. 4) GO TO 606
IF (IOPT .LT. 0 .OR. IOPT .GT. 1) GO TO 607
IF (JT .EQ. 3 .OR. JT .LT. 1 .OR. JT .GT. 5) GO TO 608
JTYP = JT
IF (JT .LE. 2) GO TO 30
ML = IWORK(1)
MU = IWORK(2)
IF (ML .LT. 0 .OR. ML .GE. N) GO TO 609
IF (MU .LT. 0 .OR. MU .GE. N) GO TO 610
30 CONTINUE
IF (NG .LT. 0) GO TO 630
IF (ISTATE .EQ. 1) GO TO 35
IF (IRFND .EQ. 0 .AND. NG .NE. NGC) GO TO 631
35 NGC = NG
C Next process and check the optional inputs. --------------------------
IF (IOPT .EQ. 1) GO TO 40
IXPR = 0
MXSTEP = MXSTP0
MXHNIL = MXHNL0
HMXI = 0.0D0
HMIN = 0.0D0
IF (ISTATE .NE. 1) GO TO 60
H0 = 0.0D0
MXORDN = MORD(1)
MXORDS = MORD(2)
GO TO 60
40 IXPR = IWORK(5)
IF (IXPR .LT. 0 .OR. IXPR .GT. 1) GO TO 611
MXSTEP = IWORK(6)
IF (MXSTEP .LT. 0) GO TO 612
IF (MXSTEP .EQ. 0) MXSTEP = MXSTP0
MXHNIL = IWORK(7)
IF (MXHNIL .LT. 0) GO TO 613
IF (MXHNIL .EQ. 0) MXHNIL = MXHNL0
IF (ISTATE .NE. 1) GO TO 50
H0 = RWORK(5)
MXORDN = IWORK(8)
IF (MXORDN .LT. 0) GO TO 628
IF (MXORDN .EQ. 0) MXORDN = 100
MXORDN = MIN(MXORDN,MORD(1))
MXORDS = IWORK(9)
IF (MXORDS .LT. 0) GO TO 629
IF (MXORDS .EQ. 0) MXORDS = 100
MXORDS = MIN(MXORDS,MORD(2))
IF ((TOUT - T)*H0 .LT. 0.0D0) GO TO 614
50 HMAX = RWORK(6)
IF (HMAX .LT. 0.0D0) GO TO 615
HMXI = 0.0D0
IF (HMAX .GT. 0.0D0) HMXI = 1.0D0/HMAX
HMIN = RWORK(7)
IF (HMIN .LT. 0.0D0) GO TO 616
C-----------------------------------------------------------------------
C Set work array pointers and check lengths LRW and LIW.
C If ISTATE = 1, METH is initialized to 1 here to facilitate the
C checking of work space lengths.
C Pointers to segments of RWORK and IWORK are named by prefixing L to
C the name of the segment. E.g., the segment YH starts at RWORK(LYH).
C Segments of RWORK (in order) are denoted G0, G1, GX, YH, WM,
C EWT, SAVF, ACOR.
C If the lengths provided are insufficient for the current method,
C an error return occurs. This is treated as illegal input on the
C first call, but as a problem interruption with ISTATE = -7 on a
C continuation call. If the lengths are sufficient for the current
C method but not for both methods, a warning message is sent.
C-----------------------------------------------------------------------
60 IF (ISTATE .EQ. 1) METH = 1
IF (ISTATE .EQ. 1) NYH = N
LG0 = 21
LG1 = LG0 + NG
LGX = LG1 + NG
LYHNEW = LGX + NG
IF (ISTATE .EQ. 1) LYH = LYHNEW
IF (LYHNEW .EQ. LYH) GO TO 62
C If ISTATE = 3 and NG was changed, shift YH to its new location. ------
LENYH = L*NYH
IF (LRW .LT. LYHNEW-1+LENYH) GO TO 62
I1 = 1
IF (LYHNEW .GT. LYH) I1 = -1
CALL DCOPY (LENYH, RWORK(LYH), I1, RWORK(LYHNEW), I1)
LYH = LYHNEW
62 CONTINUE
LEN1N = LYHNEW - 1 + (MXORDN + 1)*NYH
LEN1S = LYHNEW - 1 + (MXORDS + 1)*NYH
LWM = LEN1S + 1
IF (JT .LE. 2) LENWM = N*N + 2
IF (JT .GE. 4) LENWM = (2*ML + MU + 1)*N + 2
LEN1S = LEN1S + LENWM
LEN1C = LEN1N
IF (METH .EQ. 2) LEN1C = LEN1S
LEN1 = MAX(LEN1N,LEN1S)
LEN2 = 3*N
LENRW = LEN1 + LEN2
LENRWC = LEN1C + LEN2
IWORK(17) = LENRW
LIWM = 1
LENIW = 20 + N
LENIWC = 20
IF (METH .EQ. 2) LENIWC = LENIW
IWORK(18) = LENIW
IF (ISTATE .EQ. 1 .AND. LRW .LT. LENRWC) GO TO 617
IF (ISTATE .EQ. 1 .AND. LIW .LT. LENIWC) GO TO 618
IF (ISTATE .EQ. 3 .AND. LRW .LT. LENRWC) GO TO 550
IF (ISTATE .EQ. 3 .AND. LIW .LT. LENIWC) GO TO 555
LEWT = LEN1 + 1
INSUFR = 0
IF (LRW .GE. LENRW) GO TO 65
INSUFR = 2
LEWT = LEN1C + 1
MSG='DLSODAR- Warning.. RWORK length is sufficient for now, but '
CALL XERRWD (MSG, 60, 103, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' may not be later. Integration will proceed anyway. '
CALL XERRWD (MSG, 60, 103, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' Length needed is LENRW = I1, while LRW = I2.'
CALL XERRWD (MSG, 50, 103, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
65 LSAVF = LEWT + N
LACOR = LSAVF + N
INSUFI = 0
IF (LIW .GE. LENIW) GO TO 70
INSUFI = 2
MSG='DLSODAR- Warning.. IWORK length is sufficient for now, but '
CALL XERRWD (MSG, 60, 104, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' may not be later. Integration will proceed anyway. '
CALL XERRWD (MSG, 60, 104, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' Length needed is LENIW = I1, while LIW = I2.'
CALL XERRWD (MSG, 50, 104, 0, 2, LENIW, LIW, 0, 0.0D0, 0.0D0)
70 CONTINUE
C Check RTOL and ATOL for legality. ------------------------------------
RTOLI = RTOL(1)
ATOLI = ATOL(1)
DO 75 I = 1,N
IF (ITOL .GE. 3) RTOLI = RTOL(I)
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
IF (RTOLI .LT. 0.0D0) GO TO 619
IF (ATOLI .LT. 0.0D0) GO TO 620
75 CONTINUE
IF (ISTATE .EQ. 1) GO TO 100
C if ISTATE = 3, set flag to signal parameter changes to DSTODA. -------
JSTART = -1
IF (N .EQ. NYH) GO TO 200
C NEQ was reduced. zero part of yh to avoid undefined references. -----
I1 = LYH + L*NYH
I2 = LYH + (MAXORD + 1)*NYH - 1
IF (I1 .GT. I2) GO TO 200
DO 95 I = I1,I2
95 RWORK(I) = 0.0D0
GO TO 200
C-----------------------------------------------------------------------
C Block C.
C The next block is for the initial call only (ISTATE = 1).
C It contains all remaining initializations, the initial call to F,
C and the calculation of the initial step size.
C The error weights in EWT are inverted after being loaded.
C-----------------------------------------------------------------------
100 UROUND = DUMACH()
TN = T
TSW = T
MAXORD = MXORDN
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 110
TCRIT = RWORK(1)
IF ((TCRIT - TOUT)*(TOUT - T) .LT. 0.0D0) GO TO 625
IF (H0 .NE. 0.0D0 .AND. (T + H0 - TCRIT)*H0 .GT. 0.0D0)
1 H0 = TCRIT - T
110 JSTART = 0
NHNIL = 0
NST = 0
NJE = 0
NSLAST = 0
HU = 0.0D0
NQU = 0
MUSED = 0
MITER = 0
CCMAX = 0.3D0
MAXCOR = 3
MSBP = 20
MXNCF = 10
C Initial call to F. (LF0 points to YH(*,2).) -------------------------
LF0 = LYH + NYH
CALL F (NEQ, T, Y, RWORK(LF0))
NFE = 1
C Load the initial value vector in YH. ---------------------------------
DO 115 I = 1,N
115 RWORK(I+LYH-1) = Y(I)
C Load and invert the EWT array. (H is temporarily set to 1.0.) -------
NQ = 1
H = 1.0D0
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 120 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 621
120 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
C-----------------------------------------------------------------------
C The coding below computes the step size, H0, to be attempted on the
C first step, unless the user has supplied a value for this.
C First check that TOUT - T differs significantly from zero.
C A scalar tolerance quantity TOL is computed, as MAX(RTOL(i))
C if this is positive, or MAX(ATOL(i)/ABS(Y(i))) otherwise, adjusted
C so as to be between 100*UROUND and 1.0E-3.
C Then the computed value H0 is given by:
C
C H0**(-2) = 1./(TOL * w0**2) + TOL * (norm(F))**2
C
C where w0 = MAX ( ABS(T), ABS(TOUT) ),
C F = the initial value of the vector f(t,y), and
C norm() = the weighted vector norm used throughout, given by
C the DMNORM function routine, and weighted by the
C tolerances initially loaded into the EWT array.
C The sign of H0 is inferred from the initial values of TOUT and T.
C ABS(H0) is made .le. ABS(TOUT-T) in any case.
C-----------------------------------------------------------------------
IF (H0 .NE. 0.0D0) GO TO 180
TDIST = ABS(TOUT - T)
W0 = MAX(ABS(T),ABS(TOUT))
IF (TDIST .LT. 2.0D0*UROUND*W0) GO TO 622
TOL = RTOL(1)
IF (ITOL .LE. 2) GO TO 140
DO 130 I = 1,N
130 TOL = MAX(TOL,RTOL(I))
140 IF (TOL .GT. 0.0D0) GO TO 160
ATOLI = ATOL(1)
DO 150 I = 1,N
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
AYI = ABS(Y(I))
IF (AYI .NE. 0.0D0) TOL = MAX(TOL,ATOLI/AYI)
150 CONTINUE
160 TOL = MAX(TOL,100.0D0*UROUND)
TOL = MIN(TOL,0.001D0)
SUM = DMNORM (N, RWORK(LF0), RWORK(LEWT))
SUM = 1.0D0/(TOL*W0*W0) + TOL*SUM**2
H0 = 1.0D0/SQRT(SUM)
H0 = MIN(H0,TDIST)
H0 = SIGN(H0,TOUT-T)
C Adjust H0 if necessary to meet HMAX bound. ---------------------------
180 RH = ABS(H0)*HMXI
IF (RH .GT. 1.0D0) H0 = H0/RH
C Load H with H0 and scale YH(*,2) by H0. ------------------------------
H = H0
DO 190 I = 1,N
190 RWORK(I+LF0-1) = H0*RWORK(I+LF0-1)
C
C Check for a zero of g at T. ------------------------------------------
IRFND = 0
TOUTC = TOUT
IF (NGC .EQ. 0) GO TO 270
CALL DRCHEK (1, G, NEQ, Y, RWORK(LYH), NYH,
1 RWORK(LG0), RWORK(LG1), RWORK(LGX), JROOT, IRT)
IF (IRT .EQ. 0) GO TO 270
GO TO 632
C-----------------------------------------------------------------------
C Block D.
C The next code block is for continuation calls only (ISTATE = 2 or 3)
C and is to check stop conditions before taking a step.
C First, DRCHEK is called to check for a root within the last step
C taken, other than the last root found there, if any.
C If ITASK = 2 or 5, and y(TN) has not yet been returned to the user
C because of an intervening root, return through Block G.
C-----------------------------------------------------------------------
200 NSLAST = NST
C
IRFP = IRFND
IF (NGC .EQ. 0) GO TO 205
IF (ITASK .EQ. 1 .OR. ITASK .EQ. 4) TOUTC = TOUT
CALL DRCHEK (2, G, NEQ, Y, RWORK(LYH), NYH,
1 RWORK(LG0), RWORK(LG1), RWORK(LGX), JROOT, IRT)
IF (IRT .NE. 1) GO TO 205
IRFND = 1
ISTATE = 3
T = T0
GO TO 425
205 CONTINUE
IRFND = 0
IF (IRFP .EQ. 1 .AND. TLAST .NE. TN .AND. ITASK .EQ. 2) GO TO 400
C
GO TO (210, 250, 220, 230, 240), ITASK
210 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
220 TP = TN - HU*(1.0D0 + 100.0D0*UROUND)
IF ((TP - TOUT)*H .GT. 0.0D0) GO TO 623
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
T = TN
GO TO 400
230 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
IF ((TCRIT - TOUT)*H .LT. 0.0D0) GO TO 625
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 245
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
240 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
245 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) T = TCRIT
IF (IRFP .EQ. 1 .AND. TLAST .NE. TN .AND. ITASK .EQ. 5) GO TO 400
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
IF (ISTATE .EQ. 2 .AND. JSTART .GE. 0) JSTART = -2
C-----------------------------------------------------------------------
C Block E.
C The next block is normally executed for all calls and contains
C the call to the one-step core integrator DSTODA.
C
C This is a looping point for the integration steps.
C
C First check for too many steps being taken, update EWT (if not at
C start of problem), check for too much accuracy being requested, and
C check for H below the roundoff level in T.
C-----------------------------------------------------------------------
250 CONTINUE
IF (METH .EQ. MUSED) GO TO 255
IF (INSUFR .EQ. 1) GO TO 550
IF (INSUFI .EQ. 1) GO TO 555
255 IF ((NST-NSLAST) .GE. MXSTEP) GO TO 500
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 260 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 510
260 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
270 TOLSF = UROUND*DMNORM (N, RWORK(LYH), RWORK(LEWT))
IF (TOLSF .LE. 1.0D0) GO TO 280
TOLSF = TOLSF*2.0D0
IF (NST .EQ. 0) GO TO 626
GO TO 520
280 IF ((TN + H) .NE. TN) GO TO 290
NHNIL = NHNIL + 1
IF (NHNIL .GT. MXHNIL) GO TO 290
MSG = 'DLSODAR- Warning..Internal T(=R1) and H(=R2) are '
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' such that in the machine, T + H = T on the next step '
CALL XERRWD (MSG, 60, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' (H = step size). Solver will continue anyway.'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 2, TN, H)
IF (NHNIL .LT. MXHNIL) GO TO 290
MSG = 'DLSODAR- Above warning has been issued I1 times. '
CALL XERRWD (MSG, 50, 102, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' It will not be issued again for this problem.'
CALL XERRWD (MSG, 50, 102, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
290 CONTINUE
C-----------------------------------------------------------------------
C CALL DSTODA(NEQ,Y,YH,NYH,YH,EWT,SAVF,ACOR,WM,IWM,F,JAC,DPRJA,DSOLSY)
C-----------------------------------------------------------------------
CALL DSTODA (NEQ, Y, RWORK(LYH), NYH, RWORK(LYH), RWORK(LEWT),
1 RWORK(LSAVF), RWORK(LACOR), RWORK(LWM), IWORK(LIWM),
2 F, JAC, DPRJA, DSOLSY)
KGO = 1 - KFLAG
GO TO (300, 530, 540), KGO
C-----------------------------------------------------------------------
C Block F.
C The following block handles the case of a successful return from the
C core integrator (KFLAG = 0).
C If a method switch was just made, record TSW, reset MAXORD,
C set JSTART to -1 to signal DSTODA to complete the switch,
C and do extra printing of data if IXPR = 1.
C Then call DRCHEK to check for a root within the last step.
C Then, if no root was found, check for stop conditions.
C-----------------------------------------------------------------------
300 INIT = 1
IF (METH .EQ. MUSED) GO TO 310
TSW = TN
MAXORD = MXORDN
IF (METH .EQ. 2) MAXORD = MXORDS
IF (METH .EQ. 2) RWORK(LWM) = SQRT(UROUND)
INSUFR = MIN(INSUFR,1)
INSUFI = MIN(INSUFI,1)
JSTART = -1
IF (IXPR .EQ. 0) GO TO 310
IF (METH .EQ. 2) THEN
MSG='DLSODAR- A switch to the BDF (stiff) method has occurred '
CALL XERRWD (MSG, 60, 105, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
IF (METH .EQ. 1) THEN
MSG='DLSODAR- A switch to the Adams (nonstiff) method occurred '
CALL XERRWD (MSG, 60, 106, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
MSG=' at T = R1, tentative step size H = R2, step NST = I1 '
CALL XERRWD (MSG, 60, 107, 0, 1, NST, 0, 2, TN, H)
310 CONTINUE
C
IF (NGC .EQ. 0) GO TO 315
CALL DRCHEK (3, G, NEQ, Y, RWORK(LYH), NYH,
1 RWORK(LG0), RWORK(LG1), RWORK(LGX), JROOT, IRT)
IF (IRT .NE. 1) GO TO 315
IRFND = 1
ISTATE = 3
T = T0
GO TO 425
315 CONTINUE
C
GO TO (320, 400, 330, 340, 350), ITASK
C ITASK = 1. If TOUT has been reached, interpolate. -------------------
320 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
C ITASK = 3. Jump to exit if TOUT was reached. ------------------------
330 IF ((TN - TOUT)*H .GE. 0.0D0) GO TO 400
GO TO 250
C ITASK = 4. See if TOUT or TCRIT was reached. Adjust H if necessary.
340 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 345
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
345 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
IF (JSTART .GE. 0) JSTART = -2
GO TO 250
C ITASK = 5. See if TCRIT was reached and jump to exit. ---------------
350 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
C-----------------------------------------------------------------------
C Block G.
C The following block handles all successful returns from DLSODAR.
C If ITASK .ne. 1, Y is loaded from YH and T is set accordingly.
C ISTATE is set to 2, and the optional outputs are loaded into the
C work arrays before returning.
C-----------------------------------------------------------------------
400 DO 410 I = 1,N
410 Y(I) = RWORK(I+LYH-1)
T = TN
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 420
IF (IHIT) T = TCRIT
420 ISTATE = 2
425 CONTINUE
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
RWORK(15) = TSW
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(19) = MUSED
IWORK(20) = METH
IWORK(10) = NGE
TLAST = T
RETURN
C-----------------------------------------------------------------------
C Block H.
C The following block handles all unsuccessful returns other than
C those for illegal input. First the error message routine is called.
C If there was an error test or convergence test failure, IMXER is set.
C Then Y is loaded from YH and T is set to TN.
C The optional outputs are loaded into the work arrays before returning.
C-----------------------------------------------------------------------
C The maximum number of steps was taken before reaching TOUT. ----------
500 MSG = 'DLSODAR- At current T (=R1), MXSTEP (=I1) steps '
CALL XERRWD (MSG, 50, 201, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' taken on this call before reaching TOUT '
CALL XERRWD (MSG, 50, 201, 0, 1, MXSTEP, 0, 1, TN, 0.0D0)
ISTATE = -1
GO TO 580
C EWT(i) .le. 0.0 for some i (not at start of problem). ----------------
510 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODAR- At T(=R1), EWT(I1) has become R2 .le. 0.'
CALL XERRWD (MSG, 50, 202, 0, 1, I, 0, 2, TN, EWTI)
ISTATE = -6
GO TO 580
C Too much accuracy requested for machine precision. -------------------
520 MSG = 'DLSODAR- At T (=R1), too much accuracy requested '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' for precision of machine.. See TOLSF (=R2) '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 2, TN, TOLSF)
RWORK(14) = TOLSF
ISTATE = -2
GO TO 580
C KFLAG = -1. Error test failed repeatedly or with ABS(H) = HMIN. -----
530 MSG = 'DLSODAR- At T(=R1), step size H(=R2), the error '
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' test failed repeatedly or with ABS(H) = HMIN'
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 2, TN, H)
ISTATE = -4
GO TO 560
C KFLAG = -2. Convergence failed repeatedly or with ABS(H) = HMIN. ----
540 MSG = 'DLSODAR- At T (=R1) and step size H (=R2), the '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' corrector convergence failed repeatedly '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' or with ABS(H) = HMIN '
CALL XERRWD (MSG, 30, 205, 0, 0, 0, 0, 2, TN, H)
ISTATE = -5
GO TO 560
C RWORK length too small to proceed. -----------------------------------
550 MSG = 'DLSODAR- At current T(=R1), RWORK length too small'
CALL XERRWD (MSG, 50, 206, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' to proceed. The integration was otherwise successful.'
CALL XERRWD (MSG, 60, 206, 0, 0, 0, 0, 1, TN, 0.0D0)
ISTATE = -7
GO TO 580
C IWORK length too small to proceed. -----------------------------------
555 MSG = 'DLSODAR- At current T(=R1), IWORK length too small'
CALL XERRWD (MSG, 50, 207, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' to proceed. The integration was otherwise successful.'
CALL XERRWD (MSG, 60, 207, 0, 0, 0, 0, 1, TN, 0.0D0)
ISTATE = -7
GO TO 580
C Compute IMXER if relevant. -------------------------------------------
560 BIG = 0.0D0
IMXER = 1
DO 570 I = 1,N
SIZE = ABS(RWORK(I+LACOR-1)*RWORK(I+LEWT-1))
IF (BIG .GE. SIZE) GO TO 570
BIG = SIZE
IMXER = I
570 CONTINUE
IWORK(16) = IMXER
C Set Y vector, T, and optional outputs. -------------------------------
580 DO 590 I = 1,N
590 Y(I) = RWORK(I+LYH-1)
T = TN
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
RWORK(15) = TSW
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(19) = MUSED
IWORK(20) = METH
IWORK(10) = NGE
TLAST = T
RETURN
C-----------------------------------------------------------------------
C Block I.
C The following block handles all error returns due to illegal input
C (ISTATE = -3), as detected before calling the core integrator.
C First the error message routine is called. If the illegal input
C is a negative ISTATE, the run is aborted (apparent infinite loop).
C-----------------------------------------------------------------------
601 MSG = 'DLSODAR- ISTATE(=I1) illegal.'
CALL XERRWD (MSG, 30, 1, 0, 1, ISTATE, 0, 0, 0.0D0, 0.0D0)
IF (ISTATE .LT. 0) GO TO 800
GO TO 700
602 MSG = 'DLSODAR- ITASK (=I1) illegal.'
CALL XERRWD (MSG, 30, 2, 0, 1, ITASK, 0, 0, 0.0D0, 0.0D0)
GO TO 700
603 MSG = 'DLSODAR- ISTATE.gt.1 but DLSODAR not initialized.'
CALL XERRWD (MSG, 50, 3, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
GO TO 700
604 MSG = 'DLSODAR- NEQ (=I1) .lt. 1 '
CALL XERRWD (MSG, 30, 4, 0, 1, NEQ(1), 0, 0, 0.0D0, 0.0D0)
GO TO 700
605 MSG = 'DLSODAR- ISTATE = 3 and NEQ increased (I1 to I2).'
CALL XERRWD (MSG, 50, 5, 0, 2, N, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
606 MSG = 'DLSODAR- ITOL (=I1) illegal. '
CALL XERRWD (MSG, 30, 6, 0, 1, ITOL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
607 MSG = 'DLSODAR- IOPT (=I1) illegal. '
CALL XERRWD (MSG, 30, 7, 0, 1, IOPT, 0, 0, 0.0D0, 0.0D0)
GO TO 700
608 MSG = 'DLSODAR- JT (=I1) illegal. '
CALL XERRWD (MSG, 30, 8, 0, 1, JT, 0, 0, 0.0D0, 0.0D0)
GO TO 700
609 MSG = 'DLSODAR- ML (=I1) illegal: .lt.0 or .ge.NEQ (=I2)'
CALL XERRWD (MSG, 50, 9, 0, 2, ML, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
610 MSG = 'DLSODAR- MU (=I1) illegal: .lt.0 or .ge.NEQ (=I2)'
CALL XERRWD (MSG, 50, 10, 0, 2, MU, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
611 MSG = 'DLSODAR- IXPR (=I1) illegal. '
CALL XERRWD (MSG, 30, 11, 0, 1, IXPR, 0, 0, 0.0D0, 0.0D0)
GO TO 700
612 MSG = 'DLSODAR- MXSTEP (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 12, 0, 1, MXSTEP, 0, 0, 0.0D0, 0.0D0)
GO TO 700
613 MSG = 'DLSODAR- MXHNIL (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 13, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
614 MSG = 'DLSODAR- TOUT (=R1) behind T (=R2) '
CALL XERRWD (MSG, 40, 14, 0, 0, 0, 0, 2, TOUT, T)
MSG = ' Integration direction is given by H0 (=R1) '
CALL XERRWD (MSG, 50, 14, 0, 0, 0, 0, 1, H0, 0.0D0)
GO TO 700
615 MSG = 'DLSODAR- HMAX (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 15, 0, 0, 0, 0, 1, HMAX, 0.0D0)
GO TO 700
616 MSG = 'DLSODAR- HMIN (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 16, 0, 0, 0, 0, 1, HMIN, 0.0D0)
GO TO 700
617 MSG='DLSODAR- RWORK length needed, LENRW(=I1), exceeds LRW(=I2) '
CALL XERRWD (MSG, 60, 17, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
618 MSG='DLSODAR- IWORK length needed, LENIW(=I1), exceeds LIW(=I2) '
CALL XERRWD (MSG, 60, 18, 0, 2, LENIW, LIW, 0, 0.0D0, 0.0D0)
GO TO 700
619 MSG = 'DLSODAR- RTOL(I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 19, 0, 1, I, 0, 1, RTOLI, 0.0D0)
GO TO 700
620 MSG = 'DLSODAR- ATOL(I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 20, 0, 1, I, 0, 1, ATOLI, 0.0D0)
GO TO 700
621 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODAR- EWT(I1) is R1 .le. 0.0 '
CALL XERRWD (MSG, 40, 21, 0, 1, I, 0, 1, EWTI, 0.0D0)
GO TO 700
622 MSG='DLSODAR- TOUT(=R1) too close to T(=R2) to start integration.'
CALL XERRWD (MSG, 60, 22, 0, 0, 0, 0, 2, TOUT, T)
GO TO 700
623 MSG='DLSODAR- ITASK = I1 and TOUT (=R1) behind TCUR - HU (= R2) '
CALL XERRWD (MSG, 60, 23, 0, 1, ITASK, 0, 2, TOUT, TP)
GO TO 700
624 MSG='DLSODAR- ITASK = 4 or 5 and TCRIT (=R1) behind TCUR (=R2) '
CALL XERRWD (MSG, 60, 24, 0, 0, 0, 0, 2, TCRIT, TN)
GO TO 700
625 MSG='DLSODAR- ITASK = 4 or 5 and TCRIT (=R1) behind TOUT (=R2) '
CALL XERRWD (MSG, 60, 25, 0, 0, 0, 0, 2, TCRIT, TOUT)
GO TO 700
626 MSG = 'DLSODAR- At start of problem, too much accuracy '
CALL XERRWD (MSG, 50, 26, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' requested for precision of machine.. See TOLSF (=R1) '
CALL XERRWD (MSG, 60, 26, 0, 0, 0, 0, 1, TOLSF, 0.0D0)
RWORK(14) = TOLSF
GO TO 700
627 MSG = 'DLSODAR- Trouble in DINTDY. ITASK = I1, TOUT = R1'
CALL XERRWD (MSG, 50, 27, 0, 1, ITASK, 0, 1, TOUT, 0.0D0)
GO TO 700
628 MSG = 'DLSODAR- MXORDN (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 28, 0, 1, MXORDN, 0, 0, 0.0D0, 0.0D0)
GO TO 700
629 MSG = 'DLSODAR- MXORDS (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 29, 0, 1, MXORDS, 0, 0, 0.0D0, 0.0D0)
GO TO 700
630 MSG = 'DLSODAR- NG (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 30, 0, 1, NG, 0, 0, 0.0D0, 0.0D0)
GO TO 700
631 MSG = 'DLSODAR- NG changed (from I1 to I2) illegally, '
CALL XERRWD (MSG, 50, 31, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' i.e. not immediately after a root was found.'
CALL XERRWD (MSG, 50, 31, 0, 2, NGC, NG, 0, 0.0D0, 0.0D0)
GO TO 700
632 MSG = 'DLSODAR- One or more components of g has a root '
CALL XERRWD (MSG, 50, 32, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' too near to the initial point. '
CALL XERRWD (MSG, 40, 32, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
C
700 ISTATE = -3
RETURN
C
800 MSG = 'DLSODAR- Run aborted.. apparent infinite loop. '
CALL XERRWD (MSG, 50, 303, 2, 0, 0, 0, 0, 0.0D0, 0.0D0)
RETURN
C----------------------- End of Subroutine DLSODAR ---------------------
END
*DECK DLSODPK
SUBROUTINE DLSODPK (F, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
1 ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JAC, PSOL, MF)
EXTERNAL F, JAC, PSOL
INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK, LIW, MF
DOUBLE PRECISION Y, T, TOUT, RTOL, ATOL, RWORK
DIMENSION NEQ(*), Y(*), RTOL(*), ATOL(*), RWORK(LRW), IWORK(LIW)
C-----------------------------------------------------------------------
C This is the 18 November 2003 version of
C DLSODPK: Livermore Solver for Ordinary Differential equations,
C with Preconditioned Krylov iteration methods for the
C Newton correction linear systems.
C
C This version is in double precision.
C
C DLSODPK solves the initial value problem for stiff or nonstiff
C systems of first order ODEs,
C dy/dt = f(t,y) , or, in component form,
C dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(NEQ)) (i = 1,...,NEQ).
C-----------------------------------------------------------------------
C Introduction.
C
C This is a modification of the DLSODE package which incorporates
C various preconditioned Krylov subspace iteration methods for the
C linear algebraic systems that arise in the case of stiff systems.
C
C The linear systems that must be solved have the form
C A * x = b , where A = identity - hl0 * (df/dy) .
C Here hl0 is a scalar, and df/dy is the Jacobian matrix of partial
C derivatives of f (NEQ by NEQ).
C
C The particular Krylov method is chosen by setting the second digit,
C MITER, in the method flag MF.
C Currently, the values of MITER have the following meanings:
C
C MITER = 1 means the preconditioned Scaled Incomplete
C Orthogonalization Method (SPIOM).
C
C 2 means an incomplete version of the Preconditioned Scaled
C Generalized Minimal Residual method (SPIGMR).
C This is the best choice in general.
C
C 3 means the Preconditioned Conjugate Gradient method (PCG).
C Recommended only when df/dy is symmetric or nearly so.
C
C 4 means the scaled Preconditioned Conjugate Gradient method
C (PCGS). Recommended only when D-inverse * df/dy * D is
C symmetric or nearly so, where D is the diagonal scaling
C matrix with elements 1/EWT(i) (see RTOL/ATOL description).
C
C 9 means that only a user-supplied matrix P (approximating A)
C will be used, with no Krylov iteration done. This option
C allows the user to provide the complete linear system
C solution algorithm, if desired.
C
C The user can apply preconditioning to the linear system A*x = b,
C by means of arbitrary matrices (the preconditioners).
C In the case of SPIOM and SPIGMR, one can apply left and right
C preconditioners P1 and P2, and the basic iterative method is then
C applied to the matrix (P1-inverse)*A*(P2-inverse) instead of to the
C matrix A. The product P1*P2 should be an approximation to matrix A
C such that linear systems with P1 or P2 are easier to solve than with
C A. Preconditioning from the left only or right only means using
C P2 = identity or P1 = identity, respectively.
C In the case of the PCG and PCGS methods, there is only one
C preconditioner matrix P (but it can be the product of more than one).
C It should approximate the matrix A but allow for relatively
C easy solution of linear systems with coefficient matrix P.
C For PCG, P should be positive definite symmetric, or nearly so,
C and for PCGS, the scaled preconditioner D-inverse * P * D
C should be symmetric or nearly so.
C If the Jacobian J = df/dy splits in a natural way into a sum
C J = J1 + J2, then one possible choice of preconditioners is
C P1 = identity - hl0 * J1 and P2 = identity - hl0 * J2
C provided each of these is easy to solve (or approximately solve).
C
C-----------------------------------------------------------------------
C References:
C 1. Peter N. Brown and Alan C. Hindmarsh, Reduced Storage Matrix
C Methods in Stiff ODE Systems, J. Appl. Math. & Comp., 31 (1989),
C pp. 40-91; also L.L.N.L. Report UCRL-95088, Rev. 1, June 1987.
C 2. Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE
C Solvers, in Scientific Computing, R. S. Stepleman et al. (Eds.),
C North-Holland, Amsterdam, 1983, pp. 55-64.
C-----------------------------------------------------------------------
C Authors: Alan C. Hindmarsh and Peter N. Brown
C Center for Applied Scientific Computing, L-561
C Lawrence Livermore National Laboratory
C Livermore, CA 94551
C-----------------------------------------------------------------------
C Summary of Usage.
C
C Communication between the user and the DLSODPK package, for normal
C situations, is summarized here. This summary describes only a subset
C of the full set of options available. See the full description for
C details, including optional communication, nonstandard options,
C and instructions for special situations. See also the demonstration
C program distributed with this solver.
C
C A. First provide a subroutine of the form:
C SUBROUTINE F (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y(*), YDOT(*)
C which supplies the vector function f by loading YDOT(i) with f(i).
C
C B. Next determine (or guess) whether or not the problem is stiff.
C Stiffness occurs when the Jacobian matrix df/dy has an eigenvalue
C whose real part is negative and large in magnitude, compared to the
C reciprocal of the t span of interest. If the problem is nonstiff,
C use a method flag MF = 10. If it is stiff, MF should be between 21
C and 24, or possibly 29. MF = 22 is generally the best choice.
C Use 23 or 24 only if symmetry is present. Use MF = 29 if the
C complete linear system solution is to be provided by the user.
C The following four parameters must also be set.
C IWORK(1) = LWP = length of real array WP for preconditioning.
C IWORK(2) = LIWP = length of integer array IWP for preconditioning.
C IWORK(3) = JPRE = preconditioner type flag:
C = 0 for no preconditioning (P1 = P2 = P = identity)
C = 1 for left-only preconditioning (P2 = identity)
C = 2 for right-only preconditioning (P1 = identity)
C = 3 for two-sided preconditioning (and PCG or PCGS)
C IWORK(4) = JACFLG = flag for whether JAC is called.
C = 0 if JAC is not to be called,
C = 1 if JAC is to be called.
C Use JACFLG = 1 if JAC computes any nonconstant data for use in
C preconditioning, such as Jacobian elements.
C The arrays WP and IWP are work arrays under the user's control,
C for use in the routines that perform preconditioning operations.
C
C C. If the problem is stiff, you must supply two routines that deal
C with the preconditioning of the linear systems to be solved.
C These are as follows:
C
C SUBROUTINE JAC (F, NEQ, T, Y, YSV, REWT, FTY, V, HL0, WP,IWP, IER)
C DOUBLE PRECISION T, Y(*),YSV(*), REWT(*), FTY(*), V(*), HL0, WP(*)
C INTEGER IWP(*)
C This routine must evaluate and preprocess any parts of the
C Jacobian matrix df/dy involved in the preconditioners P1, P2, P.
C The Y and FTY arrays contain the current values of y and f(t,y),
C respectively, and YSV also contains the current value of y.
C The array V is work space of length NEQ.
C JAC must multiply all computed Jacobian elements by the scalar
C -HL0, add the identity matrix, and do any factorization
C operations called for, in preparation for solving linear systems
C with a coefficient matrix of P1, P2, or P. The matrix P1*P2 or P
C should be an approximation to identity - HL0 * (df/dy).
C JAC should return IER = 0 if successful, and IER .ne. 0 if not.
C (If IER .ne. 0, a smaller time step will be tried.)
C
C SUBROUTINE PSOL (NEQ, T, Y, FTY, WK, HL0, WP, IWP, B, LR, IER)
C DOUBLE PRECISION T, Y(*), FTY(*), WK(*), HL0, WP(*), B(*)
C INTEGER IWP(*)
C This routine must solve a linear system with B as right-hand
C side and one of the preconditioning matrices, P1, P2, or P, as
C coefficient matrix, and return the solution vector in B.
C LR is a flag concerning left vs right preconditioning, input
C to PSOL. PSOL is to use P1 if LR = 1 and P2 if LR = 2.
C In the case of the PCG or PCGS method, LR will be 3, and PSOL
C should solve the system P*x = B with the preconditioner matrix P.
C In the case MF = 29 (no Krylov iteration), LR will be 0,
C and PSOL is to return in B the desired approximate solution
C to A * x = B, where A = identity - HL0 * (df/dy).
C PSOL can use data generated in the JAC routine and stored in
C WP and IWP. WK is a work array of length NEQ.
C The argument HL0 is the current value of the scalar appearing
C in the linear system. If the old value, at the time of the last
C JAC call, is needed, it must have been saved by JAC in WP.
C On return, PSOL should set the error flag IER as follows:
C IER = 0 if PSOL was successful,
C IER .gt. 0 if a recoverable error occurred, meaning that the
C time step will be retried,
C IER .lt. 0 if an unrecoverable error occurred, meaning that the
C solver is to stop immediately.
C
C D. Write a main program which calls Subroutine DLSODPK once for
C each point at which answers are desired. This should also provide
C for possible use of logical unit 6 for output of error messages by
C DLSODPK. On the first call to DLSODPK, supply arguments as follows:
C F = name of subroutine for right-hand side vector f.
C This name must be declared External in calling program.
C NEQ = number of first order ODEs.
C Y = array of initial values, of length NEQ.
C T = the initial value of the independent variable.
C TOUT = first point where output is desired (.ne. T).
C ITOL = 1 or 2 according as ATOL (below) is a scalar or array.
C RTOL = relative tolerance parameter (scalar).
C ATOL = absolute tolerance parameter (scalar or array).
C the estimated local error in y(i) will be controlled so as
C to be roughly less (in magnitude) than
C EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
C EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
C Thus the local error test passes if, in each component,
C either the absolute error is less than ATOL (or ATOL(i)),
C or the relative error is less than RTOL.
C Use RTOL = 0.0 for pure absolute error control, and
C use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative error
C control. Caution: Actual (global) errors may exceed these
C local tolerances, so choose them conservatively.
C ITASK = 1 for normal computation of output values of y at t = TOUT.
C ISTATE = integer flag (input and output). Set ISTATE = 1.
C IOPT = 0 to indicate no optional inputs used.
C RWORK = real work array of length at least:
C 20 + 16*NEQ for MF = 10,
C 45 + 17*NEQ + LWP for MF = 21,
C 61 + 17*NEQ + LWP for MF = 22,
C 20 + 15*NEQ + LWP for MF = 23 or 24,
C 20 + 12*NEQ + LWP for MF = 29.
C LRW = declared length of RWORK (in user's dimension).
C IWORK = integer work array of length at least:
C 30 for MF = 10,
C 35 + LIWP for MF = 21,
C 30 + LIWP for MF = 22, 23, 24, or 29.
C LIW = declared length of IWORK (in user's dimension).
C JAC,PSOL = names of subroutines for preconditioning.
C These names must be declared External in the calling program.
C MF = method flag. Standard values are:
C 10 for nonstiff (Adams) method.
C 21 for stiff (BDF) method, with preconditioned SIOM.
C 22 for stiff method, with preconditioned GMRES method.
C 23 for stiff method, with preconditioned CG method.
C 24 for stiff method, with scaled preconditioned CG method.
C 29 for stiff method, with user's PSOL routine only.
C Note that the main program must declare arrays Y, RWORK, IWORK,
C and possibly ATOL.
C
C E. The output from the first call (or any call) is:
C Y = array of computed values of y(t) vector.
C T = corresponding value of independent variable (normally TOUT).
C ISTATE = 2 if DLSODPK was successful, negative otherwise.
C -1 means excess work done on this call (perhaps wrong MF).
C -2 means excess accuracy requested (tolerances too small).
C -3 means illegal input detected (see printed message).
C -4 means repeated error test failures (check all inputs).
C -5 means repeated convergence failures (perhaps bad JAC
C or PSOL routine supplied or wrong choice of MF or
C tolerances, or this solver is inappropriate).
C -6 means error weight became zero during problem. (Solution
C component i vanished, and ATOL or ATOL(i) = 0.)
C -7 means an unrecoverable error occurred in PSOL.
C
C F. To continue the integration after a successful return, simply
C reset TOUT and call DLSODPK again. No other parameters need be reset.
C
C-----------------------------------------------------------------------
C-----------------------------------------------------------------------
C Full Description of User Interface to DLSODPK.
C
C The user interface to DLSODPK consists of the following parts.
C
C 1. The call sequence to Subroutine DLSODPK, which is a driver
C routine for the solver. This includes descriptions of both
C the call sequence arguments and of user-supplied routines.
C Following these descriptions is a description of
C optional inputs available through the call sequence, and then
C a description of optional outputs (in the work arrays).
C
C 2. Descriptions of other routines in the DLSODPK package that may be
C (optionally) called by the user. These provide the ability to
C alter error message handling, save and restore the internal
C Common, and obtain specified derivatives of the solution y(t).
C
C 3. Descriptions of Common blocks to be declared in overlay
C or similar environments, or to be saved when doing an interrupt
C of the problem and continued solution later.
C
C 4. Description of two routines in the DLSODPK package, either of
C which the user may replace with his/her own version, if desired.
C These relate to the measurement of errors.
C
C-----------------------------------------------------------------------
C Part 1. Call Sequence.
C
C The call sequence parameters used for input only are
C F, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK, IOPT, LRW, LIW, JAC, PSOL, MF,
C and those used for both input and output are
C Y, T, ISTATE.
C The work arrays RWORK and IWORK are also used for conditional and
C optional inputs and optional outputs. (The term output here refers
C to the return from Subroutine DLSODPK to the user's calling program.)
C
C The legality of input parameters will be thoroughly checked on the
C initial call for the problem, but not checked thereafter unless a
C change in input parameters is flagged by ISTATE = 3 on input.
C
C The descriptions of the call arguments are as follows.
C
C F = the name of the user-supplied subroutine defining the
C ODE system. The system must be put in the first-order
C form dy/dt = f(t,y), where f is a vector-valued function
C of the scalar t and the vector y. Subroutine F is to
C compute the function f. It is to have the form
C SUBROUTINE F (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y(*), YDOT(*)
C where NEQ, T, and Y are input, and the array YDOT = f(t,y)
C is output. Y and YDOT are arrays of length NEQ.
C Subroutine F should not alter Y(1),...,Y(NEQ).
C F must be declared External in the calling program.
C
C Subroutine F may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in F) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y below.
C
C If quantities computed in the F routine are needed
C externally to DLSODPK, an extra call to F should be made
C for this purpose, for consistent and accurate results.
C If only the derivative dy/dt is needed, use DINTDY instead.
C
C NEQ = the size of the ODE system (number of first order
C ordinary differential equations). Used only for input.
C NEQ may be decreased, but not increased, during the problem.
C If NEQ is decreased (with ISTATE = 3 on input), the
C remaining components of Y should be left undisturbed, if
C these are to be accessed in the user-supplied subroutines.
C
C Normally, NEQ is a scalar, and it is generally referred to
C as a scalar in this user interface description. However,
C NEQ may be an array, with NEQ(1) set to the system size.
C (The DLSODPK package accesses only NEQ(1).) In either case,
C this parameter is passed as the NEQ argument in all calls
C to F, JAC, and PSOL. Hence, if it is an array, locations
C NEQ(2),... may be used to store other integer data and pass
C it to the user-supplied subroutines. Each such routine must
C include NEQ in a Dimension statement in that case.
C
C Y = a real array for the vector of dependent variables, of
C length NEQ or more. Used for both input and output on the
C first call (ISTATE = 1), and only for output on other calls.
C On the first call, Y must contain the vector of initial
C values. On output, Y contains the computed solution vector,
C evaluated at T. If desired, the Y array may be used
C for other purposes between calls to the solver.
C
C This array is passed as the Y argument in all calls to F,
C JAC, and PSOL. Hence its length may exceed NEQ, and locations
C Y(NEQ+1),... may be used to store other real data and
C pass it to the user-supplied subroutines. (The DLSODPK
C package accesses only Y(1),...,Y(NEQ).)
C
C T = the independent variable. On input, T is used only on the
C first call, as the initial point of the integration.
C On output, after each call, T is the value at which a
C computed solution y is evaluated (usually the same as TOUT).
C On an error return, T is the farthest point reached.
C
C TOUT = the next value of t at which a computed solution is desired.
C Used only for input.
C
C When starting the problem (ISTATE = 1), TOUT may be equal
C to T for one call, then should .ne. T for the next call.
C For the initial T, an input value of TOUT .ne. T is used
C in order to determine the direction of the integration
C (i.e. the algebraic sign of the step sizes) and the rough
C scale of the problem. Integration in either direction
C (forward or backward in t) is permitted.
C
C If ITASK = 2 or 5 (one-step modes), TOUT is ignored after
C the first call (i.e. the first call with TOUT .ne. T).
C Otherwise, TOUT is required on every call.
C
C If ITASK = 1, 3, or 4, the values of TOUT need not be
C monotone, but a value of TOUT which backs up is limited
C to the current internal T interval, whose endpoints are
C TCUR - HU and TCUR (see optional outputs, below, for
C TCUR and HU).
C
C ITOL = an indicator for the type of error control. See
C description below under ATOL. Used only for input.
C
C RTOL = a relative error tolerance parameter, either a scalar or
C an array of length NEQ. See description below under ATOL.
C Input only.
C
C ATOL = an absolute error tolerance parameter, either a scalar or
C an array of length NEQ. Input only.
C
C The input parameters ITOL, RTOL, and ATOL determine
C the error control performed by the solver. The solver will
C control the vector E = (E(i)) of estimated local errors
C in y, according to an inequality of the form
C RMS-norm of ( E(i)/EWT(i) ) .le. 1,
C where EWT(i) = RTOL(i)*ABS(Y(i)) + ATOL(i),
C and the RMS-norm (root-mean-square norm) here is
C RMS-norm(v) = SQRT(sum v(i)**2 / NEQ). Here EWT = (EWT(i))
C is a vector of weights which must always be positive, and
C the values of RTOL and ATOL should all be non-negative.
C the following table gives the types (scalar/array) of
C RTOL and ATOL, and the corresponding form of EWT(i).
C
C ITOL RTOL ATOL EWT(i)
C 1 scalar scalar RTOL*ABS(Y(i)) + ATOL
C 2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
C 3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
C 4 array array RTOL(i)*ABS(Y(i)) + ATOL(i)
C
C When either of these parameters is a scalar, it need not
C be dimensioned in the user's calling program.
C
C If none of the above choices (with ITOL, RTOL, and ATOL
C fixed throughout the problem) is suitable, more general
C error controls can be obtained by substituting
C user-supplied routines for the setting of EWT and/or for
C the norm calculation. See Part 4 below.
C
C If global errors are to be estimated by making a repeated
C run on the same problem with smaller tolerances, then all
C components of RTOL and ATOL (i.e. of EWT) should be scaled
C down uniformly.
C
C ITASK = an index specifying the task to be performed.
C Input only. ITASK has the following values and meanings.
C 1 means normal computation of output values of y(t) at
C t = TOUT (by overshooting and interpolating).
C 2 means take one step only and return.
C 3 means stop at the first internal mesh point at or
C beyond t = TOUT and return.
C 4 means normal computation of output values of y(t) at
C t = TOUT but without overshooting t = TCRIT.
C TCRIT must be input as RWORK(1). TCRIT may be equal to
C or beyond TOUT, but not behind it in the direction of
C integration. This option is useful if the problem
C has a singularity at or beyond t = TCRIT.
C 5 means take one step, without passing TCRIT, and return.
C TCRIT must be input as RWORK(1).
C
C Note: If ITASK = 4 or 5 and the solver reaches TCRIT
C (within roundoff), it will return T = TCRIT (exactly) to
C indicate this (unless ITASK = 4 and TOUT comes before TCRIT,
C in which case answers at t = TOUT are returned first).
C
C ISTATE = an index used for input and output to specify the
C the state of the calculation.
C
C On input, the values of ISTATE are as follows.
C 1 means this is the first call for the problem
C (initializations will be done). See note below.
C 2 means this is not the first call, and the calculation
C is to continue normally, with no change in any input
C parameters except possibly TOUT and ITASK.
C (If ITOL, RTOL, and/or ATOL are changed between calls
C with ISTATE = 2, the new values will be used but not
C tested for legality.)
C 3 means this is not the first call, and the
C calculation is to continue normally, but with
C a change in input parameters other than
C TOUT and ITASK. Changes are allowed in
C NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, MF,
C and any of the optional inputs except H0.
C Note: A preliminary call with TOUT = T is not counted
C as a first call here, as no initialization or checking of
C input is done. (Such a call is sometimes useful for the
C purpose of outputting the initial conditions.)
C Thus the first call for which TOUT .ne. T requires
C ISTATE = 1 on input.
C
C On output, ISTATE has the following values and meanings.
C 1 means nothing was done; TOUT = T and ISTATE = 1 on input.
C 2 means the integration was performed successfully.
C -1 means an excessive amount of work (more than MXSTEP
C steps) was done on this call, before completing the
C requested task, but the integration was otherwise
C successful as far as T. (MXSTEP is an optional input
C and is normally 500.) To continue, the user may
C simply reset ISTATE to a value .gt. 1 and call again
C (the excess work step counter will be reset to 0).
C In addition, the user may increase MXSTEP to avoid
C this error return (see below on optional inputs).
C -2 means too much accuracy was requested for the precision
C of the machine being used. This was detected before
C completing the requested task, but the integration
C was successful as far as T. To continue, the tolerance
C parameters must be reset, and ISTATE must be set
C to 3. The optional output TOLSF may be used for this
C purpose. (Note: If this condition is detected before
C taking any steps, then an illegal input return
C (ISTATE = -3) occurs instead.)
C -3 means illegal input was detected, before taking any
C integration steps. See written message for details.
C Note: If the solver detects an infinite loop of calls
C to the solver with illegal input, it will cause
C the run to stop.
C -4 means there were repeated error test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C The problem may have a singularity, or the input
C may be inappropriate.
C -5 means there were repeated convergence test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C -6 means EWT(i) became zero for some i during the
C integration. Pure relative error control (ATOL(i)=0.0)
C was requested on a variable which has now vanished.
C The integration was successful as far as T.
C -7 means the PSOL routine returned an unrecoverable error
C flag (IER .lt. 0). The integration was successful as
C far as T.
C
C Note: since the normal output value of ISTATE is 2,
C it does not need to be reset for normal continuation.
C Also, since a negative input value of ISTATE will be
C regarded as illegal, a negative output value requires the
C user to change it, and possibly other inputs, before
C calling the solver again.
C
C IOPT = an integer flag to specify whether or not any optional
C inputs are being used on this call. Input only.
C The optional inputs are listed separately below.
C IOPT = 0 means no optional inputs are being used.
C Default values will be used in all cases.
C IOPT = 1 means one or more optional inputs are being used.
C
C RWORK = a real working array (double precision).
C The length of RWORK must be at least
C 20 + NYH*(MAXORD + 1) + 3*NEQ + LENLS + LWP where
C NYH = the initial value of NEQ,
C MAXORD = 12 (if METH = 1) or 5 (if METH = 2) (unless a
C smaller value is given as an optional input),
C LENLS = length of work space for linear system (Krylov)
C method, excluding preconditioning:
C LENLS = 0 if MITER = 0,
C LENLS = NEQ*(MAXL+3) + MAXL**2 if MITER = 1,
C LENLS = NEQ*(MAXL+3+MIN(1,MAXL-KMP))
C + (MAXL+3)*MAXL + 1 if MITER = 2,
C LENLS = 6*NEQ if MITER = 3 or 4,
C LENLS = 3*NEQ if MITER = 9.
C (See the MF description for METH and MITER, and the
C list of optional inputs for MAXL and KMP.)
C LWP = length of real user work space for preconditioning
C (see JAC/PSOL).
C Thus if default values are used and NEQ is constant,
C this length is:
C 20 + 16*NEQ for MF = 10,
C 45 + 24*NEQ + LWP FOR MF = 11,
C 61 + 24*NEQ + LWP FOR MF = 12,
C 20 + 22*NEQ + LWP FOR MF = 13 OR 14,
C 20 + 19*NEQ + LWP FOR MF = 19,
C 20 + 9*NEQ FOR MF = 20,
C 45 + 17*NEQ + LWP FOR MF = 21,
C 61 + 17*NEQ + LWP FOR MF = 22,
C 20 + 15*NEQ + LWP FOR MF = 23 OR 24,
C 20 + 12*NEQ + LWP for MF = 29.
C The first 20 words of RWORK are reserved for conditional
C and optional inputs and optional outputs.
C
C The following word in RWORK is a conditional input:
C RWORK(1) = TCRIT = critical value of t which the solver
C is not to overshoot. Required if ITASK is
C 4 or 5, and ignored otherwise. (See ITASK.)
C
C LRW = the length of the array RWORK, as declared by the user.
C (This will be checked by the solver.)
C
C IWORK = an integer work array. The length of IWORK must be at least
C 30 if MITER = 0 (MF = 10 or 20),
C 30 + MAXL + LIWP if MITER = 1 (MF = 11, 21),
C 30 + LIWP if MITER = 2, 3, 4, or 9.
C MAXL = 5 unless a different optional input value is given.
C LIWP = length of integer user work space for preconditioning
C (see conditional input list following).
C The first few words of IWORK are used for conditional and
C optional inputs and optional outputs.
C
C The following 4 words in IWORK are conditional inputs,
C required if MITER .ge. 1:
C IWORK(1) = LWP = length of real array WP for use in
C preconditioning (part of RWORK array).
C IWORK(2) = LIWP = length of integer array IWP for use in
C preconditioning (part of IWORK array).
C The arrays WP and IWP are work arrays under the
C user's control, for use in the routines that
C perform preconditioning operations (JAC and PSOL).
C IWORK(3) = JPRE = preconditioner type flag:
C = 0 for no preconditioning (P1 = P2 = P = identity)
C = 1 for left-only preconditioning (P2 = identity)
C = 2 for right-only preconditioning (P1 = identity)
C = 3 for two-sided preconditioning (and PCG or PCGS)
C IWORK(4) = JACFLG = flag for whether JAC is called.
C = 0 if JAC is not to be called,
C = 1 if JAC is to be called.
C Use JACFLG = 1 if JAC computes any nonconstant
C data needed in preconditioning operations,
C such as some of the Jacobian elements.
C
C LIW = the length of the array IWORK, as declared by the user.
C (This will be checked by the solver.)
C
C Note: The work arrays must not be altered between calls to DLSODPK
C for the same problem, except possibly for the conditional and
C optional inputs, and except for the last 3*NEQ words of RWORK.
C The latter space is used for internal scratch space, and so is
C available for use by the user outside DLSODPK between calls, if
C desired (but not for use by any of the user-supplied subroutines).
C
C JAC = the name of the user-supplied routine to compute any
C Jacobian elements (or approximations) involved in the
C matrix preconditioning operations (MITER .ge. 1).
C It is to have the form
C SUBROUTINE JAC (F, NEQ, T, Y, YSV, REWT, FTY, V,
C 1 HL0, WP, IWP, IER)
C DOUBLE PRECISION T, Y(*),YSV(*), REWT(*), FTY(*), V(*),
C 1 HL0, WP(*)
C INTEGER IWP(*)
C This routine must evaluate and preprocess any parts of the
C Jacobian matrix df/dy used in the preconditioners P1, P2, P.
C the Y and FTY arrays contain the current values of y and
C f(t,y), respectively, and YSV also contains the current
C value of y. The array V is work space of length
C NEQ for use by JAC. REWT is the array of reciprocal error
C weights (1/EWT). JAC must multiply all computed Jacobian
C elements by the scalar -HL0, add the identity matrix, and do
C any factorization operations called for, in preparation
C for solving linear systems with a coefficient matrix of
C P1, P2, or P. The matrix P1*P2 or P should be an
C approximation to identity - HL0 * (df/dy). JAC should
C return IER = 0 if successful, and IER .ne. 0 if not.
C (If IER .ne. 0, a smaller time step will be tried.)
C The arrays WP (of length LWP) and IWP (of length LIWP)
C are for use by JAC and PSOL for work space and for storage
C of data needed for the solution of the preconditioner
C linear systems. Their lengths and contents are under the
C user's control.
C The JAC routine may save relevant Jacobian elements (or
C approximations) used in the preconditioners, along with the
C value of HL0, and use these to reconstruct preconditioner
C matrices later without reevaluationg those elements.
C This may be cost-effective if JAC is called with HL0
C considerably different from its earlier value, indicating
C that a corrector convergence failure has occurred because
C of the change in HL0, not because of changes in the
C value of the Jacobian. In doing this, use the saved and
C current values of HL0 to decide whether to use saved
C or reevaluated elements.
C JAC may alter V, but may not alter Y, YSV, REWT, FTY, or HL0.
C JAC must be declared External in the calling program.
C Subroutine JAC may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in JAC) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y above.
C
C PSOL = the name of the user-supplied routine for the
C solution of preconditioner linear systems.
C It is to have the form
C SUBROUTINE PSOL (NEQ, T, Y, FTY, WK,HL0, WP,IWP, B, LR,IER)
C DOUBLE PRECISION T, Y(*), FTY(*), WK(*), HL0, WP(*), B(*)
C INTEGER IWP(*)
C This routine must solve a linear system with B as right-hand
C side and one of the preconditioning matrices, P1, P2, or P,
C as coefficient matrix, and return the solution vector in B.
C LR is a flag concerning left vs right preconditioning, input
C to PSOL. PSOL is to use P1 if LR = 1 and P2 if LR = 2.
C In the case of the PCG or PCGS method, LR will be 3, and PSOL
C should solve the system P*x = B with the preconditioner P.
C In the case MITER = 9 (no Krylov iteration), LR will be 0,
C and PSOL is to return in B the desired approximate solution
C to A * x = B, where A = identity - HL0 * (df/dy).
C PSOL can use data generated in the JAC routine and stored in
C WP and IWP.
C The Y and FTY arrays contain the current values of y and
C f(t,y), respectively. The array WK is work space of length
C NEQ for use by PSOL.
C The argument HL0 is the current value of the scalar appearing
C in the linear system. If the old value, as of the last
C JAC call, is needed, it must have been saved by JAC in WP.
C On return, PSOL should set the error flag IER as follows:
C IER = 0 if PSOL was successful,
C IER .gt. 0 on a recoverable error, meaning that the
C time step will be retried,
C IER .lt. 0 on an unrecoverable error, meaning that the
C solver is to stop immediately.
C PSOL may not alter Y, FTY, or HL0.
C PSOL must be declared External in the calling program.
C Subroutine PSOL may access user-defined quantities in
C NEQ(2),... and Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in PSOL) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y above.
C
C MF = the method flag. Used only for input. The legal values of
C MF are 10, 11, 12, 13, 14, 19, 20, 21, 22, 23, 24, and 29.
C MF has decimal digits METH and MITER: MF = 10*METH + MITER.
C METH indicates the basic linear multistep method:
C METH = 1 means the implicit Adams method.
C METH = 2 means the method based on Backward
C Differentiation Formulas (BDFs).
C MITER indicates the corrector iteration method:
C MITER = 0 means functional iteration (no linear system
C is involved).
C MITER = 1 means Newton iteration with Scaled Preconditioned
C Incomplete Orthogonalization Method (SPIOM)
C for the linear systems.
C MITER = 2 means Newton iteration with Scaled Preconditioned
C Generalized Minimal Residual method (SPIGMR)
C for the linear systems.
C MITER = 3 means Newton iteration with Preconditioned
C Conjugate Gradient method (PCG)
C for the linear systems.
C MITER = 4 means Newton iteration with scaled Preconditioned
C Conjugate Gradient method (PCGS)
C for the linear systems.
C MITER = 9 means Newton iteration with only the
C user-supplied PSOL routine called (no Krylov
C iteration) for the linear systems.
C JPRE is ignored, and PSOL is called with LR = 0.
C See comments in the introduction about the choice of MITER.
C If MITER .ge. 1, the user must supply routines JAC and PSOL
C (the names are arbitrary) as described above.
C For MITER = 0, dummy arguments can be used.
C-----------------------------------------------------------------------
C Optional Inputs.
C
C The following is a list of the optional inputs provided for in the
C call sequence. (See also Part 2.) For each such input variable,
C this table lists its name as used in this documentation, its
C location in the call sequence, its meaning, and the default value.
C The use of any of these inputs requires IOPT = 1, and in that
C case all of these inputs are examined. A value of zero for any
C of these optional inputs will cause the default value to be used.
C Thus to use a subset of the optional inputs, simply preload
C locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively, and
C then set those of interest to nonzero values.
C
C Name Location Meaning and Default Value
C
C H0 RWORK(5) the step size to be attempted on the first step.
C The default value is determined by the solver.
C
C HMAX RWORK(6) the maximum absolute step size allowed.
C The default value is infinite.
C
C HMIN RWORK(7) the minimum absolute step size allowed.
C The default value is 0. (This lower bound is not
C enforced on the final step before reaching TCRIT
C when ITASK = 4 or 5.)
C
C DELT RWORK(8) convergence test constant in Krylov iteration
C algorithm. The default is .05.
C
C MAXORD IWORK(5) the maximum order to be allowed. The default
C value is 12 if METH = 1, and 5 if METH = 2.
C If MAXORD exceeds the default value, it will
C be reduced to the default value.
C If MAXORD is changed during the problem, it may
C cause the current order to be reduced.
C
C MXSTEP IWORK(6) maximum number of (internally defined) steps
C allowed during one call to the solver.
C The default value is 500.
C
C MXHNIL IWORK(7) maximum number of messages printed (per problem)
C warning that T + H = T on a step (H = step size).
C This must be positive to result in a non-default
C value. The default value is 10.
C
C MAXL IWORK(8) maximum number of iterations in the SPIOM, SPIGMR,
C PCG, or PCGS algorithm (.le. NEQ).
C The default is MAXL = MIN(5,NEQ).
C
C KMP IWORK(9) number of vectors on which orthogonalization
C is done in SPIOM or SPIGMR algorithm (.le. MAXL).
C The default is KMP = MAXL.
C Note: When KMP .lt. MAXL and MF = 22, the length
C of RWORK must be defined accordingly. See
C the definition of RWORK above.
C-----------------------------------------------------------------------
C Optional Outputs.
C
C As optional additional output from DLSODPK, the variables listed
C below are quantities related to the performance of DLSODPK
C which are available to the user. These are communicated by way of
C the work arrays, but also have internal mnemonic names as shown.
C Except where stated otherwise, all of these outputs are defined
C on any successful return from DLSODPK, and on any return with
C ISTATE = -1, -2, -4, -5, -6, or -7. On an illegal input return
C (ISTATE = -3), they will be unchanged from their existing values
C (if any), except possibly for TOLSF, LENRW, and LENIW.
C On any error return, outputs relevant to the error will be defined,
C as noted below.
C
C Name Location Meaning
C
C HU RWORK(11) the step size in t last used (successfully).
C
C HCUR RWORK(12) the step size to be attempted on the next step.
C
C TCUR RWORK(13) the current value of the independent variable
C which the solver has actually reached, i.e. the
C current internal mesh point in t. On output, TCUR
C will always be at least as far as the argument
C T, but may be farther (if interpolation was done).
C
C TOLSF RWORK(14) a tolerance scale factor, greater than 1.0,
C computed when a request for too much accuracy was
C detected (ISTATE = -3 if detected at the start of
C the problem, ISTATE = -2 otherwise). If ITOL is
C left unaltered but RTOL and ATOL are uniformly
C scaled up by a factor of TOLSF for the next call,
C then the solver is deemed likely to succeed.
C (The user may also ignore TOLSF and alter the
C tolerance parameters in any other way appropriate.)
C
C NST IWORK(11) the number of steps taken for the problem so far.
C
C NFE IWORK(12) the number of f evaluations for the problem so far.
C
C NPE IWORK(13) the number of calls to JAC so far (for Jacobian
C evaluation associated with preconditioning).
C
C NQU IWORK(14) the method order last used (successfully).
C
C NQCUR IWORK(15) the order to be attempted on the next step.
C
C IMXER IWORK(16) the index of the component of largest magnitude in
C the weighted local error vector ( E(i)/EWT(i) ),
C on an error return with ISTATE = -4 or -5.
C
C LENRW IWORK(17) the length of RWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C LENIW IWORK(18) the length of IWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C NNI IWORK(19) number of nonlinear iterations so far (each of
C which calls an iterative linear solver).
C
C NLI IWORK(20) number of linear iterations so far.
C Note: A measure of the success of algorithm is
C the average number of linear iterations per
C nonlinear iteration, given by NLI/NNI.
C If this is close to MAXL, MAXL may be too small.
C
C NPS IWORK(21) number of preconditioning solve operations
C (PSOL calls) so far.
C
C NCFN IWORK(22) number of convergence failures of the nonlinear
C (Newton) iteration so far.
C Note: A measure of success is the overall
C rate of nonlinear convergence failures, NCFN/NST.
C
C NCFL IWORK(23) number of convergence failures of the linear
C iteration so far.
C Note: A measure of success is the overall
C rate of linear convergence failures, NCFL/NNI.
C
C The following two arrays are segments of the RWORK array which
C may also be of interest to the user as optional outputs.
C For each array, the table below gives its internal name,
C its base address in RWORK, and its description.
C
C Name Base Address Description
C
C YH 21 the Nordsieck history array, of size NYH by
C (NQCUR + 1), where NYH is the initial value
C of NEQ. For j = 0,1,...,NQCUR, column j+1
C of YH contains HCUR**j/factorial(j) times
C the j-th derivative of the interpolating
C polynomial currently representing the solution,
C evaluated at t = TCUR.
C
C ACOR LENRW-NEQ+1 array of size NEQ used for the accumulated
C corrections on each step, scaled on output
C to represent the estimated local error in y
C on the last step. This is the vector E in
C the description of the error control. It is
C defined only on a successful return from
C DLSODPK.
C
C-----------------------------------------------------------------------
C Part 2. Other Routines Callable.
C
C The following are optional calls which the user may make to
C gain additional capabilities in conjunction with DLSODPK.
C (The routines XSETUN and XSETF are designed to conform to the
C SLATEC error handling package.)
C
C Form of Call Function
C CALL XSETUN(LUN) Set the logical unit number, LUN, for
C output of messages from DLSODPK, if
C the default is not desired.
C The default value of lun is 6.
C
C CALL XSETF(MFLAG) Set a flag to control the printing of
C messages by DLSODPK.
C MFLAG = 0 means do not print. (Danger:
C This risks losing valuable information.)
C MFLAG = 1 means print (the default).
C
C Either of the above calls may be made at
C any time and will take effect immediately.
C
C CALL DSRCPK(RSAV,ISAV,JOB) saves and restores the contents of
C the internal Common blocks used by
C DLSODPK (see Part 3 below).
C RSAV must be a real array of length 222
C or more, and ISAV must be an integer
C array of length 50 or more.
C JOB=1 means save Common into RSAV/ISAV.
C JOB=2 means restore Common from RSAV/ISAV.
C DSRCPK is useful if one is
C interrupting a run and restarting
C later, or alternating between two or
C more problems solved with DLSODPK.
C
C CALL DINTDY(,,,,,) Provide derivatives of y, of various
C (See below) orders, at a specified point t, if
C desired. It may be called only after
C a successful return from DLSODPK.
C
C The detailed instructions for using DINTDY are as follows.
C The form of the call is:
C
C CALL DINTDY (T, K, RWORK(21), NYH, DKY, IFLAG)
C
C The input parameters are:
C
C T = value of independent variable where answers are desired
C (normally the same as the T last returned by DLSODPK).
C for valid results, T must lie between TCUR - HU and TCUR.
C (See optional outputs for TCUR and HU.)
C K = integer order of the derivative desired. K must satisfy
C 0 .le. K .le. NQCUR, where NQCUR is the current order
C (see optional outputs). The capability corresponding
C to K = 0, i.e. computing y(T), is already provided
C by DLSODPK directly. Since NQCUR .ge. 1, the first
C derivative dy/dt is always available with DINTDY.
C RWORK(21) = the base address of the history array YH.
C NYH = column length of YH, equal to the initial value of NEQ.
C
C The output parameters are:
C
C DKY = a real array of length NEQ containing the computed value
C of the K-th derivative of y(t).
C IFLAG = integer flag, returned as 0 if K and T were legal,
C -1 if K was illegal, and -2 if T was illegal.
C On an error return, a message is also written.
C-----------------------------------------------------------------------
C Part 3. Common Blocks.
C
C If DLSODPK is to be used in an overlay situation, the user
C must declare, in the primary overlay, the variables in:
C (1) the call sequence to DLSODPK, and
C (2) the two internal Common blocks
C /DLS001/ of length 255 (218 double precision words
C followed by 37 integer words),
C /DLPK01/ of length 17 (4 double precision words
C followed by 13 integer words).
C
C If DLSODPK is used on a system in which the contents of internal
C Common blocks are not preserved between calls, the user should
C declare the above Common blocks in the calling program to insure
C that their contents are preserved.
C
C If the solution of a given problem by DLSODPK is to be interrupted
C and then later continued, such as when restarting an interrupted run
C or alternating between two or more problems, the user should save,
C following the return from the last DLSODPK call prior to the
C interruption, the contents of the call sequence variables and the
C internal Common blocks, and later restore these values before the
C next DLSODPK call for that problem. To save and restore the Common
C blocks, use Subroutine DSRCPK (see Part 2 above).
C
C-----------------------------------------------------------------------
C Part 4. Optionally Replaceable Solver Routines.
C
C below are descriptions of two routines in the DLSODPK package which
C relate to the measurement of errors. Either routine can be
C replaced by a user-supplied version, if desired. However, since such
C a replacement may have a major impact on performance, it should be
C done only when absolutely necessary, and only with great caution.
C (Note: The means by which the package version of a routine is
C superseded by the user's version may be system-dependent.)
C
C (a) DEWSET.
C The following subroutine is called just before each internal
C integration step, and sets the array of error weights, EWT, as
C described under ITOL/RTOL/ATOL above:
C SUBROUTINE DEWSET (NEQ, ITOL, RTOL, ATOL, YCUR, EWT)
C where NEQ, ITOL, RTOL, and ATOL are as in the DLSODPK call sequence,
C YCUR contains the current dependent variable vector, and
C EWT is the array of weights set by DEWSET.
C
C If the user supplies this subroutine, it must return in EWT(i)
C (i = 1,...,NEQ) a positive quantity suitable for comparing errors
C in y(i) to. The EWT array returned by DEWSET is passed to the DVNORM
C routine (see below), and also used by DLSODPK in the computation
C of the optional output IMXER, the diagonal Jacobian approximation,
C and the increments for difference quotient Jacobians.
C
C In the user-supplied version of DEWSET, it may be desirable to use
C the current values of derivatives of y. Derivatives up to order NQ
C are available from the history array YH, described above under
C optional outputs. In DEWSET, YH is identical to the YCUR array,
C extended to NQ + 1 columns with a column length of NYH and scale
C factors of H**j/factorial(j). On the first call for the problem,
C given by NST = 0, NQ is 1 and H is temporarily set to 1.0.
C NYH is the initial value of NEQ. The quantities NQ, H, and NST
C can be obtained by including in DEWSET the statements:
C DOUBLE PRECISION RLS
C COMMON /DLS001/ RLS(218),ILS(37)
C NQ = ILS(33)
C NST = ILS(34)
C H = RLS(212)
C Thus, for example, the current value of dy/dt can be obtained as
C YCUR(NYH+i)/H (i=1,...,NEQ) (and the division by H is
C unnecessary when NST = 0).
C
C (b) DVNORM.
C The following is a real function routine which computes the weighted
C root-mean-square norm of a vector v:
C D = DVNORM (N, V, W)
C where:
C N = the length of the vector,
C V = real array of length N containing the vector,
C W = real array of length N containing weights,
C D = SQRT( (1/N) * sum(V(i)*W(i))**2 ).
C DVNORM is called with N = NEQ and with W(i) = 1.0/EWT(i), where
C EWT is as set by Subroutine DEWSET.
C
C If the user supplies this function, it should return a non-negative
C value of DVNORM suitable for use in the error control in DLSODPK.
C None of the arguments should be altered by DVNORM.
C For example, a user-supplied DVNORM routine might:
C -substitute a max-norm of (V(i)*W(i)) for the RMS-norm, or
C -ignore some components of V in the norm, with the effect of
C suppressing the error control on those components of y.
C-----------------------------------------------------------------------
C
C***REVISION HISTORY (YYYYMMDD)
C 19860901 DATE WRITTEN
C 19861010 Numerous minor revisions to SPIOM and SPGMR routines;
C minor corrections to prologues and comments.
C 19870114 Changed name SPGMR to SPIGMR; revised residual norm
C calculation in SPIGMR (for incomplete case);
C revised error return logic in SPIGMR;
C 19870330 Major update: corrected comments throughout;
C removed TRET from Common; rewrote EWSET with 4 loops;
C fixed t test in INTDY; added Cray directives in STODPK;
C in STODPK, fixed DELP init. and logic around PJAC call;
C combined routines to save/restore Common;
C passed LEVEL = 0 in error message calls (except run abort).
C 19871130 Added option MITER = 9; shortened WM array by 2;
C revised early return from SPIOM and SPIGMR;
C replaced copy loops with SCOPY/DCOPY calls;
C minor corrections/revisions to SOLPK, SPIGMR, ATV, ATP;
C corrections to main prologue and internal comments.
C 19880304 Corrections to type declarations in SOLPK, SPIOM, USOL.
C 19891025 Added ISTATE = -7 return; minor revisions to USOL;
C added initialization of JACFLG in main driver;
C removed YH and NYH from PKSET call list;
C minor revisions to SPIOM and SPIGMR;
C corrections to main prologue and internal comments.
C 19900803 Added YSV to JAC call list; minor comment corrections.
C 20010425 Major update: convert source lines to upper case;
C added *DECK lines; changed from 1 to * in dummy dimensions;
C changed names R1MACH/D1MACH to RUMACH/DUMACH;
C renamed routines for uniqueness across single/double prec.;
C converted intrinsic names to generic form;
C removed ILLIN and NTREP (data loaded) from Common;
C removed all 'own' variables from Common;
C changed error messages to quoted strings;
C replaced XERRWV/XERRWD with 1993 revised version;
C converted prologues, comments, error messages to mixed case;
C numerous corrections to prologues and internal comments.
C 20010507 Converted single precision source to double precision.
C 20020502 Corrected declarations in descriptions of user routines.
C 20030603 Corrected duplicate type declaration for DUMACH.
C 20031105 Restored 'own' variables to Common blocks, to enable
C interrupt/restart feature.
C 20031112 Added SAVE statements for data-loaded constants.
C 20031117 Changed internal name NPE to NJE.
C
C-----------------------------------------------------------------------
C Other routines in the DLSODPK package.
C
C In addition to Subroutine DLSODPK, the DLSODPK package includes the
C following subroutines and function routines:
C DINTDY computes an interpolated value of the y vector at t = TOUT.
C DEWSET sets the error weight vector EWT before each step.
C DVNORM computes the weighted RMS-norm of a vector.
C DSTODPK is the core integrator, which does one step of the
C integration and the associated error control.
C DCFODE sets all method coefficients and test constants.
C DPKSET interfaces between DSTODPK and the JAC routine.
C DSOLPK manages solution of linear system in Newton iteration.
C DSPIOM performs the SPIOM algorithm.
C DATV computes a scaled, preconditioned product (I-hl0*J)*v.
C DORTHOG orthogonalizes a vector against previous basis vectors.
C DHEFA generates an LU factorization of a Hessenberg matrix.
C DHESL solves a Hessenberg square linear system.
C DSPIGMR performs the SPIGMR algorithm.
C DHEQR generates a QR factorization of a Hessenberg matrix.
C DHELS finds the least squares solution of a Hessenberg system.
C DPCG performs Preconditioned Conjugate Gradient algorithm (PCG).
C DPCGS performs the PCGS algorithm.
C DATP computes the product A*p, where A = I - hl0*df/dy.
C DUSOL interfaces to the user's PSOL routine (MITER = 9).
C DSRCPK is a user-callable routine to save and restore
C the contents of the internal Common blocks.
C DAXPY, DCOPY, DDOT, DNRM2, and DSCAL are basic linear
C algebra modules (from the BLAS collection).
C DUMACH computes the unit roundoff in a machine-independent manner.
C XERRWD, XSETUN, XSETF, IXSAV, and IUMACH handle the printing of all
C error messages and warnings. XERRWD is machine-dependent.
C Note: DVNORM, DDOT, DNRM2, DUMACH, IXSAV, and IUMACH are function
C routines. All the others are subroutines.
C
C-----------------------------------------------------------------------
DOUBLE PRECISION DUMACH, DVNORM
INTEGER INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT,
1 NNI, NLI, NPS, NCFN, NCFL
INTEGER I, I1, I2, IFLAG, IMXER, KGO, LF0, LENIW,
1 LENIWK, LENRW, LENWM, LENWK, LIWP, LWP, MORD, MXHNL0, MXSTP0,
2 NCFN0, NCFL0, NLI0, NNI0, NNID, NSTD, NWARN
DOUBLE PRECISION ROWNS,
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
DOUBLE PRECISION DELT, EPCON, SQRTN, RSQRTN
DOUBLE PRECISION ATOLI, AVDIM, AYI, BIG, EWTI, H0, HMAX, HMX,
1 RCFL, RCFN, RH, RTOLI, TCRIT,
2 TDIST, TNEXT, TOL, TOLSF, TP, SIZE, SUM, W0
DIMENSION MORD(2)
LOGICAL IHIT, LAVD, LCFN, LCFL, LWARN
CHARACTER*60 MSG
SAVE MORD, MXSTP0, MXHNL0
C-----------------------------------------------------------------------
C The following two internal Common blocks contain
C (a) variables which are local to any subroutine but whose values must
C be preserved between calls to the routine ("own" variables), and
C (b) variables which are communicated between subroutines.
C The block DLS001 is declared in subroutines DLSODPK, DINTDY, DSTODPK,
C DSOLPK, and DATV.
C The block DLPK01 is declared in subroutines DLSODPK, DSTODPK, DPKSET,
C and DSOLPK.
C Groups of variables are replaced by dummy arrays in the Common
C declarations in routines where those variables are not used.
C-----------------------------------------------------------------------
COMMON /DLS001/ ROWNS(209),
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
2 INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS(6),
3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
COMMON /DLPK01/ DELT, EPCON, SQRTN, RSQRTN,
1 JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT,
2 NNI, NLI, NPS, NCFN, NCFL
C
DATA MORD(1),MORD(2)/12,5/, MXSTP0/500/, MXHNL0/10/
C-----------------------------------------------------------------------
C Block A.
C This code block is executed on every call.
C It tests ISTATE and ITASK for legality and branches appropriately.
C If ISTATE .gt. 1 but the flag INIT shows that initialization has
C not yet been done, an error return occurs.
C If ISTATE = 1 and TOUT = T, return immediately.
C-----------------------------------------------------------------------
IF (ISTATE .LT. 1 .OR. ISTATE .GT. 3) GO TO 601
IF (ITASK .LT. 1 .OR. ITASK .GT. 5) GO TO 602
IF (ISTATE .EQ. 1) GO TO 10
IF (INIT .EQ. 0) GO TO 603
IF (ISTATE .EQ. 2) GO TO 200
GO TO 20
10 INIT = 0
IF (TOUT .EQ. T) RETURN
C-----------------------------------------------------------------------
C Block B.
C The next code block is executed for the initial call (ISTATE = 1),
C or for a continuation call with parameter changes (ISTATE = 3).
C It contains checking of all inputs and various initializations.
C
C First check legality of the non-optional inputs NEQ, ITOL, IOPT, MF.
C-----------------------------------------------------------------------
20 IF (NEQ(1) .LE. 0) GO TO 604
IF (ISTATE .EQ. 1) GO TO 25
IF (NEQ(1) .GT. N) GO TO 605
25 N = NEQ(1)
IF (ITOL .LT. 1 .OR. ITOL .GT. 4) GO TO 606
IF (IOPT .LT. 0 .OR. IOPT .GT. 1) GO TO 607
METH = MF/10
MITER = MF - 10*METH
IF (METH .LT. 1 .OR. METH .GT. 2) GO TO 608
IF (MITER .LT. 0) GO TO 608
IF (MITER .GT. 4 .AND. MITER .LT. 9) GO TO 608
IF (MITER .GE. 1) JPRE = IWORK(3)
JACFLG = 0
IF (MITER .GE. 1) JACFLG = IWORK(4)
C Next process and check the optional inputs. --------------------------
IF (IOPT .EQ. 1) GO TO 40
MAXORD = MORD(METH)
MXSTEP = MXSTP0
MXHNIL = MXHNL0
IF (ISTATE .EQ. 1) H0 = 0.0D0
HMXI = 0.0D0
HMIN = 0.0D0
MAXL = MIN(5,N)
KMP = MAXL
DELT = 0.05D0
GO TO 60
40 MAXORD = IWORK(5)
IF (MAXORD .LT. 0) GO TO 611
IF (MAXORD .EQ. 0) MAXORD = 100
MAXORD = MIN(MAXORD,MORD(METH))
MXSTEP = IWORK(6)
IF (MXSTEP .LT. 0) GO TO 612
IF (MXSTEP .EQ. 0) MXSTEP = MXSTP0
MXHNIL = IWORK(7)
IF (MXHNIL .LT. 0) GO TO 613
IF (MXHNIL .EQ. 0) MXHNIL = MXHNL0
IF (ISTATE .NE. 1) GO TO 50
H0 = RWORK(5)
IF ((TOUT - T)*H0 .LT. 0.0D0) GO TO 614
50 HMAX = RWORK(6)
IF (HMAX .LT. 0.0D0) GO TO 615
HMXI = 0.0D0
IF (HMAX .GT. 0.0D0) HMXI = 1.0D0/HMAX
HMIN = RWORK(7)
IF (HMIN .LT. 0.0D0) GO TO 616
MAXL = IWORK(8)
IF (MAXL .EQ. 0) MAXL = 5
MAXL = MIN(MAXL,N)
KMP = IWORK(9)
IF (KMP .EQ. 0 .OR. KMP .GT. MAXL) KMP = MAXL
DELT = RWORK(8)
IF (DELT .EQ. 0.0D0) DELT = 0.05D0
C-----------------------------------------------------------------------
C Set work array pointers and check lengths LRW and LIW.
C Pointers to segments of RWORK and IWORK are named by prefixing L to
C the name of the segment. E.g., the segment YH starts at RWORK(LYH).
C RWORK segments (in order) are denoted YH, WM, EWT, SAVF, SAVX, ACOR.
C-----------------------------------------------------------------------
60 LYH = 21
IF (ISTATE .EQ. 1) NYH = N
LWM = LYH + (MAXORD + 1)*NYH
IF (MITER .EQ. 0) LENWK = 0
IF (MITER .EQ. 1) LENWK = N*(MAXL+2) + MAXL*MAXL
IF (MITER .EQ. 2)
1 LENWK = N*(MAXL+2+MIN(1,MAXL-KMP)) + (MAXL+3)*MAXL + 1
IF (MITER .EQ. 3 .OR. MITER .EQ. 4) LENWK = 5*N
IF (MITER .EQ. 9) LENWK = 2*N
LWP = 0
IF (MITER .GE. 1) LWP = IWORK(1)
LENWM = LENWK + LWP
LOCWP = LENWK + 1
LEWT = LWM + LENWM
LSAVF = LEWT + N
LSAVX = LSAVF + N
LACOR = LSAVX + N
IF (MITER .EQ. 0) LACOR = LSAVF + N
LENRW = LACOR + N - 1
IWORK(17) = LENRW
LIWM = 31
LENIWK = 0
IF (MITER .EQ. 1) LENIWK = MAXL
LIWP = 0
IF (MITER .GE. 1) LIWP = IWORK(2)
LENIW = 30 + LENIWK + LIWP
LOCIWP = LENIWK + 1
IWORK(18) = LENIW
IF (LENRW .GT. LRW) GO TO 617
IF (LENIW .GT. LIW) GO TO 618
C Check RTOL and ATOL for legality. ------------------------------------
RTOLI = RTOL(1)
ATOLI = ATOL(1)
DO 70 I = 1,N
IF (ITOL .GE. 3) RTOLI = RTOL(I)
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
IF (RTOLI .LT. 0.0D0) GO TO 619
IF (ATOLI .LT. 0.0D0) GO TO 620
70 CONTINUE
C Load SQRT(N) and its reciprocal in Common. ---------------------------
SQRTN = SQRT(REAL(N))
RSQRTN = 1.0D0/SQRTN
IF (ISTATE .EQ. 1) GO TO 100
C If ISTATE = 3, set flag to signal parameter changes to DSTODPK. ------
JSTART = -1
IF (NQ .LE. MAXORD) GO TO 90
C MAXORD was reduced below NQ. Copy YH(*,MAXORD+2) into SAVF. ---------
DO 80 I = 1,N
80 RWORK(I+LSAVF-1) = RWORK(I+LWM-1)
90 CONTINUE
IF (N .EQ. NYH) GO TO 200
C NEQ was reduced. Zero part of YH to avoid undefined references. -----
I1 = LYH + L*NYH
I2 = LYH + (MAXORD + 1)*NYH - 1
IF (I1 .GT. I2) GO TO 200
DO 95 I = I1,I2
95 RWORK(I) = 0.0D0
GO TO 200
C-----------------------------------------------------------------------
C Block C.
C The next block is for the initial call only (ISTATE = 1).
C It contains all remaining initializations, the initial call to F,
C and the calculation of the initial step size.
C The error weights in EWT are inverted after being loaded.
C-----------------------------------------------------------------------
100 UROUND = DUMACH()
TN = T
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 110
TCRIT = RWORK(1)
IF ((TCRIT - TOUT)*(TOUT - T) .LT. 0.0D0) GO TO 625
IF (H0 .NE. 0.0D0 .AND. (T + H0 - TCRIT)*H0 .GT. 0.0D0)
1 H0 = TCRIT - T
110 JSTART = 0
NHNIL = 0
NST = 0
NJE = 0
NSLAST = 0
NLI0 = 0
NNI0 = 0
NCFN0 = 0
NCFL0 = 0
NWARN = 0
HU = 0.0D0
NQU = 0
CCMAX = 0.3D0
MAXCOR = 3
MSBP = 20
MXNCF = 10
NNI = 0
NLI = 0
NPS = 0
NCFN = 0
NCFL = 0
C Initial call to F. (LF0 points to YH(*,2).) -------------------------
LF0 = LYH + NYH
CALL F (NEQ, T, Y, RWORK(LF0))
NFE = 1
C Load the initial value vector in YH. ---------------------------------
DO 115 I = 1,N
115 RWORK(I+LYH-1) = Y(I)
C Load and invert the EWT array. (H is temporarily set to 1.0.) -------
NQ = 1
H = 1.0D0
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 120 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 621
120 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
C-----------------------------------------------------------------------
C The coding below computes the step size, H0, to be attempted on the
C first step, unless the user has supplied a value for this.
C First check that TOUT - T differs significantly from zero.
C A scalar tolerance quantity TOL is computed, as MAX(RTOL(i))
C if this is positive, or MAX(ATOL(i)/ABS(Y(i))) otherwise, adjusted
C so as to be between 100*UROUND and 1.0E-3.
C Then the computed value H0 is given by..
C NEQ
C H0**2 = TOL / ( w0**-2 + (1/NEQ) * Sum ( f(i)/ywt(i) )**2 )
C 1
C where w0 = MAX ( ABS(T), ABS(TOUT) ),
C f(i) = i-th component of initial value of f,
C ywt(i) = EWT(i)/TOL (a weight for y(i)).
C The sign of H0 is inferred from the initial values of TOUT and T.
C-----------------------------------------------------------------------
IF (H0 .NE. 0.0D0) GO TO 180
TDIST = ABS(TOUT - T)
W0 = MAX(ABS(T),ABS(TOUT))
IF (TDIST .LT. 2.0D0*UROUND*W0) GO TO 622
TOL = RTOL(1)
IF (ITOL .LE. 2) GO TO 140
DO 130 I = 1,N
130 TOL = MAX(TOL,RTOL(I))
140 IF (TOL .GT. 0.0D0) GO TO 160
ATOLI = ATOL(1)
DO 150 I = 1,N
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
AYI = ABS(Y(I))
IF (AYI .NE. 0.0D0) TOL = MAX(TOL,ATOLI/AYI)
150 CONTINUE
160 TOL = MAX(TOL,100.0D0*UROUND)
TOL = MIN(TOL,0.001D0)
SUM = DVNORM (N, RWORK(LF0), RWORK(LEWT))
SUM = 1.0D0/(TOL*W0*W0) + TOL*SUM**2
H0 = 1.0D0/SQRT(SUM)
H0 = MIN(H0,TDIST)
H0 = SIGN(H0,TOUT-T)
C Adjust H0 if necessary to meet HMAX bound. ---------------------------
180 RH = ABS(H0)*HMXI
IF (RH .GT. 1.0D0) H0 = H0/RH
C Load H with H0 and scale YH(*,2) by H0. ------------------------------
H = H0
DO 190 I = 1,N
190 RWORK(I+LF0-1) = H0*RWORK(I+LF0-1)
GO TO 270
C-----------------------------------------------------------------------
C Block D.
C The next code block is for continuation calls only (ISTATE = 2 or 3)
C and is to check stop conditions before taking a step.
C-----------------------------------------------------------------------
200 NSLAST = NST
NLI0 = NLI
NNI0 = NNI
NCFN0 = NCFN
NCFL0 = NCFL
NWARN = 0
GO TO (210, 250, 220, 230, 240), ITASK
210 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
220 TP = TN - HU*(1.0D0 + 100.0D0*UROUND)
IF ((TP - TOUT)*H .GT. 0.0D0) GO TO 623
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
GO TO 400
230 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
IF ((TCRIT - TOUT)*H .LT. 0.0D0) GO TO 625
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 245
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
240 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
245 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
IF (ISTATE .EQ. 2) JSTART = -2
C-----------------------------------------------------------------------
C Block E.
C The next block is normally executed for all calls and contains
C the call to the one-step core integrator DSTODPK.
C
C This is a looping point for the integration steps.
C
C First check for too many steps being taken,
C Check for poor Newton/Krylov method performance, update EWT (if not
C at start of problem), check for too much accuracy being requested,
C and check for H below the roundoff level in T.
C-----------------------------------------------------------------------
250 CONTINUE
IF ((NST-NSLAST) .GE. MXSTEP) GO TO 500
NSTD = NST - NSLAST
NNID = NNI - NNI0
IF (NSTD .LT. 10 .OR. NNID .EQ. 0) GO TO 255
AVDIM = REAL(NLI - NLI0)/REAL(NNID)
RCFN = REAL(NCFN - NCFN0)/REAL(NSTD)
RCFL = REAL(NCFL - NCFL0)/REAL(NNID)
LAVD = AVDIM .GT. (MAXL - 0.05D0)
LCFN = RCFN .GT. 0.9D0
LCFL = RCFL .GT. 0.9D0
LWARN = LAVD .OR. LCFN .OR. LCFL
IF (.NOT.LWARN) GO TO 255
NWARN = NWARN + 1
IF (NWARN .GT. 10) GO TO 255
IF (LAVD) THEN
MSG='DLSODPK- Warning. Poor iterative algorithm performance seen '
CALL XERRWD (MSG, 60, 111, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
IF (LAVD) THEN
MSG=' at T = R1 by average no. of linear iterations = R2 '
CALL XERRWD (MSG, 60, 111, 0, 0, 0, 0, 2, TN, AVDIM)
ENDIF
IF (LCFN) THEN
MSG='DLSODPK- Warning. Poor iterative algorithm performance seen '
CALL XERRWD (MSG, 60, 112, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
IF (LCFN) THEN
MSG=' at T = R1 by nonlinear convergence failure rate = R2 '
CALL XERRWD (MSG, 60, 112, 0, 0, 0, 0, 2, TN, RCFN)
ENDIF
IF (LCFL) THEN
MSG='DLSODPK- Warning. Poor iterative algorithm performance seen '
CALL XERRWD (MSG, 60, 113, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
IF (LCFL) THEN
MSG=' at T = R1 by linear convergence failure rate = R2 '
CALL XERRWD (MSG, 60, 113, 0, 0, 0, 0, 2, TN, RCFL)
ENDIF
255 CONTINUE
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 260 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 510
260 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
270 TOLSF = UROUND*DVNORM (N, RWORK(LYH), RWORK(LEWT))
IF (TOLSF .LE. 1.0D0) GO TO 280
TOLSF = TOLSF*2.0D0
IF (NST .EQ. 0) GO TO 626
GO TO 520
280 IF ((TN + H) .NE. TN) GO TO 290
NHNIL = NHNIL + 1
IF (NHNIL .GT. MXHNIL) GO TO 290
MSG = 'DLSODPK- Warning..Internal T(=R1) and H(=R2) are '
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' such that in the machine, T + H = T on the next step '
CALL XERRWD (MSG, 60, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' (H = step size). Solver will continue anyway.'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 2, TN, H)
IF (NHNIL .LT. MXHNIL) GO TO 290
MSG = 'DLSODPK- Above warning has been issued I1 times. '
CALL XERRWD (MSG, 50, 102, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' It will not be issued again for this problem.'
CALL XERRWD (MSG, 50, 102, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
290 CONTINUE
C-----------------------------------------------------------------------
C CALL DSTODPK(NEQ,Y,YH,NYH,YH,EWT,SAVF,SAVX,ACOR,WM,IWM,F,JAC,PSOL)
C-----------------------------------------------------------------------
CALL DSTODPK (NEQ, Y, RWORK(LYH), NYH, RWORK(LYH), RWORK(LEWT),
1 RWORK(LSAVF), RWORK(LSAVX), RWORK(LACOR), RWORK(LWM),
2 IWORK(LIWM), F, JAC, PSOL)
KGO = 1 - KFLAG
GO TO (300, 530, 540, 550), KGO
C-----------------------------------------------------------------------
C Block F.
C The following block handles the case of a successful return from the
C core integrator (KFLAG = 0). Test for stop conditions.
C-----------------------------------------------------------------------
300 INIT = 1
GO TO (310, 400, 330, 340, 350), ITASK
C ITASK = 1. If TOUT has been reached, interpolate. -------------------
310 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
C ITASK = 3. Jump to exit if TOUT was reached. ------------------------
330 IF ((TN - TOUT)*H .GE. 0.0D0) GO TO 400
GO TO 250
C ITASK = 4. See if TOUT or TCRIT was reached. Adjust H if necessary.
340 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 345
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
345 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
JSTART = -2
GO TO 250
C ITASK = 5. see if TCRIT was reached and jump to exit. ---------------
350 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
C-----------------------------------------------------------------------
C Block G.
C The following block handles all successful returns from DLSODPK.
C If ITASK .ne. 1, Y is loaded from YH and T is set accordingly.
C ISTATE is set to 2, and the optional outputs are loaded into the
C work arrays before returning.
C-----------------------------------------------------------------------
400 DO 410 I = 1,N
410 Y(I) = RWORK(I+LYH-1)
T = TN
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 420
IF (IHIT) T = TCRIT
420 ISTATE = 2
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(19) = NNI
IWORK(20) = NLI
IWORK(21) = NPS
IWORK(22) = NCFN
IWORK(23) = NCFL
RETURN
C-----------------------------------------------------------------------
C Block H.
C The following block handles all unsuccessful returns other than
C those for illegal input. First the error message routine is called.
C If there was an error test or convergence test failure, IMXER is set.
C Then Y is loaded from YH and T is set to TN.
C The optional outputs are loaded into the work arrays before returning.
C-----------------------------------------------------------------------
C The maximum number of steps was taken before reaching TOUT. ----------
500 MSG = 'DLSODPK- At current T (=R1), MXSTEP (=I1) steps '
CALL XERRWD (MSG, 50, 201, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' taken on this call before reaching TOUT '
CALL XERRWD (MSG, 50, 201, 0, 1, MXSTEP, 0, 1, TN, 0.0D0)
ISTATE = -1
GO TO 580
C EWT(i) .le. 0.0 for some i (not at start of problem). ----------------
510 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODPK- At T (=R1), EWT(I1) has become R2.le.0. '
CALL XERRWD (MSG, 50, 202, 0, 1, I, 0, 2, TN, EWTI)
ISTATE = -6
GO TO 580
C Too much accuracy requested for machine precision. -------------------
520 MSG = 'DLSODPK- At T (=R1), too much accuracy requested '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' for precision of machine.. See TOLSF (=R2) '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 2, TN, TOLSF)
RWORK(14) = TOLSF
ISTATE = -2
GO TO 580
C KFLAG = -1. Error test failed repeatedly or with ABS(H) = HMIN. -----
530 MSG = 'DLSODPK- At T(=R1), step size H(=R2), the error '
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' test failed repeatedly or with ABS(H) = HMIN'
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 2, TN, H)
ISTATE = -4
GO TO 560
C KFLAG = -2. Convergence failed repeatedly or with ABS(H) = HMIN. ----
540 MSG = 'DLSODPK- At T (=R1) and step size H (=R2), the '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' corrector convergence failed repeatedly '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' or with ABS(H) = HMIN '
CALL XERRWD (MSG, 30, 205, 0, 0, 0, 0, 2, TN, H)
ISTATE = -5
GO TO 560
C KFLAG = -3. Unrecoverable error from PSOL. --------------------------
550 MSG = 'DLSODPK- At T (=R1) an unrecoverable error return'
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' was made from Subroutine PSOL '
CALL XERRWD (MSG, 40, 205, 0, 0, 0, 0, 1, TN, 0.0D0)
ISTATE = -7
GO TO 580
C Compute IMXER if relevant. -------------------------------------------
560 BIG = 0.0D0
IMXER = 1
DO 570 I = 1,N
SIZE = ABS(RWORK(I+LACOR-1)*RWORK(I+LEWT-1))
IF (BIG .GE. SIZE) GO TO 570
BIG = SIZE
IMXER = I
570 CONTINUE
IWORK(16) = IMXER
C Set Y vector, T, and optional outputs. -------------------------------
580 DO 590 I = 1,N
590 Y(I) = RWORK(I+LYH-1)
T = TN
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(19) = NNI
IWORK(20) = NLI
IWORK(21) = NPS
IWORK(22) = NCFN
IWORK(23) = NCFL
RETURN
C-----------------------------------------------------------------------
C Block I.
C The following block handles all error returns due to illegal input
C (ISTATE = -3), as detected before calling the core integrator.
C First the error message routine is called. If the illegal input
C is a negative ISTATE, the run is aborted (apparent infinite loop).
C-----------------------------------------------------------------------
601 MSG = 'DLSODPK- ISTATE(=I1) illegal.'
CALL XERRWD (MSG, 30, 1, 0, 1, ISTATE, 0, 0, 0.0D0, 0.0D0)
IF (ISTATE .LT. 0) GO TO 800
GO TO 700
602 MSG = 'DLSODPK- ITASK (=I1) illegal.'
CALL XERRWD (MSG, 30, 2, 0, 1, ITASK, 0, 0, 0.0D0, 0.0D0)
GO TO 700
603 MSG = 'DLSODPK- ISTATE.gt.1 but DLSODPK not initialized.'
CALL XERRWD (MSG, 50, 3, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
GO TO 700
604 MSG = 'DLSODPK- NEQ (=I1) .lt. 1 '
CALL XERRWD (MSG, 30, 4, 0, 1, NEQ(1), 0, 0, 0.0D0, 0.0D0)
GO TO 700
605 MSG = 'DLSODPK- ISTATE = 3 and NEQ increased (I1 to I2).'
CALL XERRWD (MSG, 50, 5, 0, 2, N, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
606 MSG = 'DLSODPK- ITOL (=I1) illegal. '
CALL XERRWD (MSG, 30, 6, 0, 1, ITOL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
607 MSG = 'DLSODPK- IOPT (=I1) illegal. '
CALL XERRWD (MSG, 30, 7, 0, 1, IOPT, 0, 0, 0.0D0, 0.0D0)
GO TO 700
608 MSG = 'DLSODPK- MF (=I1) illegal. '
CALL XERRWD (MSG, 30, 8, 0, 1, MF, 0, 0, 0.0D0, 0.0D0)
GO TO 700
611 MSG = 'DLSODPK- MAXORD (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 11, 0, 1, MAXORD, 0, 0, 0.0D0, 0.0D0)
GO TO 700
612 MSG = 'DLSODPK- MXSTEP (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 12, 0, 1, MXSTEP, 0, 0, 0.0D0, 0.0D0)
GO TO 700
613 MSG = 'DLSODPK- MXHNIL (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 13, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
614 MSG = 'DLSODPK- TOUT (=R1) behind T (=R2) '
CALL XERRWD (MSG, 40, 14, 0, 0, 0, 0, 2, TOUT, T)
MSG = ' Integration direction is given by H0 (=R1) '
CALL XERRWD (MSG, 50, 14, 0, 0, 0, 0, 1, H0, 0.0D0)
GO TO 700
615 MSG = 'DLSODPK- HMAX (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 15, 0, 0, 0, 0, 1, HMAX, 0.0D0)
GO TO 700
616 MSG = 'DLSODPK- HMIN (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 16, 0, 0, 0, 0, 1, HMIN, 0.0D0)
GO TO 700
617 MSG='DLSODPK- RWORK length needed, LENRW(=I1), exceeds LRW(=I2) '
CALL XERRWD (MSG, 60, 17, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
618 MSG='DLSODPK- IWORK length needed, LENIW(=I1), exceeds LIW(=I2) '
CALL XERRWD (MSG, 60, 18, 0, 2, LENIW, LIW, 0, 0.0D0, 0.0D0)
GO TO 700
619 MSG = 'DLSODPK- RTOL(I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 19, 0, 1, I, 0, 1, RTOLI, 0.0D0)
GO TO 700
620 MSG = 'DLSODPK- ATOL(I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 20, 0, 1, I, 0, 1, ATOLI, 0.0D0)
GO TO 700
621 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODPK- EWT(I1) is R1 .le. 0.0 '
CALL XERRWD (MSG, 40, 21, 0, 1, I, 0, 1, EWTI, 0.0D0)
GO TO 700
622 MSG='DLSODPK- TOUT(=R1) too close to T(=R2) to start integration.'
CALL XERRWD (MSG, 60, 22, 0, 0, 0, 0, 2, TOUT, T)
GO TO 700
623 MSG='DLSODPK- ITASK = I1 and TOUT (=R1) behind TCUR - HU (= R2) '
CALL XERRWD (MSG, 60, 23, 0, 1, ITASK, 0, 2, TOUT, TP)
GO TO 700
624 MSG='DLSODPK- ITASK = 4 or 5 and TCRIT (=R1) behind TCUR (=R2) '
CALL XERRWD (MSG, 60, 24, 0, 0, 0, 0, 2, TCRIT, TN)
GO TO 700
625 MSG='DLSODPK- ITASK = 4 or 5 and TCRIT (=R1) behind TOUT (=R2) '
CALL XERRWD (MSG, 60, 25, 0, 0, 0, 0, 2, TCRIT, TOUT)
GO TO 700
626 MSG = 'DLSODPK- At start of problem, too much accuracy '
CALL XERRWD (MSG, 50, 26, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' requested for precision of machine.. See TOLSF (=R1) '
CALL XERRWD (MSG, 60, 26, 0, 0, 0, 0, 1, TOLSF, 0.0D0)
RWORK(14) = TOLSF
GO TO 700
627 MSG = 'DLSODPK- Trouble in DINTDY. ITASK = I1, TOUT = R1'
CALL XERRWD (MSG, 50, 27, 0, 1, ITASK, 0, 1, TOUT, 0.0D0)
C
700 ISTATE = -3
RETURN
C
800 MSG = 'DLSODPK- Run aborted.. apparent infinite loop. '
CALL XERRWD (MSG, 50, 303, 2, 0, 0, 0, 0, 0.0D0, 0.0D0)
RETURN
C----------------------- End of Subroutine DLSODPK ---------------------
END
*DECK DLSODKR
SUBROUTINE DLSODKR (F, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
1 ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JAC, PSOL,
2 MF, G, NG, JROOT)
EXTERNAL F, JAC, PSOL, G
INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK, LIW, MF,
1 NG, JROOT
DOUBLE PRECISION Y, T, TOUT, RTOL, ATOL, RWORK
DIMENSION NEQ(*), Y(*), RTOL(*), ATOL(*), RWORK(LRW), IWORK(LIW),
1 JROOT(*)
C-----------------------------------------------------------------------
C This is the 18 November 2003 version of
C DLSODKR: Livermore Solver for Ordinary Differential equations,
C with preconditioned Krylov iteration methods for the
C Newton correction linear systems, and with Rootfinding.
C
C This version is in double precision.
C
C DLSODKR solves the initial value problem for stiff or nonstiff
C systems of first order ODEs,
C dy/dt = f(t,y) , or, in component form,
C dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(NEQ)) (i = 1,...,NEQ).
C At the same time, it locates the roots of any of a set of functions
C g(i) = g(i,t,y(1),...,y(NEQ)) (i = 1,...,ng).
C
C-----------------------------------------------------------------------
C Introduction.
C
C This is a modification of the DLSODE package, and differs from it
C in five ways:
C (a) It uses various preconditioned Krylov subspace iteration methods
C for the linear algebraic systems that arise in the case of stiff
C systems. See the introductory notes below.
C (b) It does automatic switching between functional (fixpoint)
C iteration and Newton iteration in the corrector iteration.
C (c) It finds the root of at least one of a set of constraint
C functions g(i) of the independent and dependent variables.
C It finds only those roots for which some g(i), as a function
C of t, changes sign in the interval of integration.
C It then returns the solution at the root, if that occurs
C sooner than the specified stop condition, and otherwise returns
C the solution according the specified stop condition.
C (d) It supplies to JAC an input flag, JOK, which indicates whether
C JAC may (optionally) bypass the evaluation of Jacobian matrix data
C and instead process saved data (with the current value of scalar hl0).
C (e) It contains a new subroutine that calculates the initial step
C size to be attempted.
C
C
C Introduction to the Krylov methods in DLSODKR:
C
C The linear systems that must be solved have the form
C A * x = b , where A = identity - hl0 * (df/dy) .
C Here hl0 is a scalar, and df/dy is the Jacobian matrix of partial
C derivatives of f (NEQ by NEQ).
C
C The particular Krylov method is chosen by setting the second digit,
C MITER, in the method flag MF.
C Currently, the values of MITER have the following meanings:
C
C MITER = 1 means the Scaled Preconditioned Incomplete
C Orthogonalization Method (SPIOM).
C
C 2 means an incomplete version of the preconditioned scaled
C Generalized Minimal Residual method (SPIGMR).
C This is the best choice in general.
C
C 3 means the Preconditioned Conjugate Gradient method (PCG).
C Recommended only when df/dy is symmetric or nearly so.
C
C 4 means the scaled Preconditioned Conjugate Gradient method
C (PCGS). Recommended only when D-inverse * df/dy * D is
C symmetric or nearly so, where D is the diagonal scaling
C matrix with elements 1/EWT(i) (see RTOL/ATOL description).
C
C 9 means that only a user-supplied matrix P (approximating A)
C will be used, with no Krylov iteration done. This option
C allows the user to provide the complete linear system
C solution algorithm, if desired.
C
C The user can apply preconditioning to the linear system A*x = b,
C by means of arbitrary matrices (the preconditioners).
C In the case of SPIOM and SPIGMR, one can apply left and right
C preconditioners P1 and P2, and the basic iterative method is then
C applied to the matrix (P1-inverse)*A*(P2-inverse) instead of to the
C matrix A. The product P1*P2 should be an approximation to matrix A
C such that linear systems with P1 or P2 are easier to solve than with
C A. Preconditioning from the left only or right only means using
C P2 = identity or P1 = identity, respectively.
C In the case of the PCG and PCGS methods, there is only one
C preconditioner matrix P (but it can be the product of more than one).
C It should approximate the matrix A but allow for relatively
C easy solution of linear systems with coefficient matrix P.
C For PCG, P should be positive definite symmetric, or nearly so,
C and for PCGS, the scaled preconditioner D-inverse * P * D
C should be symmetric or nearly so.
C If the Jacobian J = df/dy splits in a natural way into a sum
C J = J1 + J2, then one possible choice of preconditioners is
C P1 = identity - hl0 * J1 and P2 = identity - hl0 * J2
C provided each of these is easy to solve (or approximately solve).
C
C-----------------------------------------------------------------------
C References:
C 1. Peter N. Brown and Alan C. Hindmarsh, Reduced Storage Matrix
C Methods in Stiff ODE Systems, J. Appl. Math. & Comp., 31 (1989),
C pp. 40-91; also L.L.N.L. Report UCRL-95088, Rev. 1, June 1987.
C 2. Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE
C Solvers, in Scientific Computing, R. S. Stepleman et al. (Eds.),
C North-Holland, Amsterdam, 1983, pp. 55-64.
C-----------------------------------------------------------------------
C Authors: Alan C. Hindmarsh and Peter N. Brown
C Center for Applied Scientific Computing, L-561
C Lawrence Livermore National Laboratory
C Livermore, CA 94551
C-----------------------------------------------------------------------
C Summary of Usage.
C
C Communication between the user and the DLSODKR package, for normal
C situations, is summarized here. This summary describes only a subset
C of the full set of options available. See the full description for
C details, including optional communication, nonstandard options,
C and instructions for special situations. See also the demonstration
C program distributed with this solver.
C
C A. First provide a subroutine of the form:
C SUBROUTINE F (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y(*), YDOT(*)
C which supplies the vector function f by loading YDOT(i) with f(i).
C
C B. Provide a subroutine of the form:
C SUBROUTINE G (NEQ, T, Y, NG, GOUT)
C DOUBLE PRECISION T, Y(*), GOUT(NG)
C which supplies the vector function g by loading GOUT(i) with
C g(i), the i-th constraint function whose root is sought.
C
C C. Next determine (or guess) whether or not the problem is stiff.
C Stiffness occurs when the Jacobian matrix df/dy has an eigenvalue
C whose real part is negative and large in magnitude, compared to the
C reciprocal of the t span of interest. If the problem is nonstiff,
C use a method flag MF = 10. If it is stiff, MF should be between 21
C and 24, or possibly 29. MF = 22 is generally the best choice.
C Use 23 or 24 only if symmetry is present. Use MF = 29 if the
C complete linear system solution is to be provided by the user.
C The following four parameters must also be set.
C IWORK(1) = LWP = length of real array WP for preconditioning.
C IWORK(2) = LIWP = length of integer array IWP for preconditioning.
C IWORK(3) = JPRE = preconditioner type flag:
C = 0 for no preconditioning (P1 = P2 = P = identity)
C = 1 for left-only preconditioning (P2 = identity)
C = 2 for right-only preconditioning (P1 = identity)
C = 3 for two-sided preconditioning (and PCG or PCGS)
C IWORK(4) = JACFLG = flag for whether JAC is called.
C = 0 if JAC is not to be called,
C = 1 if JAC is to be called.
C Use JACFLG = 1 if JAC computes any nonconstant data for use in
C preconditioning, such as Jacobian elements.
C The arrays WP and IWP are work arrays under the user's control,
C for use in the routines that perform preconditioning operations.
C
C D. If the problem is stiff, you must supply two routines that deal
C with the preconditioning of the linear systems to be solved.
C These are as follows:
C
C SUBROUTINE JAC (F, NEQ, T, Y, YSV, REWT, FTY,V,HL0,JOK,WP,IWP,IER)
C DOUBLE PRECISION T, Y(*), YSV(*), REWT(*), FTY(*), V(*), HL0,WP(*)
C INTEGER IWP(*)
C This routine must evaluate and preprocess any parts of the
C Jacobian matrix df/dy involved in the preconditioners P1, P2, P.
C The Y and FTY arrays contain the current values of y and f(t,y),
C respectively, and YSV also contains the current value of y.
C The array V is work space of length NEQ.
C JAC must multiply all computed Jacobian elements by the scalar
C -HL0, add the identity matrix, and do any factorization
C operations called for, in preparation for solving linear systems
C with a coefficient matrix of P1, P2, or P. The matrix P1*P2 or P
C should be an approximation to identity - hl0 * (df/dy).
C JAC should return IER = 0 if successful, and IER .ne. 0 if not.
C (If IER .ne. 0, a smaller time step will be tried.)
C JAC may alter Y and V, but not YSV, REWT, FTY, or HL0.
C The JOK argument can be ignored (or see full description below).
C
C SUBROUTINE PSOL (NEQ, T, Y, FTY, WK, HL0, WP, IWP, B, LR, IER)
C DOUBLE PRECISION T, Y(*), FTY(*), WK(*), HL0, WP(*), B(*)
C INTEGER IWP(*)
C This routine must solve a linear system with B as right-hand
C side and one of the preconditioning matrices, P1, P2, or P, as
C coefficient matrix, and return the solution vector in B.
C LR is a flag concerning left vs right preconditioning, input
C to PSOL. PSOL is to use P1 if LR = 1 and P2 if LR = 2.
C In the case of the PCG or PCGS method, LR will be 3, and PSOL
C should solve the system P*x = B with the preconditioner matrix P.
C In the case MF = 29 (no Krylov iteration), LR will be 0,
C and PSOL is to return in B the desired approximate solution
C to A * x = B, where A = identity - hl0 * (df/dy).
C PSOL can use data generated in the JAC routine and stored in
C WP and IWP. WK is a work array of length NEQ.
C The argument HL0 is the current value of the scalar appearing
C in the linear system. If the old value, at the time of the last
C JAC call, is needed, it must have been saved by JAC in WP.
C on return, PSOL should set the error flag IER as follows:
C IER = 0 if PSOL was successful,
C IER .gt. 0 if a recoverable error occurred, meaning that the
C time step will be retried,
C IER .lt. 0 if an unrecoverable error occurred, meaning that the
C solver is to stop immediately.
C
C E. Write a main program which calls Subroutine DLSODKR once for
C each point at which answers are desired. This should also provide
C for possible use of logical unit 6 for output of error messages
C by DLSODKR. On the first call to DLSODKR, supply arguments as
C follows:
C F = name of subroutine for right-hand side vector f.
C This name must be declared External in calling program.
C NEQ = number of first order ODEs.
C Y = array of initial values, of length NEQ.
C T = the initial value of the independent variable.
C TOUT = first point where output is desired (.ne. T).
C ITOL = 1 or 2 according as ATOL (below) is a scalar or array.
C RTOL = relative tolerance parameter (scalar).
C ATOL = absolute tolerance parameter (scalar or array).
C The estimated local error in y(i) will be controlled so as
C to be roughly less (in magnitude) than
C EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
C EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
C Thus the local error test passes if, in each component,
C either the absolute error is less than ATOL (or ATOL(i)),
C or the relative error is less than RTOL.
C Use RTOL = 0.0 for pure absolute error control, and
C use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative error
C control. Caution: Actual (global) errors may exceed these
C local tolerances, so choose them conservatively.
C ITASK = 1 for normal computation of output values of y at t = TOUT.
C ISTATE = integer flag (input and output). Set ISTATE = 1.
C IOPT = 0 to indicate no optional inputs used.
C RWORK = real work array of length at least:
C 20 + 16*NEQ + 3*NG for MF = 10,
C 45 + 17*NEQ + 3*NG + LWP for MF = 21,
C 61 + 17*NEQ + 3*NG + LWP for MF = 22,
C 20 + 15*NEQ + 3*NG + LWP for MF = 23 or 24,
C 20 + 12*NEQ + 3*NG + LWP for MF = 29.
C LRW = declared length of RWORK (in user's dimension).
C IWORK = integer work array of length at least:
C 30 for MF = 10,
C 35 + LIWP for MF = 21,
C 30 + LIWP for MF = 22, 23, 24, or 29.
C LIW = declared length of IWORK (in user's dimension).
C JAC,PSOL = names of subroutines for preconditioning.
C These names must be declared External in the calling program.
C MF = method flag. Standard values are:
C 10 for nonstiff (Adams) method.
C 21 for stiff (BDF) method, with preconditioned SIOM.
C 22 for stiff method, with preconditioned GMRES method.
C 23 for stiff method, with preconditioned CG method.
C 24 for stiff method, with scaled preconditioned CG method.
C 29 for stiff method, with user's PSOL routine only.
C G = name of subroutine for constraint functions, whose
C roots are desired during the integration.
C This name must be declared External in calling program.
C NG = number of constraint functions g(i). If there are none,
C set NG = 0, and pass a dummy name for G.
C JROOT = integer array of length NG for output of root information.
C See next paragraph.
C Note that the main program must declare arrays Y, RWORK, IWORK,
C JROOT, and possibly ATOL.
C
C F. The output from the first call (or any call) is:
C Y = array of computed values of y(t) vector.
C T = corresponding value of independent variable (normally TOUT).
C ISTATE = 2 or 3 if DLSODKR was successful, negative otherwise.
C 2 means no root was found, and TOUT was reached as desired.
C 3 means a root was found prior to reaching TOUT.
C -1 means excess work done on this call (perhaps wrong MF).
C -2 means excess accuracy requested (tolerances too small).
C -3 means illegal input detected (see printed message).
C -4 means repeated error test failures (check all inputs).
C -5 means repeated convergence failures (perhaps bad JAC
C or PSOL routine supplied or wrong choice of MF or
C tolerances, or this solver is inappropriate).
C -6 means error weight became zero during problem. (Solution
C component i vanished, and ATOL or ATOL(i) = 0.)
C -7 means an unrecoverable error occurred in PSOL.
C JROOT = array showing roots found if ISTATE = 3 on return.
C JROOT(i) = 1 if g(i) has a root at T, or 0 otherwise.
C
C G. To continue the integration after a successful return, proceed
C as follows:
C (a) If ISTATE = 2 on return, reset TOUT and call DLSODKR again.
C (b) If ISTATE = 3 on return, reset ISTATE to 2 and call DLSODKR again.
C In either case, no other parameters need be reset.
C
C-----------------------------------------------------------------------
C-----------------------------------------------------------------------
C Full Description of User Interface to DLSODKR.
C
C The user interface to DLSODKR consists of the following parts.
C
C 1. The call sequence to Subroutine DLSODKR, which is a driver
C routine for the solver. This includes descriptions of both
C the call sequence arguments and of user-supplied routines.
C Following these descriptions is a description of
C optional inputs available through the call sequence, and then
C a description of optional outputs (in the work arrays).
C
C 2. Descriptions of other routines in the DLSODKR package that may be
C (optionally) called by the user. These provide the ability to
C alter error message handling, save and restore the internal
C Common, and obtain specified derivatives of the solution y(t).
C
C 3. Descriptions of Common blocks to be declared in overlay
C or similar environments, or to be saved when doing an interrupt
C of the problem and continued solution later.
C
C 4. Description of two routines in the DLSODKR package, either of
C which the user may replace with his/her own version, if desired.
C These relate to the measurement of errors.
C
C-----------------------------------------------------------------------
C Part 1. Call Sequence.
C
C The call sequence parameters used for input only are
C F, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK, IOPT, LRW, LIW, JAC, PSOL, MF,
C G, and NG,
C that used only for output is JROOT,
C and those used for both input and output are
C Y, T, ISTATE.
C The work arrays RWORK and IWORK are also used for conditional and
C optional inputs and optional outputs. (The term output here refers
C to the return from Subroutine DLSODKR to the user's calling program.)
C
C The legality of input parameters will be thoroughly checked on the
C initial call for the problem, but not checked thereafter unless a
C change in input parameters is flagged by ISTATE = 3 on input.
C
C The descriptions of the call arguments are as follows.
C
C F = the name of the user-supplied subroutine defining the
C ODE system. The system must be put in the first-order
C form dy/dt = f(t,y), where f is a vector-valued function
C of the scalar t and the vector y. Subroutine F is to
C compute the function f. It is to have the form
C SUBROUTINE F (NEQ, T, Y, YDOT)
C DOUBLE PRECISION T, Y(*), YDOT(*)
C where NEQ, T, and Y are input, and the array YDOT = f(t,y)
C is output. Y and YDOT are arrays of length NEQ.
C Subroutine F should not alter Y(1),...,Y(NEQ).
C F must be declared External in the calling program.
C
C Subroutine F may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in F) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y below.
C
C If quantities computed in the F routine are needed
C externally to DLSODKR, an extra call to F should be made
C for this purpose, for consistent and accurate results.
C If only the derivative dy/dt is needed, use DINTDY instead.
C
C NEQ = the size of the ODE system (number of first order
C ordinary differential equations). Used only for input.
C NEQ may be decreased, but not increased, during the problem.
C If NEQ is decreased (with ISTATE = 3 on input), the
C remaining components of Y should be left undisturbed, if
C these are to be accessed in the user-supplied routines.
C
C Normally, NEQ is a scalar, and it is generally referred to
C as a scalar in this user interface description. However,
C NEQ may be an array, with NEQ(1) set to the system size.
C (The DLSODKR package accesses only NEQ(1).) In either case,
C this parameter is passed as the NEQ argument in all calls
C to the user-supplied routines. Hence, if it is an array,
C locations NEQ(2),... may be used to store other integer data
C and pass it to the user-supplied routines. Each such routine
C must include NEQ in a Dimension statement in that case.
C
C Y = a real array for the vector of dependent variables, of
C length NEQ or more. Used for both input and output on the
C first call (ISTATE = 1), and only for output on other calls.
C On the first call, Y must contain the vector of initial
C values. On output, Y contains the computed solution vector,
C evaluated at T. If desired, the Y array may be used
C for other purposes between calls to the solver.
C
C This array is passed as the Y argument in all calls to F, G,
C JAC, and PSOL. Hence its length may exceed NEQ, and
C locations Y(NEQ+1),... may be used to store other real data
C and pass it to the user-supplied routines.
C (The DLSODKR package accesses only Y(1),...,Y(NEQ).)
C
C T = the independent variable. On input, T is used only on the
C first call, as the initial point of the integration.
C On output, after each call, T is the value at which a
C computed solution y is evaluated (usually the same as TOUT).
C If a root was found, T is the computed location of the
C root reached first, on output.
C On an error return, T is the farthest point reached.
C
C TOUT = the next value of t at which a computed solution is desired.
C Used only for input.
C
C When starting the problem (ISTATE = 1), TOUT may be equal
C to T for one call, then should .ne. T for the next call.
C For the initial T, an input value of TOUT .ne. T is used
C in order to determine the direction of the integration
C (i.e. the algebraic sign of the step sizes) and the rough
C scale of the problem. Integration in either direction
C (forward or backward in t) is permitted.
C
C If ITASK = 2 or 5 (one-step modes), TOUT is ignored after
C the first call (i.e. the first call with TOUT .ne. T).
C Otherwise, TOUT is required on every call.
C
C If ITASK = 1, 3, or 4, the values of TOUT need not be
C monotone, but a value of TOUT which backs up is limited
C to the current internal T interval, whose endpoints are
C TCUR - HU and TCUR (see optional outputs, below, for
C TCUR and HU).
C
C ITOL = an indicator for the type of error control. See
C description below under ATOL. Used only for input.
C
C RTOL = a relative error tolerance parameter, either a scalar or
C an array of length NEQ. See description below under ATOL.
C Input only.
C
C ATOL = an absolute error tolerance parameter, either a scalar or
C an array of length NEQ. Input only.
C
C The input parameters ITOL, RTOL, and ATOL determine
C the error control performed by the solver. The solver will
C control the vector E = (E(i)) of estimated local errors
C in y, according to an inequality of the form
C RMS-norm of ( E(i)/EWT(i) ) .le. 1,
C where EWT(i) = RTOL(i)*ABS(Y(i)) + ATOL(i),
C and the RMS-norm (root-mean-square norm) here is
C RMS-norm(v) = SQRT(sum v(i)**2 / NEQ). Here EWT = (EWT(i))
C is a vector of weights which must always be positive, and
C the values of RTOL and ATOL should all be non-negative.
C The following table gives the types (scalar/array) of
C RTOL and ATOL, and the corresponding form of EWT(i).
C
C ITOL RTOL ATOL EWT(i)
C 1 scalar scalar RTOL*ABS(Y(i)) + ATOL
C 2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
C 3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
C 4 array array RTOL(i)*ABS(Y(i)) + ATOL(i)
C
C When either of these parameters is a scalar, it need not
C be dimensioned in the user's calling program.
C
C If none of the above choices (with ITOL, RTOL, and ATOL
C fixed throughout the problem) is suitable, more general
C error controls can be obtained by substituting
C user-supplied routines for the setting of EWT and/or for
C the norm calculation. See Part 4 below.
C
C If global errors are to be estimated by making a repeated
C run on the same problem with smaller tolerances, then all
C components of RTOL and ATOL (i.e. of EWT) should be scaled
C down uniformly.
C
C ITASK = an index specifying the task to be performed.
C Input only. ITASK has the following values and meanings.
C 1 means normal computation of output values of y(t) at
C t = TOUT (by overshooting and interpolating).
C 2 means take one step only and return.
C 3 means stop at the first internal mesh point at or
C beyond t = TOUT and return.
C 4 means normal computation of output values of y(t) at
C t = TOUT but without overshooting t = TCRIT.
C TCRIT must be input as RWORK(1). TCRIT may be equal to
C or beyond TOUT, but not behind it in the direction of
C integration. This option is useful if the problem
C has a singularity at or beyond t = TCRIT.
C 5 means take one step, without passing TCRIT, and return.
C TCRIT must be input as RWORK(1).
C
C Note: If ITASK = 4 or 5 and the solver reaches TCRIT
C (within roundoff), it will return T = TCRIT (exactly) to
C indicate this (unless ITASK = 4 and TOUT comes before TCRIT,
C in which case answers at T = TOUT are returned first).
C
C ISTATE = an index used for input and output to specify the
C the state of the calculation.
C
C On input, the values of ISTATE are as follows.
C 1 means this is the first call for the problem
C (initializations will be done). See note below.
C 2 means this is not the first call, and the calculation
C is to continue normally, with no change in any input
C parameters except possibly TOUT and ITASK.
C (If ITOL, RTOL, and/or ATOL are changed between calls
C with ISTATE = 2, the new values will be used but not
C tested for legality.)
C 3 means this is not the first call, and the
C calculation is to continue normally, but with
C a change in input parameters other than
C TOUT and ITASK. Changes are allowed in
C NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, MF,
C and any of the optional inputs except H0.
C In addition, immediately following a return with
C ISTATE = 3 (root found), NG and G may be changed.
C (But changing NG from 0 to .gt. 0 is not allowed.)
C Note: A preliminary call with TOUT = T is not counted
C as a first call here, as no initialization or checking of
C input is done. (Such a call is sometimes useful for the
C purpose of outputting the initial conditions.)
C Thus the first call for which TOUT .ne. T requires
C ISTATE = 1 on input.
C
C On output, ISTATE has the following values and meanings.
C 1 means nothing was done; TOUT = T and ISTATE = 1 on input.
C 2 means the integration was performed successfully.
C 3 means the integration was successful, and one or more
C roots were found before satisfying the stop condition
C specified by ITASK. See JROOT.
C -1 means an excessive amount of work (more than MXSTEP
C steps) was done on this call, before completing the
C requested task, but the integration was otherwise
C successful as far as T. (MXSTEP is an optional input
C and is normally 500.) To continue, the user may
C simply reset ISTATE to a value .gt. 1 and call again
C (the excess work step counter will be reset to 0).
C In addition, the user may increase MXSTEP to avoid
C this error return (see below on optional inputs).
C -2 means too much accuracy was requested for the precision
C of the machine being used. This was detected before
C completing the requested task, but the integration
C was successful as far as T. To continue, the tolerance
C parameters must be reset, and ISTATE must be set
C to 3. The optional output TOLSF may be used for this
C purpose. (Note: If this condition is detected before
C taking any steps, then an illegal input return
C (ISTATE = -3) occurs instead.)
C -3 means illegal input was detected, before taking any
C integration steps. See written message for details.
C Note: If the solver detects an infinite loop of calls
C to the solver with illegal input, it will cause
C the run to stop.
C -4 means there were repeated error test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C The problem may have a singularity, or the input
C may be inappropriate.
C -5 means there were repeated convergence test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C -6 means EWT(i) became zero for some i during the
C integration. Pure relative error control (ATOL(i)=0.0)
C was requested on a variable which has now vanished.
C The integration was successful as far as T.
C -7 means the PSOL routine returned an unrecoverable error
C flag (IER .lt. 0). The integration was successful as
C far as T.
C
C Note: Since the normal output value of ISTATE is 2,
C it does not need to be reset for normal continuation.
C Also, since a negative input value of ISTATE will be
C regarded as illegal, a negative output value requires the
C user to change it, and possibly other inputs, before
C calling the solver again.
C
C IOPT = an integer flag to specify whether or not any optional
C inputs are being used on this call. Input only.
C The optional inputs are listed separately below.
C IOPT = 0 means no optional inputs are being used.
C Default values will be used in all cases.
C IOPT = 1 means one or more optional inputs are being used.
C
C RWORK = a real working array (double precision).
C The length of RWORK must be at least
C 20 + NYH*(MAXORD+1) + 3*NEQ + 3*NG + LENLS + LWP where
C NYH = the initial value of NEQ,
C MAXORD = 12 (if METH = 1) or 5 (if METH = 2) (unless a
C smaller value is given as an optional input),
C LENLS = length of work space for linear system (Krylov)
C method, excluding preconditioning:
C LENLS = 0 if MITER = 0,
C LENLS = NEQ*(MAXL+3) + MAXL**2 if MITER = 1,
C LENLS = NEQ*(MAXL+3+MIN(1,MAXL-KMP))
C + (MAXL+3)*MAXL + 1 if MITER = 2,
C LENLS = 6*NEQ if MITER = 3 or 4,
C LENLS = 3*NEQ if MITER = 9.
C (See the MF description for METH and MITER, and the
C list of optional inputs for MAXL and KMP.)
C LWP = length of real user work space for preconditioning
C (see JAC/PSOL).
C Thus if default values are used and NEQ is constant,
C this length is:
C 20 + 16*NEQ + 3*NG for MF = 10,
C 45 + 24*NEQ + 3*NG + LWP for MF = 11,
C 61 + 24*NEQ + 3*NG + LWP for MF = 12,
C 20 + 22*NEQ + 3*NG + LWP for MF = 13 or 14,
C 20 + 19*NEQ + 3*NG + LWP for MF = 19,
C 20 + 9*NEQ + 3*NG for MF = 20,
C 45 + 17*NEQ + 3*NG + LWP for MF = 21,
C 61 + 17*NEQ + 3*NG + LWP for MF = 22,
C 20 + 15*NEQ + 3*NG + LWP for MF = 23 or 24,
C 20 + 12*NEQ + 3*NG + LWP for MF = 29.
C The first 20 words of RWORK are reserved for conditional
C and optional inputs and optional outputs.
C
C The following word in RWORK is a conditional input:
C RWORK(1) = TCRIT = critical value of t which the solver
C is not to overshoot. Required if ITASK is
C 4 or 5, and ignored otherwise. (See ITASK.)
C
C LRW = the length of the array RWORK, as declared by the user.
C (This will be checked by the solver.)
C
C IWORK = an integer work array. The length of IWORK must be at least
C 30 if MITER = 0 (MF = 10 or 20),
C 30 + MAXL + LIWP if MITER = 1 (MF = 11, 21),
C 30 + LIWP if MITER = 2, 3, 4, or 9.
C MAXL = 5 unless a different optional input value is given.
C LIWP = length of integer user work space for preconditioning
C (see conditional input list following).
C The first few words of IWORK are used for conditional and
C optional inputs and optional outputs.
C
C The following 4 words in IWORK are conditional inputs,
C required if MITER .ge. 1:
C IWORK(1) = LWP = length of real array WP for use in
C preconditioning (part of RWORK array).
C IWORK(2) = LIWP = length of integer array IWP for use in
C preconditioning (part of IWORK array).
C The arrays WP and IWP are work arrays under the
C user's control, for use in the routines that
C perform preconditioning operations (JAC and PSOL).
C IWORK(3) = JPRE = preconditioner type flag:
C = 0 for no preconditioning (P1 = P2 = P = identity)
C = 1 for left-only preconditioning (P2 = identity)
C = 2 for right-only preconditioning (P1 = identity)
C = 3 for two-sided preconditioning (and PCG or PCGS)
C IWORK(4) = JACFLG = flag for whether JAC is called.
C = 0 if JAC is not to be called,
C = 1 if JAC is to be called.
C Use JACFLG = 1 if JAC computes any nonconstant
C data needed in preconditioning operations,
C such as some of the Jacobian elements.
C
C LIW = the length of the array IWORK, as declared by the user.
C (This will be checked by the solver.)
C
C Note: The work arrays must not be altered between calls to DLSODKR
C for the same problem, except possibly for the conditional and
C optional inputs, and except for the last 3*NEQ words of RWORK.
C The latter space is used for internal scratch space, and so is
C available for use by the user outside DLSODKR between calls, if
C desired (but not for use by any of the user-supplied routines).
C
C JAC = the name of the user-supplied routine to compute any
C Jacobian elements (or approximations) involved in the
C matrix preconditioning operations (MITER .ge. 1).
C It is to have the form
C SUBROUTINE JAC (F, NEQ, T, Y, YSV, REWT, FTY, V,
C 1 HL0, JOK, WP, IWP, IER)
C DOUBLE PRECISION T, Y(*), YSV(*), REWT(*), FTY(*), V(*),
C 1 HL0, WP(*)
C INTEGER IWP(*)
C This routine must evaluate and preprocess any parts of the
C Jacobian matrix df/dy used in the preconditioners P1, P2, P.
C The Y and FTY arrays contain the current values of y and
C f(t,y), respectively, and the YSV array also contains
C the current y vector. The array V is work space of length
C NEQ for use by JAC. REWT is the array of reciprocal error
C weights (1/EWT). JAC must multiply all computed Jacobian
C elements by the scalar -HL0, add the identity matrix, and do
C any factorization operations called for, in preparation
C for solving linear systems with a coefficient matrix of
C P1, P2, or P. The matrix P1*P2 or P should be an
C approximation to identity - hl0 * (df/dy). JAC should
C return IER = 0 if successful, and IER .ne. 0 if not.
C (If IER .ne. 0, a smaller time step will be tried.)
C The arrays WP (of length LWP) and IWP (of length LIWP)
C are for use by JAC and PSOL for work space and for storage
C of data needed for the solution of the preconditioner
C linear systems. Their lengths and contents are under the
C user's control.
C The argument JOK is an input flag for optional use
C by JAC in deciding whether to recompute Jacobian elements
C or use saved values. If JOK = -1, then JAC must compute
C any relevant Jacobian elements (or approximations) used in
C the preconditioners. Optionally, JAC may also save these
C elements for later reuse. If JOK = 1, the integrator has
C made a judgement (based on the convergence history and the
C value of HL0) that JAC need not recompute Jacobian elements,
C but instead use saved values, and the current value of HL0,
C to reconstruct the preconditioner matrices, followed by
C any required factorizations. This may be cost-effective if
C Jacobian elements are costly and storage is available.
C JAC may alter Y and V, but not YSV, REWT, FTY, or HL0.
C JAC must be declared External in the calling program.
C Subroutine JAC may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in JAC) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y above.
C
C PSOL = the name of the user-supplied routine for the
C solution of preconditioner linear systems.
C It is to have the form
C SUBROUTINE PSOL (NEQ, T, Y, FTY, WK,HL0, WP,IWP, B, LR,IER)
C DOUBLE PRECISION T, Y(*), FTY(*), WK(*), HL0, WP(*), B(*)
C INTEGER IWP(*)
C This routine must solve a linear system with B as right-hand
C side and one of the preconditioning matrices, P1, P2, or P,
C as coefficient matrix, and return the solution vector in B.
C LR is a flag concerning left vs right preconditioning, input
C to PSOL. PSOL is to use P1 if LR = 1 and P2 if LR = 2.
C In the case of the PCG or PCGS method, LR will be 3, and PSOL
C should solve the system P*x = B with the preconditioner P.
C In the case MITER = 9 (no Krylov iteration), LR will be 0,
C and PSOL is to return in B the desired approximate solution
C to A * x = B, where A = identity - hl0 * (df/dy).
C PSOL can use data generated in the JAC routine and stored in
C WP and IWP.
C The Y and FTY arrays contain the current values of y and
C f(t,y), respectively. The array WK is work space of length
C NEQ for use by PSOL.
C The argument HL0 is the current value of the scalar appearing
C in the linear system. If the old value, as of the last
C JAC call, is needed, it must have been saved by JAC in WP.
C On return, PSOL should set the error flag IER as follows:
C IER = 0 if PSOL was successful,
C IER .gt. 0 on a recoverable error, meaning that the
C time step will be retried,
C IER .lt. 0 on an unrecoverable error, meaning that the
C solver is to stop immediately.
C PSOL may not alter Y, FTY, or HL0.
C PSOL must be declared External in the calling program.
C Subroutine PSOL may access user-defined quantities in
C NEQ(2),... and Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in PSOL) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y above.
C
C MF = the method flag. Used only for input. The legal values of
C MF are 10, 11, 12, 13, 14, 19, 20, 21, 22, 23, 24, and 29.
C MF has decimal digits METH and MITER: MF = 10*METH + MITER.
C METH indicates the basic linear multistep method:
C METH = 1 means the implicit Adams method.
C METH = 2 means the method based on Backward
C Differentiation Formulas (BDFs).
C MITER indicates the corrector iteration method:
C MITER = 0 means functional iteration (no linear system
C is involved).
C MITER = 1 means Newton iteration with Scaled Preconditioned
C Incomplete Orthogonalization Method (SPIOM)
C for the linear systems.
C MITER = 2 means Newton iteration with Scaled Preconditioned
C Incomplete Generalized Minimal Residual method
C (SPIGMR) for the linear systems.
C MITER = 3 means Newton iteration with Preconditioned
C Conjugate Gradient method (PCG)
C for the linear systems.
C MITER = 4 means Newton iteration with scaled preconditioned
C Conjugate Gradient method (PCGS)
C for the linear systems.
C MITER = 9 means Newton iteration with only the
C user-supplied PSOL routine called (no Krylov
C iteration) for the linear systems.
C JPRE is ignored, and PSOL is called with LR = 0.
C See comments in the introduction about the choice of MITER.
C If MITER .ge. 1, the user must supply routines JAC and PSOL
C (the names are arbitrary) as described above.
C For MITER = 0, a dummy argument can be used.
C
C G = the name of subroutine for constraint functions, whose
C roots are desired during the integration. It is to have
C the form
C SUBROUTINE G (NEQ, T, Y, NG, GOUT)
C DOUBLE PRECISION T, Y(*), GOUT(NG)
C where NEQ, T, Y, and NG are input, and the array GOUT
C is output. NEQ, T, and Y have the same meaning as in
C the F routine, and GOUT is an array of length NG.
C For i = 1,...,NG, this routine is to load into GOUT(i)
C the value at (t,y) of the i-th constraint function g(i).
C DLSODKR will find roots of the g(i) of odd multiplicity
C (i.e. sign changes) as they occur during the integration.
C G must be declared External in the calling program.
C
C Caution: Because of numerical errors in the functions
C g(i) due to roundoff and integration error, DLSODKR may
C return false roots, or return the same root at two or more
C nearly equal values of t. If such false roots are
C suspected, the user should consider smaller error tolerances
C and/or higher precision in the evaluation of the g(i).
C
C If a root of some g(i) defines the end of the problem,
C the input to DLSODKR should nevertheless allow integration
C to a point slightly past that root, so that DLSODKR can
C locate the root by interpolation.
C
C Subroutine G may access user-defined quantities in
C NEQ(2),... and Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in G) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y above.
C
C NG = number of constraint functions g(i). If there are none,
C set NG = 0, and pass a dummy name for G.
C
C JROOT = integer array of length NG. Used only for output.
C On a return with ISTATE = 3 (one or more roots found),
C JROOT(i) = 1 if g(i) has a root at t, or JROOT(i) = 0 if not.
C-----------------------------------------------------------------------
C Optional Inputs.
C
C The following is a list of the optional inputs provided for in the
C call sequence. (See also Part 2.) For each such input variable,
C this table lists its name as used in this documentation, its
C location in the call sequence, its meaning, and the default value.
C The use of any of these inputs requires IOPT = 1, and in that
C case all of these inputs are examined. A value of zero for any
C of these optional inputs will cause the default value to be used.
C Thus to use a subset of the optional inputs, simply preload
C locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively, and
C then set those of interest to nonzero values.
C
C Name Location Meaning and Default Value
C
C H0 RWORK(5) the step size to be attempted on the first step.
C The default value is determined by the solver.
C
C HMAX RWORK(6) the maximum absolute step size allowed.
C The default value is infinite.
C
C HMIN RWORK(7) the minimum absolute step size allowed.
C The default value is 0. (This lower bound is not
C enforced on the final step before reaching TCRIT
C when ITASK = 4 or 5.)
C
C DELT RWORK(8) convergence test constant in Krylov iteration
C algorithm. The default is .05.
C
C MAXORD IWORK(5) the maximum order to be allowed. The default
C value is 12 if METH = 1, and 5 if METH = 2.
C If MAXORD exceeds the default value, it will
C be reduced to the default value.
C If MAXORD is changed during the problem, it may
C cause the current order to be reduced.
C
C MXSTEP IWORK(6) maximum number of (internally defined) steps
C allowed during one call to the solver.
C The default value is 500.
C
C MXHNIL IWORK(7) maximum number of messages printed (per problem)
C warning that T + H = T on a step (H = step size).
C This must be positive to result in a non-default
C value. The default value is 10.
C
C MAXL IWORK(8) maximum number of iterations in the SPIOM, SPIGMR,
C PCG, or PCGS algorithm (.le. NEQ).
C The default is MAXL = MIN(5,NEQ).
C
C KMP IWORK(9) number of vectors on which orthogonalization
C is done in SPIOM or SPIGMR algorithm (.le. MAXL).
C The default is KMP = MAXL.
C Note: When KMP .lt. MAXL and MF = 22, the length
C of RWORK must be defined accordingly. See
C the definition of RWORK above.
C-----------------------------------------------------------------------
C Optional Outputs.
C
C As optional additional output from DLSODKR, the variables listed
C below are quantities related to the performance of DLSODKR
C which are available to the user. These are communicated by way of
C the work arrays, but also have internal mnemonic names as shown.
C Except where stated otherwise, all of these outputs are defined
C on any successful return from DLSODKR, and on any return with
C ISTATE = -1, -2, -4, -5, -6, or -7. On an illegal input return
C (ISTATE = -3), they will be unchanged from their existing values
C (if any), except possibly for TOLSF, LENRW, and LENIW.
C On any error return, outputs relevant to the error will be defined,
C as noted below.
C
C Name Location Meaning
C
C HU RWORK(11) the step size in t last used (successfully).
C
C HCUR RWORK(12) the step size to be attempted on the next step.
C
C TCUR RWORK(13) the current value of the independent variable
C which the solver has actually reached, i.e. the
C current internal mesh point in t. On output, TCUR
C will always be at least as far as the argument
C T, but may be farther (if interpolation was done).
C
C TOLSF RWORK(14) a tolerance scale factor, greater than 1.0,
C computed when a request for too much accuracy was
C detected (ISTATE = -3 if detected at the start of
C the problem, ISTATE = -2 otherwise). If ITOL is
C left unaltered but RTOL and ATOL are uniformly
C scaled up by a factor of TOLSF for the next call,
C then the solver is deemed likely to succeed.
C (The user may also ignore TOLSF and alter the
C tolerance parameters in any other way appropriate.)
C
C NGE IWORK(10) the number of g evaluations for the problem so far.
C
C NST IWORK(11) the number of steps taken for the problem so far.
C
C NFE IWORK(12) the number of f evaluations for the problem so far.
C
C NPE IWORK(13) the number of calls to JAC so far (for evaluation
C of preconditioners).
C
C NQU IWORK(14) the method order last used (successfully).
C
C NQCUR IWORK(15) the order to be attempted on the next step.
C
C IMXER IWORK(16) the index of the component of largest magnitude in
C the weighted local error vector ( E(i)/EWT(i) ),
C on an error return with ISTATE = -4 or -5.
C
C LENRW IWORK(17) the length of RWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C LENIW IWORK(18) the length of IWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C NNI IWORK(19) number of nonlinear iterations so far (each of
C which calls an iterative linear solver).
C
C NLI IWORK(20) number of linear iterations so far.
C Note: A measure of the success of algorithm is
C the average number of linear iterations per
C nonlinear iteration, given by NLI/NNI.
C If this is close to MAXL, MAXL may be too small.
C
C NPS IWORK(21) number of preconditioning solve operations
C (PSOL calls) so far.
C
C NCFN IWORK(22) number of convergence failures of the nonlinear
C (Newton) iteration so far.
C Note: A measure of success is the overall
C rate of nonlinear convergence failures, NCFN/NST.
C
C NCFL IWORK(23) number of convergence failures of the linear
C iteration so far.
C Note: A measure of success is the overall
C rate of linear convergence failures, NCFL/NNI.
C
C NSFI IWORK(24) number of functional iteration steps so far.
C Note: A measure of the extent to which the
C problem is nonstiff is the ratio NSFI/NST.
C
C NJEV IWORK(25) number of JAC calls with JOK = -1 so far
C (number of evaluations of Jacobian data).
C
C The following two arrays are segments of the RWORK array which
C may also be of interest to the user as optional outputs.
C For each array, the table below gives its internal name,
C its base address in RWORK, and its description.
C
C Name Base Address Description
C
C YH 21 + 3*NG the Nordsieck history array, of size NYH by
C (NQCUR + 1), where NYH is the initial value
C of NEQ. For j = 0,1,...,NQCUR, column j+1
C of YH contains HCUR**j/factorial(j) times
C the j-th derivative of the interpolating
C polynomial currently representing the solution,
C evaluated at t = TCUR.
C
C ACOR LENRW-NEQ+1 array of size NEQ used for the accumulated
C corrections on each step, scaled on output
C to represent the estimated local error in y
C on the last step. This is the vector E in
C the description of the error control. It is
C defined only on a successful return from
C DLSODKR.
C
C-----------------------------------------------------------------------
C Part 2. Other Routines Callable.
C
C The following are optional calls which the user may make to
C gain additional capabilities in conjunction with DLSODKR.
C (The routines XSETUN and XSETF are designed to conform to the
C SLATEC error handling package.)
C
C Form of Call Function
C CALL XSETUN(LUN) Set the logical unit number, LUN, for
C output of messages from DLSODKR, if
C the default is not desired.
C The default value of LUN is 6.
C
C CALL XSETF(MFLAG) Set a flag to control the printing of
C messages by DLSODKR.
C MFLAG = 0 means do not print. (Danger:
C This risks losing valuable information.)
C MFLAG = 1 means print (the default).
C
C Either of the above calls may be made at
C any time and will take effect immediately.
C
C CALL DSRCKR(RSAV,ISAV,JOB) saves and restores the contents of
C the internal Common blocks used by
C DLSODKR (see Part 3 below).
C RSAV must be a real array of length 228
C or more, and ISAV must be an integer
C array of length 63 or more.
C JOB=1 means save Common into RSAV/ISAV.
C JOB=2 means restore Common from RSAV/ISAV.
C DSRCKR is useful if one is
C interrupting a run and restarting
C later, or alternating between two or
C more problems solved with DLSODKR.
C
C CALL DINTDY(,,,,,) Provide derivatives of y, of various
C (see below) orders, at a specified point t, if
C desired. It may be called only after
C a successful return from DLSODKR.
C
C The detailed instructions for using DINTDY are as follows.
C The form of the call is:
C
C LYH = 21 + 3*NG
C CALL DINTDY (T, K, RWORK(LYH), NYH, DKY, IFLAG)
C
C The input parameters are:
C
C T = value of independent variable where answers are desired
C (normally the same as the T last returned by DLSODKR).
C For valid results, T must lie between TCUR - HU and TCUR.
C (See optional outputs for TCUR and HU.)
C K = integer order of the derivative desired. K must satisfy
C 0 .le. K .le. NQCUR, where NQCUR is the current order
C (see optional outputs). The capability corresponding
C to K = 0, i.e. computing y(T), is already provided
C by DLSODKR directly. Since NQCUR .ge. 1, the first
C derivative dy/dt is always available with DINTDY.
C LYH = 21 + 3*NG = base address in RWORK of the history array YH.
C NYH = column length of YH, equal to the initial value of NEQ.
C
C The output parameters are:
C
C DKY = a real array of length NEQ containing the computed value
C of the K-th derivative of y(t).
C IFLAG = integer flag, returned as 0 if K and T were legal,
C -1 if K was illegal, and -2 if T was illegal.
C On an error return, a message is also written.
C-----------------------------------------------------------------------
C Part 3. Common Blocks.
C
C If DLSODKR is to be used in an overlay situation, the user
C must declare, in the primary overlay, the variables in:
C (1) the call sequence to DLSODKR, and
C (2) the four internal Common blocks
C /DLS001/ of length 255 (218 double precision words
C followed by 37 integer words),
C /DLS002/ of length 5 (1 double precision word
C followed by 4 integer words),
C /DLPK01/ of length 17 (4 double precision words
C followed by 13 integer words),
C /DLSR01/ of length 14 (5 double precision words
C followed by 9 integer words).
C
C If DLSODKR is used on a system in which the contents of internal
C Common blocks are not preserved between calls, the user should
C declare the above Common blocks in the calling program to insure
C that their contents are preserved.
C
C If the solution of a given problem by DLSODKR is to be interrupted
C and then later continued, such as when restarting an interrupted run
C or alternating between two or more problems, the user should save,
C following the return from the last DLSODKR call prior to the
C interruption, the contents of the call sequence variables and the
C internal Common blocks, and later restore these values before the
C next DLSODKR call for that problem. To save and restore the Common
C blocks, use Subroutine DSRCKR (see Part 2 above).
C
C-----------------------------------------------------------------------
C Part 4. Optionally Replaceable Solver Routines.
C
C Below are descriptions of two routines in the DLSODKR package which
C relate to the measurement of errors. Either routine can be
C replaced by a user-supplied version, if desired. However, since such
C a replacement may have a major impact on performance, it should be
C done only when absolutely necessary, and only with great caution.
C (Note: The means by which the package version of a routine is
C superseded by the user's version may be system-dependent.)
C
C (a) DEWSET.
C The following subroutine is called just before each internal
C integration step, and sets the array of error weights, EWT, as
C described under ITOL/RTOL/ATOL above:
C SUBROUTINE DEWSET (NEQ, ITOL, RTOL, ATOL, YCUR, EWT)
C where NEQ, ITOL, RTOL, and ATOL are as in the DLSODKR call sequence,
C YCUR contains the current dependent variable vector, and
C EWT is the array of weights set by DEWSET.
C
C If the user supplies this subroutine, it must return in EWT(i)
C (i = 1,...,NEQ) a positive quantity suitable for comparing errors
C in y(i) to. The EWT array returned by DEWSET is passed to the DVNORM
C routine (see below), and also used by DLSODKR in the computation
C of the optional output IMXER, the diagonal Jacobian approximation,
C and the increments for difference quotient Jacobians.
C
C In the user-supplied version of DEWSET, it may be desirable to use
C the current values of derivatives of y. Derivatives up to order NQ
C are available from the history array YH, described above under
C optional outputs. In DEWSET, YH is identical to the YCUR array,
C extended to NQ + 1 columns with a column length of NYH and scale
C factors of H**j/factorial(j). On the first call for the problem,
C given by NST = 0, NQ is 1 and H is temporarily set to 1.0.
C NYH is the initial value of NEQ. The quantities NQ, H, and NST
C can be obtained by including in DEWSET the statements:
C DOUBLE PRECISION RLS
C COMMON /DLS001/ RLS(218),ILS(37)
C NQ = ILS(33)
C NST = ILS(34)
C H = RLS(212)
C Thus, for example, the current value of dy/dt can be obtained as
C YCUR(NYH+i)/H (i=1,...,NEQ) (and the division by H is
C unnecessary when NST = 0).
C
C (b) DVNORM.
C The following is a real function routine which computes the weighted
C root-mean-square norm of a vector v:
C D = DVNORM (N, V, W)
C where:
C N = the length of the vector,
C V = real array of length N containing the vector,
C W = real array of length N containing weights,
C D = SQRT( (1/N) * sum(V(i)*W(i))**2 ).
C DVNORM is called with N = NEQ and with W(i) = 1.0/EWT(i), where
C EWT is as set by Subroutine DEWSET.
C
C If the user supplies this function, it should return a non-negative
C value of DVNORM suitable for use in the error control in DLSODKR.
C None of the arguments should be altered by DVNORM.
C For example, a user-supplied DVNORM routine might:
C -substitute a max-norm of (V(i)*W(i)) for the RMS-norm, or
C -ignore some components of V in the norm, with the effect of
C suppressing the error control on those components of y.
C-----------------------------------------------------------------------
C
C***REVISION HISTORY (YYYYMMDD)
C 19900117 DATE WRITTEN
C 19900503 Added iteration switching (functional/Newton).
C 19900802 Added flag for Jacobian-saving in user preconditioner.
C 19900910 Added new initial stepsize routine LHIN.
C 19901019 Corrected LHIN - y array restored.
C 19910909 Changed names STOPK to STOKA, PKSET to SETPK;
C removed unused variables in driver declarations;
C minor corrections to main prologue.
C 20010425 Major update: convert source lines to upper case;
C added *DECK lines; changed from 1 to * in dummy dimensions;
C changed names R1MACH/D1MACH to RUMACH/DUMACH;
C renamed routines for uniqueness across single/double prec.;
C converted intrinsic names to generic form;
C removed ILLIN and NTREP (data loaded) from Common;
C removed all 'own' variables from Common;
C changed error messages to quoted strings;
C replaced XERRWV/XERRWD with 1993 revised version;
C converted prologues, comments, error messages to mixed case;
C numerous corrections to prologues and internal comments.
C 20010507 Converted single precision source to double precision.
C 20020502 Corrected declarations in descriptions of user routines.
C 20030603 Corrected duplicate type declaration for DUMACH.
C 20031105 Restored 'own' variables to Common blocks, to enable
C interrupt/restart feature.
C 20031112 Added SAVE statements for data-loaded constants.
C 20031117 Changed internal name NPE to NJE.
C
C-----------------------------------------------------------------------
C Other routines in the DLSODKR package.
C
C In addition to Subroutine DLSODKR, the DLSODKR package includes the
C following subroutines and function routines:
C DLHIN calculates a step size to be attempted initially.
C DRCHEK does preliminary checking for roots, and serves as an
C interface between Subroutine DLSODKR and Subroutine DROOTS.
C DROOTS finds the leftmost root of a set of functions.
C DINTDY computes an interpolated value of the y vector at t = TOUT.
C DEWSET sets the error weight vector EWT before each step.
C DVNORM computes the weighted RMS-norm of a vector.
C DSTOKA is the core integrator, which does one step of the
C integration and the associated error control.
C DCFODE sets all method coefficients and test constants.
C DSETPK interfaces between DSTOKA and the JAC routine.
C DSOLPK manages solution of linear system in Newton iteration.
C DSPIOM performs the SPIOM algorithm.
C DATV computes a scaled, preconditioned product (I-hl0*J)*v.
C DORTHOG orthogonalizes a vector against previous basis vectors.
C DHEFA generates an LU factorization of a Hessenberg matrix.
C DHESL solves a Hessenberg square linear system.
C DSPIGMR performs the SPIGMR algorithm.
C DHEQR generates a QR factorization of a Hessenberg matrix.
C DHELS finds the least squares solution of a Hessenberg system.
C DPCG performs preconditioned conjugate gradient algorithm (PCG).
C DPCGS performs the PCGS algorithm.
C DATP computes the product A*p, where A = I - hl0*df/dy.
C DUSOL interfaces to the user's PSOL routine (MITER = 9).
C DSRCKR is a user-callable routine to save and restore
C the contents of the internal Common blocks.
C DAXPY, DCOPY, DDOT, DNRM2, and DSCAL are basic linear
C algebra modules (from the BLAS collection).
C DUMACH computes the unit roundoff in a machine-independent manner.
C XERRWD, XSETUN, XSETF, IXSAV, and IUMACH handle the printing of all
C error messages and warnings. XERRWD is machine-dependent.
C Note: DVNORM, DDOT, DNRM2, DUMACH, IXSAV, and IUMACH are function
C routines. All the others are subroutines.
C
C-----------------------------------------------------------------------
DOUBLE PRECISION DUMACH, DVNORM
INTEGER INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER NEWT, NSFI, NSLJ, NJEV
INTEGER LG0, LG1, LGX, IOWNR3, IRFND, ITASKC, NGC, NGE
INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT,
1 NNI, NLI, NPS, NCFN, NCFL
INTEGER I, I1, I2, IER, IFLAG, IMXER, KGO, LF0,
1 LENIW, LENIWK, LENRW, LENWM, LENWK, LIWP, LWP, MORD, MXHNL0,
2 MXSTP0, NCFN0, NCFL0, NITER, NLI0, NNI0, NNID, NSTD, NWARN
INTEGER IRFP, IRT, LENYH, LYHNEW
DOUBLE PRECISION ROWNS,
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
DOUBLE PRECISION STIFR
DOUBLE PRECISION ROWNR3, T0, TLAST, TOUTC
DOUBLE PRECISION DELT, EPCON, SQRTN, RSQRTN
DOUBLE PRECISION ATOLI, AVDIM, BIG, EWTI, H0, HMAX, HMX, RCFL,
1 RCFN, RH, RTOLI, TCRIT, TNEXT, TOLSF, TP, SIZE
DIMENSION MORD(2)
LOGICAL IHIT, LAVD, LCFN, LCFL, LWARN
CHARACTER*60 MSG
SAVE MORD, MXSTP0, MXHNL0
C-----------------------------------------------------------------------
C The following four internal Common blocks contain
C (a) variables which are local to any subroutine but whose values must
C be preserved between calls to the routine ("own" variables), and
C (b) variables which are communicated between subroutines.
C The block DLS001 is declared in subroutines DLSODKR, DINTDY,
C DSTOKA, DSOLPK, and DATV.
C The block DLS002 is declared in subroutines DLSODKR and DSTOKA.
C The block DLSR01 is declared in subroutines DLSODKR, DRCHEK, DROOTS.
C The block DLPK01 is declared in subroutines DLSODKR, DSTOKA, DSETPK,
C and DSOLPK.
C Groups of variables are replaced by dummy arrays in the Common
C declarations in routines where those variables are not used.
C-----------------------------------------------------------------------
COMMON /DLS001/ ROWNS(209),
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
2 INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS(6),
3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
COMMON /DLS002/ STIFR, NEWT, NSFI, NSLJ, NJEV
C
COMMON /DLSR01/ ROWNR3(2), T0, TLAST, TOUTC,
1 LG0, LG1, LGX, IOWNR3(2), IRFND, ITASKC, NGC, NGE
C
COMMON /DLPK01/ DELT, EPCON, SQRTN, RSQRTN,
1 JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT,
2 NNI, NLI, NPS, NCFN, NCFL
C
DATA MORD(1),MORD(2)/12,5/, MXSTP0/500/, MXHNL0/10/
C-----------------------------------------------------------------------
C Block A.
C This code block is executed on every call.
C It tests ISTATE and ITASK for legality and branches appropriately.
C If ISTATE .gt. 1 but the flag INIT shows that initialization has
C not yet been done, an error return occurs.
C If ISTATE = 1 and TOUT = T, return immediately.
C-----------------------------------------------------------------------
IF (ISTATE .LT. 1 .OR. ISTATE .GT. 3) GO TO 601
IF (ITASK .LT. 1 .OR. ITASK .GT. 5) GO TO 602
ITASKC = ITASK
IF (ISTATE .EQ. 1) GO TO 10
IF (INIT .EQ. 0) GO TO 603
IF (ISTATE .EQ. 2) GO TO 200
GO TO 20
10 INIT = 0
IF (TOUT .EQ. T) RETURN
C-----------------------------------------------------------------------
C Block B.
C The next code block is executed for the initial call (ISTATE = 1),
C or for a continuation call with parameter changes (ISTATE = 3).
C It contains checking of all inputs and various initializations.
C
C First check legality of the non-optional inputs NEQ, ITOL, IOPT, MF,
C and NG.
C-----------------------------------------------------------------------
20 IF (NEQ(1) .LE. 0) GO TO 604
IF (ISTATE .EQ. 1) GO TO 25
IF (NEQ(1) .GT. N) GO TO 605
25 N = NEQ(1)
IF (ITOL .LT. 1 .OR. ITOL .GT. 4) GO TO 606
IF (IOPT .LT. 0 .OR. IOPT .GT. 1) GO TO 607
METH = MF/10
MITER = MF - 10*METH
IF (METH .LT. 1 .OR. METH .GT. 2) GO TO 608
IF (MITER .LT. 0) GO TO 608
IF (MITER .GT. 4 .AND. MITER .LT. 9) GO TO 608
IF (MITER .GE. 1) JPRE = IWORK(3)
JACFLG = 0
IF (MITER .GE. 1) JACFLG = IWORK(4)
IF (NG .LT. 0) GO TO 630
IF (ISTATE .EQ. 1) GO TO 35
IF (IRFND .EQ. 0 .AND. NG .NE. NGC) GO TO 631
35 NGC = NG
C Next process and check the optional inputs. --------------------------
IF (IOPT .EQ. 1) GO TO 40
MAXORD = MORD(METH)
MXSTEP = MXSTP0
MXHNIL = MXHNL0
IF (ISTATE .EQ. 1) H0 = 0.0D0
HMXI = 0.0D0
HMIN = 0.0D0
MAXL = MIN(5,N)
KMP = MAXL
DELT = 0.05D0
GO TO 60
40 MAXORD = IWORK(5)
IF (MAXORD .LT. 0) GO TO 611
IF (MAXORD .EQ. 0) MAXORD = 100
MAXORD = MIN(MAXORD,MORD(METH))
MXSTEP = IWORK(6)
IF (MXSTEP .LT. 0) GO TO 612
IF (MXSTEP .EQ. 0) MXSTEP = MXSTP0
MXHNIL = IWORK(7)
IF (MXHNIL .LT. 0) GO TO 613
IF (MXHNIL .EQ. 0) MXHNIL = MXHNL0
IF (ISTATE .NE. 1) GO TO 50
H0 = RWORK(5)
IF ((TOUT - T)*H0 .LT. 0.0D0) GO TO 614
50 HMAX = RWORK(6)
IF (HMAX .LT. 0.0D0) GO TO 615
HMXI = 0.0D0
IF (HMAX .GT. 0.0D0) HMXI = 1.0D0/HMAX
HMIN = RWORK(7)
IF (HMIN .LT. 0.0D0) GO TO 616
MAXL = IWORK(8)
IF (MAXL .EQ. 0) MAXL = 5
MAXL = MIN(MAXL,N)
KMP = IWORK(9)
IF (KMP .EQ. 0 .OR. KMP .GT. MAXL) KMP = MAXL
DELT = RWORK(8)
IF (DELT .EQ. 0.0D0) DELT = 0.05D0
C-----------------------------------------------------------------------
C Set work array pointers and check lengths LRW and LIW.
C Pointers to segments of RWORK and IWORK are named by prefixing L to
C the name of the segment. E.g., the segment YH starts at RWORK(LYH).
C RWORK segments (in order) are denoted G0, G1, GX, YH, WM,
C EWT, SAVF, SAVX, ACOR.
C-----------------------------------------------------------------------
60 IF (ISTATE .EQ. 1) NYH = N
LG0 = 21
LG1 = LG0 + NG
LGX = LG1 + NG
LYHNEW = LGX + NG
IF (ISTATE .EQ. 1) LYH = LYHNEW
IF (LYHNEW .EQ. LYH) GO TO 62
C If ISTATE = 3 and NG was changed, shift YH to its new location. ------
LENYH = L*NYH
IF (LRW .LT. LYHNEW-1+LENYH) GO TO 62
I1 = 1
IF (LYHNEW .GT. LYH) I1 = -1
CALL DCOPY (LENYH, RWORK(LYH), I1, RWORK(LYHNEW), I1)
LYH = LYHNEW
62 CONTINUE
LWM = LYH + (MAXORD + 1)*NYH
IF (MITER .EQ. 0) LENWK = 0
IF (MITER .EQ. 1) LENWK = N*(MAXL+2) + MAXL*MAXL
IF (MITER .EQ. 2)
1 LENWK = N*(MAXL+2+MIN(1,MAXL-KMP)) + (MAXL+3)*MAXL + 1
IF (MITER .EQ. 3 .OR. MITER .EQ. 4) LENWK = 5*N
IF (MITER .EQ. 9) LENWK = 2*N
LWP = 0
IF (MITER .GE. 1) LWP = IWORK(1)
LENWM = LENWK + LWP
LOCWP = LENWK + 1
LEWT = LWM + LENWM
LSAVF = LEWT + N
LSAVX = LSAVF + N
LACOR = LSAVX + N
IF (MITER .EQ. 0) LACOR = LSAVF + N
LENRW = LACOR + N - 1
IWORK(17) = LENRW
LIWM = 31
LENIWK = 0
IF (MITER .EQ. 1) LENIWK = MAXL
LIWP = 0
IF (MITER .GE. 1) LIWP = IWORK(2)
LENIW = 30 + LENIWK + LIWP
LOCIWP = LENIWK + 1
IWORK(18) = LENIW
IF (LENRW .GT. LRW) GO TO 617
IF (LENIW .GT. LIW) GO TO 618
C Check RTOL and ATOL for legality. ------------------------------------
RTOLI = RTOL(1)
ATOLI = ATOL(1)
DO 70 I = 1,N
IF (ITOL .GE. 3) RTOLI = RTOL(I)
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
IF (RTOLI .LT. 0.0D0) GO TO 619
IF (ATOLI .LT. 0.0D0) GO TO 620
70 CONTINUE
C Load SQRT(N) and its reciprocal in Common. ---------------------------
SQRTN = SQRT(REAL(N))
RSQRTN = 1.0D0/SQRTN
IF (ISTATE .EQ. 1) GO TO 100
C If ISTATE = 3, set flag to signal parameter changes to DSTOKA.--------
JSTART = -1
IF (NQ .LE. MAXORD) GO TO 90
C MAXORD was reduced below NQ. Copy YH(*,MAXORD+2) into SAVF. ---------
DO 80 I = 1,N
80 RWORK(I+LSAVF-1) = RWORK(I+LWM-1)
90 CONTINUE
IF (N .EQ. NYH) GO TO 200
C NEQ was reduced. Zero part of YH to avoid undefined references. -----
I1 = LYH + L*NYH
I2 = LYH + (MAXORD + 1)*NYH - 1
IF (I1 .GT. I2) GO TO 200
DO 95 I = I1,I2
95 RWORK(I) = 0.0D0
GO TO 200
C-----------------------------------------------------------------------
C Block C.
C The next block is for the initial call only (ISTATE = 1).
C It contains all remaining initializations, the initial call to F,
C and the calculation of the initial step size.
C The error weights in EWT are inverted after being loaded.
C-----------------------------------------------------------------------
100 UROUND = DUMACH()
TN = T
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 110
TCRIT = RWORK(1)
IF ((TCRIT - TOUT)*(TOUT - T) .LT. 0.0D0) GO TO 625
IF (H0 .NE. 0.0D0 .AND. (T + H0 - TCRIT)*H0 .GT. 0.0D0)
1 H0 = TCRIT - T
110 JSTART = 0
NHNIL = 0
NST = 0
NJE = 0
NSLAST = 0
NLI0 = 0
NNI0 = 0
NCFN0 = 0
NCFL0 = 0
NWARN = 0
HU = 0.0D0
NQU = 0
CCMAX = 0.3D0
MAXCOR = 3
MSBP = 20
MXNCF = 10
NNI = 0
NLI = 0
NPS = 0
NCFN = 0
NCFL = 0
NSFI = 0
NJEV = 0
C Initial call to F. (LF0 points to YH(*,2).) -------------------------
LF0 = LYH + NYH
CALL F (NEQ, T, Y, RWORK(LF0))
NFE = 1
C Load the initial value vector in YH. ---------------------------------
DO 115 I = 1,N
115 RWORK(I+LYH-1) = Y(I)
C Load and invert the EWT array. (H is temporarily set to 1.0.) -------
NQ = 1
H = 1.0D0
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 120 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 621
120 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
IF (H0 .NE. 0.0D0) GO TO 180
C Call DLHIN to set initial step size H0 to be attempted. --------------
CALL DLHIN (NEQ, N, T, RWORK(LYH), RWORK(LF0), F, TOUT, UROUND,
1 RWORK(LEWT), ITOL, ATOL, Y, RWORK(LACOR), H0, NITER, IER)
NFE = NFE + NITER
IF (IER .NE. 0) GO TO 622
C Adjust H0 if necessary to meet HMAX bound. ---------------------------
180 RH = ABS(H0)*HMXI
IF (RH .GT. 1.0D0) H0 = H0/RH
C Load H with H0 and scale YH(*,2) by H0. ------------------------------
H = H0
DO 190 I = 1,N
190 RWORK(I+LF0-1) = H0*RWORK(I+LF0-1)
C Check for a zero of g at T. ------------------------------------------
IRFND = 0
TOUTC = TOUT
IF (NGC .EQ. 0) GO TO 270
CALL DRCHEK (1, G, NEQ, Y, RWORK(LYH), NYH,
1 RWORK(LG0), RWORK(LG1), RWORK(LGX), JROOT, IRT)
IF (IRT .EQ. 0) GO TO 270
GO TO 632
C-----------------------------------------------------------------------
C Block D.
C The next code block is for continuation calls only (ISTATE = 2 or 3)
C and is to check stop conditions before taking a step.
C First, DRCHEK is called to check for a root within the last step
C taken, other than the last root found there, if any.
C If ITASK = 2 or 5, and y(TN) has not yet been returned to the user
C because of an intervening root, return through Block G.
C-----------------------------------------------------------------------
200 NSLAST = NST
C
IRFP = IRFND
IF (NGC .EQ. 0) GO TO 205
IF (ITASK .EQ. 1 .OR. ITASK .EQ. 4) TOUTC = TOUT
CALL DRCHEK (2, G, NEQ, Y, RWORK(LYH), NYH,
1 RWORK(LG0), RWORK(LG1), RWORK(LGX), JROOT, IRT)
IF (IRT .NE. 1) GO TO 205
IRFND = 1
ISTATE = 3
T = T0
GO TO 425
205 CONTINUE
IRFND = 0
IF (IRFP .EQ. 1 .AND. TLAST .NE. TN .AND. ITASK .EQ. 2) GO TO 400
C
NLI0 = NLI
NNI0 = NNI
NCFN0 = NCFN
NCFL0 = NCFL
NWARN = 0
GO TO (210, 250, 220, 230, 240), ITASK
210 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
220 TP = TN - HU*(1.0D0 + 100.0D0*UROUND)
IF ((TP - TOUT)*H .GT. 0.0D0) GO TO 623
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
GO TO 400
230 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
IF ((TCRIT - TOUT)*H .LT. 0.0D0) GO TO 625
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 245
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
240 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
245 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) T = TCRIT
IF (IRFP .EQ. 1 .AND. TLAST .NE. TN .AND. ITASK .EQ. 5) GO TO 400
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
IF (ISTATE .EQ. 2) JSTART = -2
C-----------------------------------------------------------------------
C Block E.
C The next block is normally executed for all calls and contains
C the call to the one-step core integrator DSTOKA.
C
C This is a looping point for the integration steps.
C
C First check for too many steps being taken,
C check for poor Newton/Krylov method performance, update EWT (if not
C at start of problem), check for too much accuracy being requested,
C and check for H below the roundoff level in T.
C-----------------------------------------------------------------------
250 CONTINUE
IF ((NST-NSLAST) .GE. MXSTEP) GO TO 500
NSTD = NST - NSLAST
NNID = NNI - NNI0
IF (NSTD .LT. 10 .OR. NNID .EQ. 0) GO TO 255
AVDIM = REAL(NLI - NLI0)/REAL(NNID)
RCFN = REAL(NCFN - NCFN0)/REAL(NSTD)
RCFL = REAL(NCFL - NCFL0)/REAL(NNID)
LAVD = AVDIM .GT. (MAXL - 0.05D0)
LCFN = RCFN .GT. 0.9D0
LCFL = RCFL .GT. 0.9D0
LWARN = LAVD .OR. LCFN .OR. LCFL
IF (.NOT.LWARN) GO TO 255
NWARN = NWARN + 1
IF (NWARN .GT. 10) GO TO 255
IF (LAVD) THEN
MSG='DLSODKR- Warning. Poor iterative algorithm performance seen '
CALL XERRWD (MSG, 60, 111, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
IF (LAVD) THEN
MSG=' at T = R1 by average no. of linear iterations = R2 '
CALL XERRWD (MSG, 60, 111, 0, 0, 0, 0, 2, TN, AVDIM)
ENDIF
IF (LCFN) THEN
MSG='DLSODKR- Warning. Poor iterative algorithm performance seen '
CALL XERRWD (MSG, 60, 112, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
IF (LCFN) THEN
MSG=' at T = R1 by nonlinear convergence failure rate = R2 '
CALL XERRWD (MSG, 60, 112, 0, 0, 0, 0, 2, TN, RCFN)
ENDIF
IF (LCFL) THEN
MSG='DLSODKR- Warning. Poor iterative algorithm performance seen '
CALL XERRWD (MSG, 60, 113, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
IF (LCFL) THEN
MSG=' at T = R1 by linear convergence failure rate = R2 '
CALL XERRWD (MSG, 60, 113, 0, 0, 0, 0, 2, TN, RCFL)
ENDIF
255 CONTINUE
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 260 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 510
260 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
270 TOLSF = UROUND*DVNORM (N, RWORK(LYH), RWORK(LEWT))
IF (TOLSF .LE. 1.0D0) GO TO 280
TOLSF = TOLSF*2.0D0
IF (NST .EQ. 0) GO TO 626
GO TO 520
280 IF ((TN + H) .NE. TN) GO TO 290
NHNIL = NHNIL + 1
IF (NHNIL .GT. MXHNIL) GO TO 290
MSG = 'DLSODKR- Warning.. Internal T(=R1) and H(=R2) are'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' such that in the machine, T + H = T on the next step '
CALL XERRWD (MSG, 60, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' (H = step size). Solver will continue anyway.'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 2, TN, H)
IF (NHNIL .LT. MXHNIL) GO TO 290
MSG = 'DLSODKR- Above warning has been issued I1 times. '
CALL XERRWD (MSG, 50, 102, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' It will not be issued again for this problem.'
CALL XERRWD (MSG, 50, 102, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
290 CONTINUE
C-----------------------------------------------------------------------
C CALL DSTOKA(NEQ,Y,YH,NYH,YH,EWT,SAVF,SAVX,ACOR,WM,IWM,F,JAC,PSOL)
C-----------------------------------------------------------------------
CALL DSTOKA (NEQ, Y, RWORK(LYH), NYH, RWORK(LYH), RWORK(LEWT),
1 RWORK(LSAVF), RWORK(LSAVX), RWORK(LACOR), RWORK(LWM),
2 IWORK(LIWM), F, JAC, PSOL)
KGO = 1 - KFLAG
GO TO (300, 530, 540, 550), KGO
C-----------------------------------------------------------------------
C Block F.
C The following block handles the case of a successful return from the
C core integrator (KFLAG = 0).
C Call DRCHEK to check for a root within the last step.
C Then, if no root was found, check for stop conditions.
C-----------------------------------------------------------------------
300 INIT = 1
C
IF (NGC .EQ. 0) GO TO 315
CALL DRCHEK (3, G, NEQ, Y, RWORK(LYH), NYH,
1 RWORK(LG0), RWORK(LG1), RWORK(LGX), JROOT, IRT)
IF (IRT .NE. 1) GO TO 315
IRFND = 1
ISTATE = 3
T = T0
GO TO 425
315 CONTINUE
C
GO TO (310, 400, 330, 340, 350), ITASK
C ITASK = 1. If TOUT has been reached, interpolate. -------------------
310 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
C ITASK = 3. Jump to exit if TOUT was reached. ------------------------
330 IF ((TN - TOUT)*H .GE. 0.0D0) GO TO 400
GO TO 250
C ITASK = 4. See if TOUT or TCRIT was reached. Adjust H if necessary.
340 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 345
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
345 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
JSTART = -2
GO TO 250
C ITASK = 5. See if TCRIT was reached and jump to exit. ---------------
350 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
C-----------------------------------------------------------------------
C Block G.
C The following block handles all successful returns from DLSODKR.
C If ITASK .ne. 1, Y is loaded from YH and T is set accordingly.
C ISTATE is set to 2, and the optional outputs are loaded into the
C work arrays before returning.
C-----------------------------------------------------------------------
400 DO 410 I = 1,N
410 Y(I) = RWORK(I+LYH-1)
T = TN
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 420
IF (IHIT) T = TCRIT
420 ISTATE = 2
425 CONTINUE
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(19) = NNI
IWORK(20) = NLI
IWORK(21) = NPS
IWORK(22) = NCFN
IWORK(23) = NCFL
IWORK(24) = NSFI
IWORK(25) = NJEV
IWORK(10) = NGE
TLAST = T
RETURN
C-----------------------------------------------------------------------
C Block H.
C The following block handles all unsuccessful returns other than
C those for illegal input. First the error message routine is called.
C If there was an error test or convergence test failure, IMXER is set.
C Then Y is loaded from YH and T is set to TN.
C The optional outputs are loaded into the work arrays before returning.
C-----------------------------------------------------------------------
C The maximum number of steps was taken before reaching TOUT. ----------
500 MSG = 'DLSODKR- At current T (=R1), MXSTEP (=I1) steps '
CALL XERRWD (MSG, 50, 201, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' taken on this call before reaching TOUT '
CALL XERRWD (MSG, 50, 201, 0, 1, MXSTEP, 0, 1, TN, 0.0D0)
ISTATE = -1
GO TO 580
C EWT(i) .le. 0.0 for some i (not at start of problem). ----------------
510 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODKR- At T(=R1), EWT(I1) has become R2 .le. 0.'
CALL XERRWD (MSG, 50, 202, 0, 1, I, 0, 2, TN, EWTI)
ISTATE = -6
GO TO 580
C Too much accuracy requested for machine precision. -------------------
520 MSG = 'DLSODKR- At T (=R1), too much accuracy requested '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' for precision of machine.. See TOLSF (=R2) '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 2, TN, TOLSF)
RWORK(14) = TOLSF
ISTATE = -2
GO TO 580
C KFLAG = -1. Error test failed repeatedly or with ABS(H) = HMIN. -----
530 MSG = 'DLSODKR- At T(=R1) and step size H(=R2), the error'
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' test failed repeatedly or with ABS(H) = HMIN'
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 2, TN, H)
ISTATE = -4
GO TO 560
C KFLAG = -2. Convergence failed repeatedly or with ABS(H) = HMIN. ----
540 MSG = 'DLSODKR- At T (=R1) and step size H (=R2), the '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' corrector convergence failed repeatedly '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' or with ABS(H) = HMIN '
CALL XERRWD (MSG, 30, 205, 0, 0, 0, 0, 2, TN, H)
ISTATE = -5
GO TO 580
C KFLAG = -3. Unrecoverable error from PSOL. --------------------------
550 MSG = 'DLSODKR- At T (=R1) an unrecoverable error return'
CALL XERRWD (MSG, 50, 206, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' was made from Subroutine PSOL '
CALL XERRWD (MSG, 40, 206, 0, 0, 0, 0, 1, TN, 0.0D0)
ISTATE = -7
GO TO 580
C Compute IMXER if relevant. -------------------------------------------
560 BIG = 0.0D0
IMXER = 1
DO 570 I = 1,N
SIZE = ABS(RWORK(I+LACOR-1)*RWORK(I+LEWT-1))
IF (BIG .GE. SIZE) GO TO 570
BIG = SIZE
IMXER = I
570 CONTINUE
IWORK(16) = IMXER
C Set Y vector, T, and optional outputs. -------------------------------
580 DO 590 I = 1,N
590 Y(I) = RWORK(I+LYH-1)
T = TN
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(19) = NNI
IWORK(20) = NLI
IWORK(21) = NPS
IWORK(22) = NCFN
IWORK(23) = NCFL
IWORK(24) = NSFI
IWORK(25) = NJEV
IWORK(10) = NGE
TLAST = T
RETURN
C-----------------------------------------------------------------------
C Block I.
C The following block handles all error returns due to illegal input
C (ISTATE = -3), as detected before calling the core integrator.
C First the error message routine is called. If the illegal input
C is a negative ISTATE, the run is aborted (apparent infinite loop).
C-----------------------------------------------------------------------
601 MSG = 'DLSODKR- ISTATE(=I1) illegal.'
CALL XERRWD (MSG, 30, 1, 0, 1, ISTATE, 0, 0, 0.0D0, 0.0D0)
IF (ISTATE .LT. 0) GO TO 800
GO TO 700
602 MSG = 'DLSODKR- ITASK (=I1) illegal.'
CALL XERRWD (MSG, 30, 2, 0, 1, ITASK, 0, 0, 0.0D0, 0.0D0)
GO TO 700
603 MSG = 'DLSODKR- ISTATE.gt.1 but DLSODKR not initialized. '
CALL XERRWD (MSG, 50, 3, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
GO TO 700
604 MSG = 'DLSODKR- NEQ (=I1) .lt. 1 '
CALL XERRWD (MSG, 30, 4, 0, 1, NEQ(1), 0, 0, 0.0D0, 0.0D0)
GO TO 700
605 MSG = 'DLSODKR- ISTATE = 3 and NEQ increased (I1 to I2).'
CALL XERRWD (MSG, 50, 5, 0, 2, N, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
606 MSG = 'DLSODKR- ITOL (=I1) illegal. '
CALL XERRWD (MSG, 30, 6, 0, 1, ITOL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
607 MSG = 'DLSODKR- IOPT (=I1) illegal. '
CALL XERRWD (MSG, 30, 7, 0, 1, IOPT, 0, 0, 0.0D0, 0.0D0)
GO TO 700
608 MSG = 'DLSODKR- MF (=I1) illegal. '
CALL XERRWD (MSG, 30, 8, 0, 1, MF, 0, 0, 0.0D0, 0.0D0)
GO TO 700
611 MSG = 'DLSODKR- MAXORD (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 11, 0, 1, MAXORD, 0, 0, 0.0D0, 0.0D0)
GO TO 700
612 MSG = 'DLSODKR- MXSTEP (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 12, 0, 1, MXSTEP, 0, 0, 0.0D0, 0.0D0)
GO TO 700
613 MSG = 'DLSODKR- MXHNIL (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 13, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
614 MSG = 'DLSODKR- TOUT (=R1) behind T (=R2) '
CALL XERRWD (MSG, 40, 14, 0, 0, 0, 0, 2, TOUT, T)
MSG = ' Integration direction is given by H0 (=R1) '
CALL XERRWD (MSG, 50, 14, 0, 0, 0, 0, 1, H0, 0.0D0)
GO TO 700
615 MSG = 'DLSODKR- HMAX (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 15, 0, 0, 0, 0, 1, HMAX, 0.0D0)
GO TO 700
616 MSG = 'DLSODKR- HMIN (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 16, 0, 0, 0, 0, 1, HMIN, 0.0D0)
GO TO 700
617 MSG='DLSODKR- RWORK length needed, LENRW(=I1), exceeds LRW(=I2) '
CALL XERRWD (MSG, 60, 17, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
618 MSG='DLSODKR- IWORK length needed, LENIW(=I1), exceeds LIW(=I2) '
CALL XERRWD (MSG, 60, 18, 0, 2, LENIW, LIW, 0, 0.0D0, 0.0D0)
GO TO 700
619 MSG = 'DLSODKR- RTOL(I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 19, 0, 1, I, 0, 1, RTOLI, 0.0D0)
GO TO 700
620 MSG = 'DLSODKR- ATOL(I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 20, 0, 1, I, 0, 1, ATOLI, 0.0D0)
GO TO 700
621 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODKR- EWT(I1) is R1 .le. 0.0 '
CALL XERRWD (MSG, 40, 21, 0, 1, I, 0, 1, EWTI, 0.0D0)
GO TO 700
622 MSG='DLSODKR- TOUT(=R1) too close to T(=R2) to start integration.'
CALL XERRWD (MSG, 60, 22, 0, 0, 0, 0, 2, TOUT, T)
GO TO 700
623 MSG='DLSODKR- ITASK = I1 and TOUT (=R1) behind TCUR - HU (= R2) '
CALL XERRWD (MSG, 60, 23, 0, 1, ITASK, 0, 2, TOUT, TP)
GO TO 700
624 MSG='DLSODKR- ITASK = 4 or 5 and TCRIT (=R1) behind TCUR (=R2) '
CALL XERRWD (MSG, 60, 24, 0, 0, 0, 0, 2, TCRIT, TN)
GO TO 700
625 MSG='DLSODKR- ITASK = 4 or 5 and TCRIT (=R1) behind TOUT (=R2) '
CALL XERRWD (MSG, 60, 25, 0, 0, 0, 0, 2, TCRIT, TOUT)
GO TO 700
626 MSG = 'DLSODKR- At start of problem, too much accuracy '
CALL XERRWD (MSG, 50, 26, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' requested for precision of machine.. See TOLSF (=R1) '
CALL XERRWD (MSG, 60, 26, 0, 0, 0, 0, 1, TOLSF, 0.0D0)
RWORK(14) = TOLSF
GO TO 700
627 MSG = 'DLSODKR- Trouble in DINTDY. ITASK = I1, TOUT = R1'
CALL XERRWD (MSG, 50, 27, 0, 1, ITASK, 0, 1, TOUT, 0.0D0)
GO TO 700
630 MSG = 'DLSODKR- NG (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 30, 0, 1, NG, 0, 0, 0.0D0, 0.0D0)
GO TO 700
631 MSG = 'DLSODKR- NG changed (from I1 to I2) illegally, '
CALL XERRWD (MSG, 50, 31, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' i.e. not immediately after a root was found.'
CALL XERRWD (MSG, 50, 31, 0, 2, NGC, NG, 0, 0.0D0, 0.0D0)
GO TO 700
632 MSG = 'DLSODKR- One or more components of g has a root '
CALL XERRWD (MSG, 50, 32, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' too near to the initial point. '
CALL XERRWD (MSG, 40, 32, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
C
700 ISTATE = -3
RETURN
C
800 MSG = 'DLSODKR- Run aborted.. apparent infinite loop. '
CALL XERRWD (MSG, 50, 303, 2, 0, 0, 0, 0, 0.0D0, 0.0D0)
RETURN
C----------------------- End of Subroutine DLSODKR ---------------------
END
*DECK DLSODI
SUBROUTINE DLSODI (RES, ADDA, JAC, NEQ, Y, YDOTI, T, TOUT, ITOL,
1 RTOL, ATOL, ITASK, ISTATE, IOPT, RWORK, LRW, IWORK, LIW, MF )
EXTERNAL RES, ADDA, JAC
INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK, LIW, MF
DOUBLE PRECISION Y, YDOTI, T, TOUT, RTOL, ATOL, RWORK
DIMENSION NEQ(*), Y(*), YDOTI(*), RTOL(*), ATOL(*), RWORK(LRW),
1 IWORK(LIW)
C-----------------------------------------------------------------------
C This is the 18 November 2003 version of
C DLSODI: Livermore Solver for Ordinary Differential Equations
C (Implicit form).
C
C This version is in double precision.
C
C DLSODI solves the initial value problem for linearly implicit
C systems of first order ODEs,
C A(t,y) * dy/dt = g(t,y) , where A(t,y) is a square matrix,
C or, in component form,
C ( a * ( dy / dt )) + ... + ( a * ( dy / dt )) =
C i,1 1 i,NEQ NEQ
C
C = g ( t, y , y ,..., y ) ( i = 1,...,NEQ )
C i 1 2 NEQ
C
C If A is singular, this is a differential-algebraic system.
C
C DLSODI is a variant version of the DLSODE package.
C-----------------------------------------------------------------------
C Reference:
C Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE
C Solvers, in Scientific Computing, R. S. Stepleman et al. (Eds.),
C North-Holland, Amsterdam, 1983, pp. 55-64.
C-----------------------------------------------------------------------
C Authors: Alan C. Hindmarsh and Jeffrey F. Painter
C Center for Applied Scientific Computing, L-561
C Lawrence Livermore National Laboratory
C Livermore, CA 94551
C-----------------------------------------------------------------------
C Summary of Usage.
C
C Communication between the user and the DLSODI package, for normal
C situations, is summarized here. This summary describes only a subset
C of the full set of options available. See the full description for
C details, including optional communication, nonstandard options,
C and instructions for special situations. See also the example
C problem (with program and output) following this summary.
C
C A. First, provide a subroutine of the form:
C SUBROUTINE RES (NEQ, T, Y, S, R, IRES)
C DOUBLE PRECISION T, Y(*), S(*), R(*)
C which computes the residual function
C r = g(t,y) - A(t,y) * s ,
C as a function of t and the vectors y and s. (s is an internally
C generated approximation to dy/dt.) The arrays Y and S are inputs
C to the RES routine and should not be altered. The residual
C vector is to be stored in the array R. The argument IRES should be
C ignored for casual use of DLSODI. (For uses of IRES, see the
C paragraph on RES in the full description below.)
C
C B. Next, decide whether full or banded form is more economical
C for the storage of matrices. DLSODI must deal internally with the
C matrices A and dr/dy, where r is the residual function defined above.
C DLSODI generates a linear combination of these two matrices, and
C this is treated in either full or banded form.
C The matrix structure is communicated by a method flag MF,
C which is 21 or 22 for the full case, and 24 or 25 in the band case.
C In the banded case, DLSODI requires two half-bandwidth
C parameters ML and MU. These are, respectively, the widths of the
C lower and upper parts of the band, excluding the main diagonal.
C Thus the band consists of the locations (i,j) with
C i-ML .le. j .le. i+MU, and the full bandwidth is ML+MU+1.
C Note that the band must accommodate the nonzero elements of
C A(t,y), dg/dy, and d(A*s)/dy (s fixed). Alternatively, one
C can define a band that encloses only the elements that are relatively
C large in magnitude, and gain some economy in storage and possibly
C also efficiency, although the appropriate threshhold for
C retaining matrix elements is highly problem-dependent.
C
C C. You must also provide a subroutine of the form:
C SUBROUTINE ADDA (NEQ, T, Y, ML, MU, P, NROWP)
C DOUBLE PRECISION T, Y(*), P(NROWP,*)
C which adds the matrix A = A(t,y) to the contents of the array P.
C T and the Y array are input and should not be altered.
C In the full matrix case, this routine should add elements of
C to P in the usual order. I.e., add A(i,j) to P(i,j). (Ignore the
C ML and MU arguments in this case.)
C In the band matrix case, this routine should add element A(i,j)
C to P(i-j+MU+1,j). I.e., add the diagonal lines of A to the rows of
C P from the top down (the top line of A added to the first row of P).
C
C D. For the sake of efficiency, you are encouraged to supply the
C Jacobian matrix dr/dy in closed form, where r = g(t,y) - A(t,y)*s
C (s = a fixed vector) as above. If dr/dy is being supplied,
C use MF = 21 or 24, and provide a subroutine of the form:
C SUBROUTINE JAC (NEQ, T, Y, S, ML, MU, P, NROWP)
C DOUBLE PRECISION T, Y(*), S(*), P(NROWP,*)
C which computes dr/dy as a function of t, y, and s. Here T, Y, and
C S are inputs, and the routine is to load dr/dy into P as follows:
C In the full matrix case (MF = 21), load P(i,j) with dr(i)/dy(j),
C the partial derivative of r(i) with respect to y(j). (Ignore the
C ML and MU arguments in this case.)
C In the band matrix case (MF = 24), load P(i-j+mu+1,j) with
C dr(i)/dy(j), i.e. load the diagonal lines of dr/dy into the rows of
C P from the top down.
C In either case, only nonzero elements need be loaded, and the
C indexing of P is the same as in the ADDA routine.
C Note that if A is independent of y (or this dependence
C is weak enough to be ignored) then JAC is to compute dg/dy.
C If it is not feasible to provide a JAC routine, use
C MF = 22 or 25, and DLSODI will compute an approximate Jacobian
C internally by difference quotients.
C
C E. Next decide whether or not to provide the initial value of the
C derivative vector dy/dt. If the initial value of A(t,y) is
C nonsingular (and not too ill-conditioned), you may let DLSODI compute
C this vector (ISTATE = 0). (DLSODI will solve the system A*s = g for
C s, with initial values of A and g.) If A(t,y) is initially
C singular, then the system is a differential-algebraic system, and
C you must make use of the particular form of the system to compute the
C initial values of y and dy/dt. In that case, use ISTATE = 1 and
C load the initial value of dy/dt into the array YDOTI.
C The input array YDOTI and the initial Y array must be consistent with
C the equations A*dy/dt = g. This implies that the initial residual
C r = g(t,y) - A(t,y)*YDOTI must be approximately zero.
C
C F. Write a main program which calls Subroutine DLSODI once for
C each point at which answers are desired. This should also provide
C for possible use of logical unit 6 for output of error messages
C by DLSODI. On the first call to DLSODI, supply arguments as follows:
C RES = name of user subroutine for residual function r.
C ADDA = name of user subroutine for computing and adding A(t,y).
C JAC = name of user subroutine for Jacobian matrix dr/dy
C (MF = 21 or 24). If not used, pass a dummy name.
C Note: the names for the RES and ADDA routines and (if used) the
C JAC routine must be declared External in the calling program.
C NEQ = number of scalar equations in the system.
C Y = array of initial values, of length NEQ.
C YDOTI = array of length NEQ (containing initial dy/dt if ISTATE = 1).
C T = the initial value of the independent variable.
C TOUT = first point where output is desired (.ne. T).
C ITOL = 1 or 2 according as ATOL (below) is a scalar or array.
C RTOL = relative tolerance parameter (scalar).
C ATOL = absolute tolerance parameter (scalar or array).
C the estimated local error in y(i) will be controlled so as
C to be roughly less (in magnitude) than
C EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
C EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
C Thus the local error test passes if, in each component,
C either the absolute error is less than ATOL (or ATOL(i)),
C or the relative error is less than RTOL.
C Use RTOL = 0.0 for pure absolute error control, and
C use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative error
C control. Caution: Actual (global) errors may exceed these
C local tolerances, so choose them conservatively.
C ITASK = 1 for normal computation of output values of y at t = TOUT.
C ISTATE = integer flag (input and output). Set ISTATE = 1 if the
C initial dy/dt is supplied, and 0 otherwise.
C IOPT = 0 to indicate no optional inputs used.
C RWORK = real work array of length at least:
C 22 + 9*NEQ + NEQ**2 for MF = 21 or 22,
C 22 + 10*NEQ + (2*ML + MU)*NEQ for MF = 24 or 25.
C LRW = declared length of RWORK (in user's dimension).
C IWORK = integer work array of length at least 20 + NEQ.
C If MF = 24 or 25, input in IWORK(1),IWORK(2) the lower
C and upper half-bandwidths ML,MU.
C LIW = declared length of IWORK (in user's dimension).
C MF = method flag. Standard values are:
C 21 for a user-supplied full Jacobian.
C 22 for an internally generated full Jacobian.
C 24 for a user-supplied banded Jacobian.
C 25 for an internally generated banded Jacobian.
C for other choices of MF, see the paragraph on MF in
C the full description below.
C Note that the main program must declare arrays Y, YDOTI, RWORK, IWORK,
C and possibly ATOL.
C
C G. The output from the first call (or any call) is:
C Y = array of computed values of y(t) vector.
C T = corresponding value of independent variable (normally TOUT).
C ISTATE = 2 if DLSODI was successful, negative otherwise.
C -1 means excess work done on this call (check all inputs).
C -2 means excess accuracy requested (tolerances too small).
C -3 means illegal input detected (see printed message).
C -4 means repeated error test failures (check all inputs).
C -5 means repeated convergence failures (perhaps bad Jacobian
C supplied or wrong choice of tolerances).
C -6 means error weight became zero during problem. (Solution
C component i vanished, and ATOL or ATOL(i) = 0.)
C -7 cannot occur in casual use.
C -8 means DLSODI was unable to compute the initial dy/dt.
C In casual use, this means A(t,y) is initially singular.
C Supply YDOTI and use ISTATE = 1 on the first call.
C
C If DLSODI returns ISTATE = -1, -4, or -5, then the output of
C DLSODI also includes YDOTI = array containing residual vector
C r = g - A * dy/dt evaluated at the current t, y, and dy/dt.
C
C H. To continue the integration after a successful return, simply
C reset TOUT and call DLSODI again. No other parameters need be reset.
C
C-----------------------------------------------------------------------
C Example Problem.
C
C The following is a simple example problem, with the coding
C needed for its solution by DLSODI. The problem is from chemical
C kinetics, and consists of the following three equations:
C dy1/dt = -.04*y1 + 1.e4*y2*y3
C dy2/dt = .04*y1 - 1.e4*y2*y3 - 3.e7*y2**2
C 0. = y1 + y2 + y3 - 1.
C on the interval from t = 0.0 to t = 4.e10, with initial conditions
C y1 = 1.0, y2 = y3 = 0.
C
C The following coding solves this problem with DLSODI, using MF = 21
C and printing results at t = .4, 4., ..., 4.e10. It uses
C ITOL = 2 and ATOL much smaller for y2 than y1 or y3 because
C y2 has much smaller values. dy/dt is supplied in YDOTI. We had
C obtained the initial value of dy3/dt by differentiating the
C third equation and evaluating the first two at t = 0.
C At the end of the run, statistical quantities of interest are
C printed (see optional outputs in the full description below).
C
C EXTERNAL RESID, APLUSP, DGBYDY
C DOUBLE PRECISION ATOL, RTOL, RWORK, T, TOUT, Y, YDOTI
C DIMENSION Y(3), YDOTI(3), ATOL(3), RWORK(58), IWORK(23)
C NEQ = 3
C Y(1) = 1.
C Y(2) = 0.
C Y(3) = 0.
C YDOTI(1) = -.04
C YDOTI(2) = .04
C YDOTI(3) = 0.
C T = 0.
C TOUT = .4
C ITOL = 2
C RTOL = 1.D-4
C ATOL(1) = 1.D-6
C ATOL(2) = 1.D-10
C ATOL(3) = 1.D-6
C ITASK = 1
C ISTATE = 1
C IOPT = 0
C LRW = 58
C LIW = 23
C MF = 21
C DO 40 IOUT = 1,12
C CALL DLSODI(RESID, APLUSP, DGBYDY, NEQ, Y, YDOTI, T, TOUT, ITOL,
C 1 RTOL, ATOL, ITASK, ISTATE, IOPT, RWORK, LRW, IWORK, LIW, MF)
C WRITE (6,20) T, Y(1), Y(2), Y(3)
C 20 FORMAT(' At t =',D12.4,' Y =',3D14.6)
C IF (ISTATE .LT. 0 ) GO TO 80
C 40 TOUT = TOUT*10.
C WRITE (6,60) IWORK(11), IWORK(12), IWORK(13)
C 60 FORMAT(/' No. steps =',I4,' No. r-s =',I4,' No. J-s =',I4)
C STOP
C 80 WRITE (6,90) ISTATE
C 90 FORMAT(///' Error halt.. ISTATE =',I3)
C STOP
C END
C
C SUBROUTINE RESID(NEQ, T, Y, S, R, IRES)
C DOUBLE PRECISION T, Y, S, R
C DIMENSION Y(3), S(3), R(3)
C R(1) = -.04*Y(1) + 1.D4*Y(2)*Y(3) - S(1)
C R(2) = .04*Y(1) - 1.D4*Y(2)*Y(3) - 3.D7*Y(2)*Y(2) - S(2)
C R(3) = Y(1) + Y(2) + Y(3) - 1.
C RETURN
C END
C
C SUBROUTINE APLUSP(NEQ, T, Y, ML, MU, P, NROWP)
C DOUBLE PRECISION T, Y, P
C DIMENSION Y(3), P(NROWP,3)
C P(1,1) = P(1,1) + 1.
C P(2,2) = P(2,2) + 1.
C RETURN
C END
C
C SUBROUTINE DGBYDY(NEQ, T, Y, S, ML, MU, P, NROWP)
C DOUBLE PRECISION T, Y, S, P
C DIMENSION Y(3), S(3), P(NROWP,3)
C P(1,1) = -.04
C P(1,2) = 1.D4*Y(3)
C P(1,3) = 1.D4*Y(2)
C P(2,1) = .04
C P(2,2) = -1.D4*Y(3) - 6.D7*Y(2)
C P(2,3) = -1.D4*Y(2)
C P(3,1) = 1.
C P(3,2) = 1.
C P(3,3) = 1.
C RETURN
C END
C
C The output of this program (on a CDC-7600 in single precision)
C is as follows:
C
C At t = 4.0000e-01 Y = 9.851726e-01 3.386406e-05 1.479357e-02
C At t = 4.0000e+00 Y = 9.055142e-01 2.240418e-05 9.446344e-02
C At t = 4.0000e+01 Y = 7.158050e-01 9.184616e-06 2.841858e-01
C At t = 4.0000e+02 Y = 4.504846e-01 3.222434e-06 5.495122e-01
C At t = 4.0000e+03 Y = 1.831701e-01 8.940379e-07 8.168290e-01
C At t = 4.0000e+04 Y = 3.897016e-02 1.621193e-07 9.610297e-01
C At t = 4.0000e+05 Y = 4.935213e-03 1.983756e-08 9.950648e-01
C At t = 4.0000e+06 Y = 5.159269e-04 2.064759e-09 9.994841e-01
C At t = 4.0000e+07 Y = 5.306413e-05 2.122677e-10 9.999469e-01
C At t = 4.0000e+08 Y = 5.494532e-06 2.197826e-11 9.999945e-01
C At t = 4.0000e+09 Y = 5.129457e-07 2.051784e-12 9.999995e-01
C At t = 4.0000e+10 Y = -7.170472e-08 -2.868188e-13 1.000000e+00
C
C No. steps = 330 No. r-s = 404 No. J-s = 69
C
C-----------------------------------------------------------------------
C Full Description of User Interface to DLSODI.
C
C The user interface to DLSODI consists of the following parts.
C
C 1. The call sequence to Subroutine DLSODI, which is a driver
C routine for the solver. This includes descriptions of both
C the call sequence arguments and of user-supplied routines.
C Following these descriptions is a description of
C optional inputs available through the call sequence, and then
C a description of optional outputs (in the work arrays).
C
C 2. Descriptions of other routines in the DLSODI package that may be
C (optionally) called by the user. These provide the ability to
C alter error message handling, save and restore the internal
C Common, and obtain specified derivatives of the solution y(t).
C
C 3. Descriptions of Common blocks to be declared in overlay
C or similar environments, or to be saved when doing an interrupt
C of the problem and continued solution later.
C
C 4. Description of two routines in the DLSODI package, either of
C which the user may replace with his/her own version, if desired.
C These relate to the measurement of errors.
C
C-----------------------------------------------------------------------
C Part 1. Call Sequence.
C
C The call sequence parameters used for input only are
C RES, ADDA, JAC, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK,
C IOPT, LRW, LIW, MF,
C and those used for both input and output are
C Y, T, ISTATE, YDOTI.
C The work arrays RWORK and IWORK are also used for conditional and
C optional inputs and optional outputs. (The term output here refers
C to the return from Subroutine DLSODI to the user's calling program.)
C
C The legality of input parameters will be thoroughly checked on the
C initial call for the problem, but not checked thereafter unless a
C change in input parameters is flagged by ISTATE = 3 on input.
C
C The descriptions of the call arguments are as follows.
C
C RES = the name of the user-supplied subroutine which supplies
C the residual vector for the ODE system, defined by
C r = g(t,y) - A(t,y) * s
C as a function of the scalar t and the vectors
C s and y (s approximates dy/dt). This subroutine
C is to have the form
C SUBROUTINE RES (NEQ, T, Y, S, R, IRES)
C DOUBLE PRECISION T, Y(*), S(*), R(*)
C where NEQ, T, Y, S, and IRES are input, and R and
C IRES are output. Y, S, and R are arrays of length NEQ.
C On input, IRES indicates how DLSODI will use the
C returned array R, as follows:
C IRES = 1 means that DLSODI needs the full residual,
C r = g - A*s, exactly.
C IRES = -1 means that DLSODI is using R only to compute
C the Jacobian dr/dy by difference quotients.
C The RES routine can ignore IRES, or it can omit some terms
C if IRES = -1. If A does not depend on y, then RES can
C just return R = g when IRES = -1. If g - A*s contains other
C additive terms that are independent of y, these can also be
C dropped, if done consistently, when IRES = -1.
C The subroutine should set the flag IRES if it
C encounters a halt condition or illegal input.
C Otherwise, it should not reset IRES. On output,
C IRES = 1 or -1 represents a normal return, and
C DLSODI continues integrating the ODE. Leave IRES
C unchanged from its input value.
C IRES = 2 tells DLSODI to immediately return control
C to the calling program, with ISTATE = 3. This lets
C the calling program change parameters of the problem,
C if necessary.
C IRES = 3 represents an error condition (for example, an
C illegal value of y). DLSODI tries to integrate the system
C without getting IRES = 3 from RES. If it cannot, DLSODI
C returns with ISTATE = -7 or -1.
C On an DLSODI return with ISTATE = 3, -1, or -7, the values
C of T and Y returned correspond to the last point reached
C successfully without getting the flag IRES = 2 or 3.
C The flag values IRES = 2 and 3 should not be used to
C handle switches or root-stop conditions. This is better
C done by calling DLSODI in a one-step mode and checking the
C stopping function for a sign change at each step.
C If quantities computed in the RES routine are needed
C externally to DLSODI, an extra call to RES should be made
C for this purpose, for consistent and accurate results.
C To get the current dy/dt for the S argument, use DINTDY.
C RES must be declared External in the calling
C program. See note below for more about RES.
C
C ADDA = the name of the user-supplied subroutine which adds the
C matrix A = A(t,y) to another matrix stored in the same form
C as A. The storage form is determined by MITER (see MF).
C This subroutine is to have the form
C SUBROUTINE ADDA (NEQ, T, Y, ML, MU, P, NROWP)
C DOUBLE PRECISION T, Y(*), P(NROWP,*)
C where NEQ, T, Y, ML, MU, and NROWP are input and P is
C output. Y is an array of length NEQ, and the matrix P is
C stored in an NROWP by NEQ array.
C In the full matrix case ( MITER = 1 or 2) ADDA should
C add A to P(i,j). ML and MU are ignored.
C i,j
C In the band matrix case ( MITER = 4 or 5) ADDA should
C add A to P(i-j+MU+1,j).
C i,j
C See JAC for details on this band storage form.
C ADDA must be declared External in the calling program.
C See note below for more information about ADDA.
C
C JAC = the name of the user-supplied subroutine which supplies the
C Jacobian matrix, dr/dy, where r = g - A*s. The form of the
C Jacobian matrix is determined by MITER. JAC is required
C if MITER = 1 or 4 -- otherwise a dummy name can be
C passed. This subroutine is to have the form
C SUBROUTINE JAC ( NEQ, T, Y, S, ML, MU, P, NROWP )
C DOUBLE PRECISION T, Y(*), S(*), P(NROWP,*)
C where NEQ, T, Y, S, ML, MU, and NROWP are input and P
C is output. Y and S are arrays of length NEQ, and the
C matrix P is stored in an NROWP by NEQ array.
C P is to be loaded with partial derivatives (elements
C of the Jacobian matrix) on output.
C In the full matrix case (MITER = 1), ML and MU
C are ignored and the Jacobian is to be loaded into P
C by columns-- i.e., dr(i)/dy(j) is loaded into P(i,j).
C In the band matrix case (MITER = 4), the elements
C within the band are to be loaded into P by columns,
C with diagonal lines of dr/dy loaded into the
C rows of P. Thus dr(i)/dy(j) is to be loaded
C into P(i-j+MU+1,j). The locations in P in the two
C triangular areas which correspond to nonexistent matrix
C elements can be ignored or loaded arbitrarily, as they
C they are overwritten by DLSODI. ML and MU are the
C half-bandwidth parameters (see IWORK).
C In either case, P is preset to zero by the solver,
C so that only the nonzero elements need be loaded by JAC.
C Each call to JAC is preceded by a call to RES with the same
C arguments NEQ, T, Y, and S. Thus to gain some efficiency,
C intermediate quantities shared by both calculations may be
C saved in a user Common block by RES and not recomputed by JAC
C if desired. Also, JAC may alter the Y array, if desired.
C JAC need not provide dr/dy exactly. A crude
C approximation (possibly with a smaller bandwidth) will do.
C JAC must be declared External in the calling program.
C See note below for more about JAC.
C
C Note on RES, ADDA, and JAC:
C These subroutines may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in the subroutines) and/or Y has length
C exceeding NEQ(1). However, these routines should not alter
C NEQ(1), Y(1),...,Y(NEQ) or any other input variables.
C See the descriptions of NEQ and Y below.
C
C NEQ = the size of the system (number of first order ordinary
C differential equations or scalar algebraic equations).
C Used only for input.
C NEQ may be decreased, but not increased, during the problem.
C If NEQ is decreased (with ISTATE = 3 on input), the
C remaining components of Y should be left undisturbed, if
C these are to be accessed in RES, ADDA, or JAC.
C
C Normally, NEQ is a scalar, and it is generally referred to
C as a scalar in this user interface description. However,
C NEQ may be an array, with NEQ(1) set to the system size.
C (The DLSODI package accesses only NEQ(1).) In either case,
C this parameter is passed as the NEQ argument in all calls
C to RES, ADDA, and JAC. Hence, if it is an array,
C locations NEQ(2),... may be used to store other integer data
C and pass it to RES, ADDA, or JAC. Each such subroutine
C must include NEQ in a Dimension statement in that case.
C
C Y = a real array for the vector of dependent variables, of
C length NEQ or more. Used for both input and output on the
C first call (ISTATE = 0 or 1), and only for output on other
C calls. On the first call, Y must contain the vector of
C initial values. On output, Y contains the computed solution
C vector, evaluated at T. If desired, the Y array may be used
C for other purposes between calls to the solver.
C
C This array is passed as the Y argument in all calls to RES,
C ADDA, and JAC. Hence its length may exceed NEQ,
C and locations Y(NEQ+1),... may be used to store other real
C data and pass it to RES, ADDA, or JAC. (The DLSODI
C package accesses only Y(1),...,Y(NEQ). )
C
C YDOTI = a real array for the initial value of the vector
C dy/dt and for work space, of dimension at least NEQ.
C
C On input:
C If ISTATE = 0, then DLSODI will compute the initial value
C of dy/dt, if A is nonsingular. Thus YDOTI will
C serve only as work space and may have any value.
C If ISTATE = 1, then YDOTI must contain the initial value
C of dy/dt.
C If ISTATE = 2 or 3 (continuation calls), then YDOTI
C may have any value.
C Note: If the initial value of A is singular, then
C DLSODI cannot compute the initial value of dy/dt, so
C it must be provided in YDOTI, with ISTATE = 1.
C
C On output, when DLSODI terminates abnormally with ISTATE =
C -1, -4, or -5, YDOTI will contain the residual
C r = g(t,y) - A(t,y)*(dy/dt). If r is large, t is near
C its initial value, and YDOTI is supplied with ISTATE = 1,
C then there may have been an incorrect input value of
C YDOTI = dy/dt, or the problem (as given to DLSODI)
C may not have a solution.
C
C If desired, the YDOTI array may be used for other
C purposes between calls to the solver.
C
C T = the independent variable. On input, T is used only on the
C first call, as the initial point of the integration.
C On output, after each call, T is the value at which a
C computed solution Y is evaluated (usually the same as TOUT).
C on an error return, T is the farthest point reached.
C
C TOUT = the next value of t at which a computed solution is desired.
C Used only for input.
C
C When starting the problem (ISTATE = 0 or 1), TOUT may be
C equal to T for one call, then should .ne. T for the next
C call. For the initial T, an input value of TOUT .ne. T is
C used in order to determine the direction of the integration
C (i.e. the algebraic sign of the step sizes) and the rough
C scale of the problem. Integration in either direction
C (forward or backward in t) is permitted.
C
C If ITASK = 2 or 5 (one-step modes), TOUT is ignored after
C the first call (i.e. the first call with TOUT .ne. T).
C Otherwise, TOUT is required on every call.
C
C If ITASK = 1, 3, or 4, the values of TOUT need not be
C monotone, but a value of TOUT which backs up is limited
C to the current internal T interval, whose endpoints are
C TCUR - HU and TCUR (see optional outputs, below, for
C TCUR and HU).
C
C ITOL = an indicator for the type of error control. See
C description below under ATOL. Used only for input.
C
C RTOL = a relative error tolerance parameter, either a scalar or
C an array of length NEQ. See description below under ATOL.
C Input only.
C
C ATOL = an absolute error tolerance parameter, either a scalar or
C an array of length NEQ. Input only.
C
C The input parameters ITOL, RTOL, and ATOL determine
C the error control performed by the solver. The solver will
C control the vector E = (E(i)) of estimated local errors
C in y, according to an inequality of the form
C RMS-norm of ( E(i)/EWT(i) ) .le. 1,
C where EWT(i) = RTOL(i)*ABS(Y(i)) + ATOL(i),
C and the RMS-norm (root-mean-square norm) here is
C RMS-norm(v) = SQRT(sum v(i)**2 / NEQ). Here EWT = (EWT(i))
C is a vector of weights which must always be positive, and
C the values of RTOL and ATOL should all be non-negative.
C The following table gives the types (scalar/array) of
C RTOL and ATOL, and the corresponding form of EWT(i).
C
C ITOL RTOL ATOL EWT(i)
C 1 scalar scalar RTOL*ABS(Y(i)) + ATOL
C 2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
C 3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
C 4 array scalar RTOL(i)*ABS(Y(i)) + ATOL(i)
C
C When either of these parameters is a scalar, it need not
C be dimensioned in the user's calling program.
C
C If none of the above choices (with ITOL, RTOL, and ATOL
C fixed throughout the problem) is suitable, more general
C error controls can be obtained by substituting
C user-supplied routines for the setting of EWT and/or for
C the norm calculation. See Part 4 below.
C
C If global errors are to be estimated by making a repeated
C run on the same problem with smaller tolerances, then all
C components of RTOL and ATOL (i.e. of EWT) should be scaled
C down uniformly.
C
C ITASK = an index specifying the task to be performed.
C Input only. ITASK has the following values and meanings.
C 1 means normal computation of output values of y(t) at
C t = TOUT (by overshooting and interpolating).
C 2 means take one step only and return.
C 3 means stop at the first internal mesh point at or
C beyond t = TOUT and return.
C 4 means normal computation of output values of y(t) at
C t = TOUT but without overshooting t = TCRIT.
C TCRIT must be input as RWORK(1). TCRIT may be equal to
C or beyond TOUT, but not behind it in the direction of
C integration. This option is useful if the problem
C has a singularity at or beyond t = TCRIT.
C 5 means take one step, without passing TCRIT, and return.
C TCRIT must be input as RWORK(1).
C
C Note: If ITASK = 4 or 5 and the solver reaches TCRIT
C (within roundoff), it will return T = TCRIT (exactly) to
C indicate this (unless ITASK = 4 and TOUT comes before TCRIT,
C in which case answers at t = TOUT are returned first).
C
C ISTATE = an index used for input and output to specify the
C state of the calculation.
C
C On input, the values of ISTATE are as follows.
C 0 means this is the first call for the problem, and
C DLSODI is to compute the initial value of dy/dt
C (while doing other initializations). See note below.
C 1 means this is the first call for the problem, and
C the initial value of dy/dt has been supplied in
C YDOTI (DLSODI will do other initializations). See note
C below.
C 2 means this is not the first call, and the calculation
C is to continue normally, with no change in any input
C parameters except possibly TOUT and ITASK.
C (If ITOL, RTOL, and/or ATOL are changed between calls
C with ISTATE = 2, the new values will be used but not
C tested for legality.)
C 3 means this is not the first call, and the
C calculation is to continue normally, but with
C a change in input parameters other than
C TOUT and ITASK. Changes are allowed in
C NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, MF, ML, MU,
C and any of the optional inputs except H0.
C (See IWORK description for ML and MU.)
C Note: A preliminary call with TOUT = T is not counted
C as a first call here, as no initialization or checking of
C input is done. (Such a call is sometimes useful for the
C purpose of outputting the initial conditions.)
C Thus the first call for which TOUT .ne. T requires
C ISTATE = 0 or 1 on input.
C
C On output, ISTATE has the following values and meanings.
C 0 or 1 means nothing was done; TOUT = t and
C ISTATE = 0 or 1 on input.
C 2 means that the integration was performed successfully.
C 3 means that the user-supplied Subroutine RES signalled
C DLSODI to halt the integration and return (IRES = 2).
C Integration as far as T was achieved with no occurrence
C of IRES = 2, but this flag was set on attempting the
C next step.
C -1 means an excessive amount of work (more than MXSTEP
C steps) was done on this call, before completing the
C requested task, but the integration was otherwise
C successful as far as T. (MXSTEP is an optional input
C and is normally 500.) To continue, the user may
C simply reset ISTATE to a value .gt. 1 and call again
C (the excess work step counter will be reset to 0).
C In addition, the user may increase MXSTEP to avoid
C this error return (see below on optional inputs).
C -2 means too much accuracy was requested for the precision
C of the machine being used. This was detected before
C completing the requested task, but the integration
C was successful as far as T. To continue, the tolerance
C parameters must be reset, and ISTATE must be set
C to 3. The optional output TOLSF may be used for this
C purpose. (Note: If this condition is detected before
C taking any steps, then an illegal input return
C (ISTATE = -3) occurs instead.)
C -3 means illegal input was detected, before taking any
C integration steps. See written message for details.
C Note: If the solver detects an infinite loop of calls
C to the solver with illegal input, it will cause
C the run to stop.
C -4 means there were repeated error test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C The problem may have a singularity, or the input
C may be inappropriate.
C -5 means there were repeated convergence test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C This may be caused by an inaccurate Jacobian matrix.
C -6 means EWT(i) became zero for some i during the
C integration. pure relative error control (ATOL(i)=0.0)
C was requested on a variable which has now vanished.
C the integration was successful as far as T.
C -7 means that the user-supplied Subroutine RES set
C its error flag (IRES = 3) despite repeated tries by
C DLSODI to avoid that condition.
C -8 means that ISTATE was 0 on input but DLSODI was unable
C to compute the initial value of dy/dt. See the
C printed message for details.
C
C Note: Since the normal output value of ISTATE is 2,
C it does not need to be reset for normal continuation.
C Similarly, ISTATE (= 3) need not be reset if RES told
C DLSODI to return because the calling program must change
C the parameters of the problem.
C Also, since a negative input value of ISTATE will be
C regarded as illegal, a negative output value requires the
C user to change it, and possibly other inputs, before
C calling the solver again.
C
C IOPT = an integer flag to specify whether or not any optional
C inputs are being used on this call. Input only.
C The optional inputs are listed separately below.
C IOPT = 0 means no optional inputs are being used.
C Default values will be used in all cases.
C IOPT = 1 means one or more optional inputs are being used.
C
C RWORK = a real working array (double precision).
C The length of RWORK must be at least
C 20 + NYH*(MAXORD + 1) + 3*NEQ + LENWM where
C NYH = the initial value of NEQ,
C MAXORD = 12 (if METH = 1) or 5 (if METH = 2) (unless a
C smaller value is given as an optional input),
C LENWM = NEQ**2 + 2 if MITER is 1 or 2, and
C LENWM = (2*ML+MU+1)*NEQ + 2 if MITER is 4 or 5.
C (See MF description for the definition of METH and MITER.)
C Thus if MAXORD has its default value and NEQ is constant,
C this length is
C 22 + 16*NEQ + NEQ**2 for MF = 11 or 12,
C 22 + 17*NEQ + (2*ML+MU)*NEQ for MF = 14 or 15,
C 22 + 9*NEQ + NEQ**2 for MF = 21 or 22,
C 22 + 10*NEQ + (2*ML+MU)*NEQ for MF = 24 or 25.
C The first 20 words of RWORK are reserved for conditional
C and optional inputs and optional outputs.
C
C The following word in RWORK is a conditional input:
C RWORK(1) = TCRIT = critical value of t which the solver
C is not to overshoot. Required if ITASK is
C 4 or 5, and ignored otherwise. (See ITASK.)
C
C LRW = the length of the array RWORK, as declared by the user.
C (This will be checked by the solver.)
C
C IWORK = an integer work array. The length of IWORK must be at least
C 20 + NEQ . The first few words of IWORK are used for
C conditional and optional inputs and optional outputs.
C
C The following 2 words in IWORK are conditional inputs:
C IWORK(1) = ML These are the lower and upper
C IWORK(2) = MU half-bandwidths, respectively, of the
C matrices in the problem-- the Jacobian dr/dy
C and the left-hand side matrix A. These
C half-bandwidths exclude the main diagonal,
C so the total bandwidth is ML + MU + 1 .
C The band is defined by the matrix locations
C (i,j) with i-ML .le. j .le. i+MU. ML and MU
C must satisfy 0 .le. ML,MU .le. NEQ-1.
C These are required if MITER is 4 or 5, and
C ignored otherwise.
C ML and MU may in fact be the band parameters
C for matrices to which dr/dy and A are only
C approximately equal.
C
C LIW = the length of the array IWORK, as declared by the user.
C (This will be checked by the solver.)
C
C Note: The work arrays must not be altered between calls to DLSODI
C for the same problem, except possibly for the conditional and
C optional inputs, and except for the last 3*NEQ words of RWORK.
C The latter space is used for internal scratch space, and so is
C available for use by the user outside DLSODI between calls, if
C desired (but not for use by RES, ADDA, or JAC).
C
C MF = the method flag. Used only for input. The legal values of
C MF are 11, 12, 14, 15, 21, 22, 24, and 25.
C MF has decimal digits METH and MITER: MF = 10*METH + MITER.
C METH indicates the basic linear multistep method:
C METH = 1 means the implicit Adams method.
C METH = 2 means the method based on Backward
C Differentiation Formulas (BDFs).
C The BDF method is strongly preferred for stiff
C problems, while the Adams method is preferred when
C the problem is not stiff. If the matrix A(t,y) is
C nonsingular, stiffness here can be taken to mean that of
C the explicit ODE system dy/dt = A-inverse * g. If A is
C singular, the concept of stiffness is not well defined.
C If you do not know whether the problem is stiff, we
C recommend using METH = 2. If it is stiff, the advantage
C of METH = 2 over METH = 1 will be great, while if it is
C not stiff, the advantage of METH = 1 will be slight.
C If maximum efficiency is important, some experimentation
C with METH may be necessary.
C MITER indicates the corrector iteration method:
C MITER = 1 means chord iteration with a user-supplied
C full (NEQ by NEQ) Jacobian.
C MITER = 2 means chord iteration with an internally
C generated (difference quotient) full Jacobian.
C This uses NEQ+1 extra calls to RES per dr/dy
C evaluation.
C MITER = 4 means chord iteration with a user-supplied
C banded Jacobian.
C MITER = 5 means chord iteration with an internally
C generated banded Jacobian (using ML+MU+2
C extra calls to RES per dr/dy evaluation).
C If MITER = 1 or 4, the user must supply a Subroutine JAC
C (the name is arbitrary) as described above under JAC.
C For other values of MITER, a dummy argument can be used.
C-----------------------------------------------------------------------
C Optional Inputs.
C
C The following is a list of the optional inputs provided for in the
C call sequence. (See also Part 2.) For each such input variable,
C this table lists its name as used in this documentation, its
C location in the call sequence, its meaning, and the default value.
C the use of any of these inputs requires IOPT = 1, and in that
C case all of these inputs are examined. A value of zero for any
C of these optional inputs will cause the default value to be used.
C Thus to use a subset of the optional inputs, simply preload
C locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively, and
C then set those of interest to nonzero values.
C
C Name Location Meaning and Default Value
C
C H0 RWORK(5) the step size to be attempted on the first step.
C The default value is determined by the solver.
C
C HMAX RWORK(6) the maximum absolute step size allowed.
C The default value is infinite.
C
C HMIN RWORK(7) the minimum absolute step size allowed.
C The default value is 0. (This lower bound is not
C enforced on the final step before reaching TCRIT
C when ITASK = 4 or 5.)
C
C MAXORD IWORK(5) the maximum order to be allowed. The default
C value is 12 if METH = 1, and 5 if METH = 2.
C If MAXORD exceeds the default value, it will
C be reduced to the default value.
C If MAXORD is changed during the problem, it may
C cause the current order to be reduced.
C
C MXSTEP IWORK(6) maximum number of (internally defined) steps
C allowed during one call to the solver.
C The default value is 500.
C
C MXHNIL IWORK(7) maximum number of messages printed (per problem)
C warning that T + H = T on a step (H = step size).
C This must be positive to result in a non-default
C value. The default value is 10.
C-----------------------------------------------------------------------
C Optional Outputs.
C
C As optional additional output from DLSODI, the variables listed
C below are quantities related to the performance of DLSODI
C which are available to the user. These are communicated by way of
C the work arrays, but also have internal mnemonic names as shown.
C Except where stated otherwise, all of these outputs are defined
C on any successful return from DLSODI, and on any return with
C ISTATE = -1, -2, -4, -5, -6, or -7. On a return with -3 (illegal
C input) or -8, they will be unchanged from their existing values
C (if any), except possibly for TOLSF, LENRW, and LENIW.
C On any error return, outputs relevant to the error will be defined,
C as noted below.
C
C Name Location Meaning
C
C HU RWORK(11) the step size in t last used (successfully).
C
C HCUR RWORK(12) the step size to be attempted on the next step.
C
C TCUR RWORK(13) the current value of the independent variable
C which the solver has actually reached, i.e. the
C current internal mesh point in t. On output, TCUR
C will always be at least as far as the argument
C T, but may be farther (if interpolation was done).
C
C TOLSF RWORK(14) a tolerance scale factor, greater than 1.0,
C computed when a request for too much accuracy was
C detected (ISTATE = -3 if detected at the start of
C the problem, ISTATE = -2 otherwise). If ITOL is
C left unaltered but RTOL and ATOL are uniformly
C scaled up by a factor of TOLSF for the next call,
C then the solver is deemed likely to succeed.
C (The user may also ignore TOLSF and alter the
C tolerance parameters in any other way appropriate.)
C
C NST IWORK(11) the number of steps taken for the problem so far.
C
C NRE IWORK(12) the number of residual evaluations (RES calls)
C for the problem so far.
C
C NJE IWORK(13) the number of Jacobian evaluations (each involving
C an evaluation of A and dr/dy) for the problem so
C far. This equals the number of calls to ADDA and
C (if MITER = 1 or 4) JAC, and the number of matrix
C LU decompositions.
C
C NQU IWORK(14) the method order last used (successfully).
C
C NQCUR IWORK(15) the order to be attempted on the next step.
C
C IMXER IWORK(16) the index of the component of largest magnitude in
C the weighted local error vector ( E(i)/EWT(i) ),
C on an error return with ISTATE = -4 or -5.
C
C LENRW IWORK(17) the length of RWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C LENIW IWORK(18) the length of IWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C
C The following two arrays are segments of the RWORK array which
C may also be of interest to the user as optional outputs.
C For each array, the table below gives its internal name,
C its base address in RWORK, and its description.
C
C Name Base Address Description
C
C YH 21 the Nordsieck history array, of size NYH by
C (NQCUR + 1), where NYH is the initial value
C of NEQ. For j = 0,1,...,NQCUR, column j+1
C of YH contains HCUR**j/factorial(j) times
C the j-th derivative of the interpolating
C polynomial currently representing the solution,
C evaluated at t = TCUR.
C
C ACOR LENRW-NEQ+1 array of size NEQ used for the accumulated
C corrections on each step, scaled on output to
C represent the estimated local error in y on the
C last step. This is the vector E in the descrip-
C tion of the error control. It is defined only
C on a return from DLSODI with ISTATE = 2.
C
C-----------------------------------------------------------------------
C Part 2. Other Routines Callable.
C
C The following are optional calls which the user may make to
C gain additional capabilities in conjunction with DLSODI.
C (The routines XSETUN and XSETF are designed to conform to the
C SLATEC error handling package.)
C
C Form of Call Function
C CALL XSETUN(LUN) Set the logical unit number, LUN, for
C output of messages from DLSODI, if
C the default is not desired.
C The default value of LUN is 6.
C
C CALL XSETF(MFLAG) Set a flag to control the printing of
C messages by DLSODI.
C MFLAG = 0 means do not print. (Danger:
C This risks losing valuable information.)
C MFLAG = 1 means print (the default).
C
C Either of the above calls may be made at
C any time and will take effect immediately.
C
C CALL DSRCOM(RSAV,ISAV,JOB) saves and restores the contents of
C the internal Common blocks used by
C DLSODI (see Part 3 below).
C RSAV must be a real array of length 218
C or more, and ISAV must be an integer
C array of length 37 or more.
C JOB=1 means save Common into RSAV/ISAV.
C JOB=2 means restore Common from RSAV/ISAV.
C DSRCOM is useful if one is
C interrupting a run and restarting
C later, or alternating between two or
C more problems solved with DLSODI.
C
C CALL DINTDY(,,,,,) Provide derivatives of y, of various
C (see below) orders, at a specified point t, if
C desired. It may be called only after
C a successful return from DLSODI.
C
C The detailed instructions for using DINTDY are as follows.
C The form of the call is:
C
C CALL DINTDY (T, K, RWORK(21), NYH, DKY, IFLAG)
C
C The input parameters are:
C
C T = value of independent variable where answers are desired
C (normally the same as the T last returned by DLSODI).
C For valid results, T must lie between TCUR - HU and TCUR.
C (See optional outputs for TCUR and HU.)
C K = integer order of the derivative desired. K must satisfy
C 0 .le. K .le. NQCUR, where NQCUR is the current order
C (see optional outputs). The capability corresponding
C to K = 0, i.e. computing y(T), is already provided
C by DLSODI directly. Since NQCUR .ge. 1, the first
C derivative dy/dt is always available with DINTDY.
C RWORK(21) = the base address of the history array YH.
C NYH = column length of YH, equal to the initial value of NEQ.
C
C The output parameters are:
C
C DKY = a real array of length NEQ containing the computed value
C of the K-th derivative of y(t).
C IFLAG = integer flag, returned as 0 if K and T were legal,
C -1 if K was illegal, and -2 if T was illegal.
C On an error return, a message is also written.
C-----------------------------------------------------------------------
C Part 3. Common Blocks.
C
C If DLSODI is to be used in an overlay situation, the user
C must declare, in the primary overlay, the variables in:
C (1) the call sequence to DLSODI, and
C (2) the internal Common block
C /DLS001/ of length 255 (218 double precision words
C followed by 37 integer words),
C
C If DLSODI is used on a system in which the contents of internal
C Common blocks are not preserved between calls, the user should
C declare the above Common block in the calling program to insure
C that their contents are preserved.
C
C If the solution of a given problem by DLSODI is to be interrupted
C and then later continued, such as when restarting an interrupted run
C or alternating between two or more problems, the user should save,
C following the return from the last DLSODI call prior to the
C interruption, the contents of the call sequence variables and the
C internal Common blocks, and later restore these values before the
C next DLSODI call for that problem. To save and restore the Common
C blocks, use Subroutine DSRCOM (see Part 2 above).
C
C-----------------------------------------------------------------------
C Part 4. Optionally Replaceable Solver Routines.
C
C Below are descriptions of two routines in the DLSODI package which
C relate to the measurement of errors. Either routine can be
C replaced by a user-supplied version, if desired. However, since such
C a replacement may have a major impact on performance, it should be
C done only when absolutely necessary, and only with great caution.
C (Note: The means by which the package version of a routine is
C superseded by the user's version may be system-dependent.)
C
C (a) DEWSET.
C The following subroutine is called just before each internal
C integration step, and sets the array of error weights, EWT, as
C described under ITOL/RTOL/ATOL above:
C SUBROUTINE DEWSET (NEQ, ITOL, RTOL, ATOL, YCUR, EWT)
C where NEQ, ITOL, RTOL, and ATOL are as in the DLSODI call sequence,
C YCUR contains the current dependent variable vector, and
C EWT is the array of weights set by DEWSET.
C
C If the user supplies this subroutine, it must return in EWT(i)
C (i = 1,...,NEQ) a positive quantity suitable for comparing errors
C in y(i) to. The EWT array returned by DEWSET is passed to the DVNORM
C routine (see below), and also used by DLSODI in the computation
C of the optional output IMXER, the diagonal Jacobian approximation,
C and the increments for difference quotient Jacobians.
C
C In the user-supplied version of DEWSET, it may be desirable to use
C the current values of derivatives of y. Derivatives up to order NQ
C are available from the history array YH, described above under
C optional outputs. In DEWSET, YH is identical to the YCUR array,
C extended to NQ + 1 columns with a column length of NYH and scale
C factors of H**j/factorial(j). On the first call for the problem,
C given by NST = 0, NQ is 1 and H is temporarily set to 1.0.
C NYH is the initial value of NEQ. The quantities NQ, H, and NST
C can be obtained by including in DEWSET the statements:
C DOUBLE PRECISION RLS
C COMMON /DLS001/ RLS(218),ILS(37)
C NQ = ILS(33)
C NST = ILS(34)
C H = RLS(212)
C Thus, for example, the current value of dy/dt can be obtained as
C YCUR(NYH+i)/H (i=1,...,NEQ) (and the division by H is
C unnecessary when NST = 0).
C
C (b) DVNORM.
C The following is a real function routine which computes the weighted
C root-mean-square norm of a vector v:
C D = DVNORM (N, V, W)
C where:
C N = the length of the vector,
C V = real array of length N containing the vector,
C W = real array of length N containing weights,
C D = SQRT( (1/N) * sum(V(i)*W(i))**2 ).
C DVNORM is called with N = NEQ and with W(i) = 1.0/EWT(i), where
C EWT is as set by Subroutine DEWSET.
C
C If the user supplies this function, it should return a non-negative
C value of DVNORM suitable for use in the error control in DLSODI.
C None of the arguments should be altered by DVNORM.
C For example, a user-supplied DVNORM routine might:
C -substitute a max-norm of (V(i)*W(i)) for the RMS-norm, or
C -ignore some components of V in the norm, with the effect of
C suppressing the error control on those components of y.
C-----------------------------------------------------------------------
C
C***REVISION HISTORY (YYYYMMDD)
C 19800424 DATE WRITTEN
C 19800519 Corrected access of YH on forced order reduction;
C numerous corrections to prologues and other comments.
C 19800617 In main driver, added loading of SQRT(UROUND) in RWORK;
C minor corrections to main prologue.
C 19800903 Corrected ISTATE logic; minor changes in prologue.
C 19800923 Added zero initialization of HU and NQU.
C 19801028 Reorganized RES calls in AINVG, STODI, and PREPJI;
C in LSODI, corrected NRE increment and reset LDY0 at 580;
C numerous corrections to main prologue.
C 19801218 Revised XERRWD routine; minor corrections to main prologue.
C 19810330 Added Common block /LSI001/; use LSODE's INTDY and SOLSY;
C minor corrections to XERRWD and error message at 604;
C minor corrections to declarations; corrections to prologues.
C 19810818 Numerous revisions: replaced EWT by 1/EWT; used flags
C JCUR, ICF, IERPJ, IERSL between STODI and subordinates;
C added tuning parameters CCMAX, MAXCOR, MSBP, MXNCF;
C reorganized returns from STODI; reorganized type decls.;
C fixed message length in XERRWD; changed default LUNIT to 6;
C changed Common lengths; changed comments throughout.
C 19820906 Corrected use of ABS(H) in STODI; minor comment fixes.
C 19830510 Numerous revisions: revised diff. quotient increment;
C eliminated block /LSI001/, using IERPJ flag;
C revised STODI logic after PJAC return;
C revised tuning of H change and step attempts in STODI;
C corrections to main prologue and internal comments.
C 19870330 Major update: corrected comments throughout;
C removed TRET from Common; rewrote EWSET with 4 loops;
C fixed t test in INTDY; added Cray directives in STODI;
C in STODI, fixed DELP init. and logic around PJAC call;
C combined routines to save/restore Common;
C passed LEVEL = 0 in error message calls (except run abort).
C 20010425 Major update: convert source lines to upper case;
C added *DECK lines; changed from 1 to * in dummy dimensions;
C changed names R1MACH/D1MACH to RUMACH/DUMACH;
C renamed routines for uniqueness across single/double prec.;
C converted intrinsic names to generic form;
C removed ILLIN and NTREP (data loaded) from Common;
C removed all 'own' variables from Common;
C changed error messages to quoted strings;
C replaced XERRWV/XERRWD with 1993 revised version;
C converted prologues, comments, error messages to mixed case;
C converted arithmetic IF statements to logical IF statements;
C numerous corrections to prologues and internal comments.
C 20010507 Converted single precision source to double precision.
C 20020502 Corrected declarations in descriptions of user routines.
C 20031105 Restored 'own' variables to Common block, to enable
C interrupt/restart feature.
C 20031112 Added SAVE statements for data-loaded constants.
C 20031117 Changed internal names NRE, LSAVR to NFE, LSAVF resp.
C
C-----------------------------------------------------------------------
C Other routines in the DLSODI package.
C
C In addition to Subroutine DLSODI, the DLSODI package includes the
C following subroutines and function routines:
C DAINVG computes the initial value of the vector
C dy/dt = A-inverse * g
C DINTDY computes an interpolated value of the y vector at t = TOUT.
C DSTODI is the core integrator, which does one step of the
C integration and the associated error control.
C DCFODE sets all method coefficients and test constants.
C DPREPJI computes and preprocesses the Jacobian matrix
C and the Newton iteration matrix P.
C DSOLSY manages solution of linear system in chord iteration.
C DEWSET sets the error weight vector EWT before each step.
C DVNORM computes the weighted RMS-norm of a vector.
C DSRCOM is a user-callable routine to save and restore
C the contents of the internal Common blocks.
C DGEFA and DGESL are routines from LINPACK for solving full
C systems of linear algebraic equations.
C DGBFA and DGBSL are routines from LINPACK for solving banded
C linear systems.
C DUMACH computes the unit roundoff in a machine-independent manner.
C XERRWD, XSETUN, XSETF, IXSAV, and IUMACH handle the printing of all
C error messages and warnings. XERRWD is machine-dependent.
C Note: DVNORM, DUMACH, IXSAV, and IUMACH are function routines.
C All the others are subroutines.
C
C-----------------------------------------------------------------------
EXTERNAL DPREPJI, DSOLSY
DOUBLE PRECISION DUMACH, DVNORM
INTEGER INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER I, I1, I2, IER, IFLAG, IMXER, IRES, KGO,
1 LENIW, LENRW, LENWM, LP, LYD0, ML, MORD, MU, MXHNL0, MXSTP0
DOUBLE PRECISION ROWNS,
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
DOUBLE PRECISION ATOLI, AYI, BIG, EWTI, H0, HMAX, HMX, RH, RTOLI,
1 TCRIT, TDIST, TNEXT, TOL, TOLSF, TP, SIZE, SUM, W0
DIMENSION MORD(2)
LOGICAL IHIT
CHARACTER*60 MSG
SAVE MORD, MXSTP0, MXHNL0
C-----------------------------------------------------------------------
C The following internal Common block contains
C (a) variables which are local to any subroutine but whose values must
C be preserved between calls to the routine ("own" variables), and
C (b) variables which are communicated between subroutines.
C The block DLS001 is declared in subroutines DLSODI, DINTDY, DSTODI,
C DPREPJI, and DSOLSY.
C Groups of variables are replaced by dummy arrays in the Common
C declarations in routines where those variables are not used.
C-----------------------------------------------------------------------
COMMON /DLS001/ ROWNS(209),
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
2 INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS(6),
3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
DATA MORD(1),MORD(2)/12,5/, MXSTP0/500/, MXHNL0/10/
C-----------------------------------------------------------------------
C Block A.
C This code block is executed on every call.
C It tests ISTATE and ITASK for legality and branches appropriately.
C If ISTATE .gt. 1 but the flag INIT shows that initialization has
C not yet been done, an error return occurs.
C If ISTATE = 0 or 1 and TOUT = T, return immediately.
C-----------------------------------------------------------------------
IF (ISTATE .LT. 0 .OR. ISTATE .GT. 3) GO TO 601
IF (ITASK .LT. 1 .OR. ITASK .GT. 5) GO TO 602
IF (ISTATE .LE. 1) GO TO 10
IF (INIT .EQ. 0) GO TO 603
IF (ISTATE .EQ. 2) GO TO 200
GO TO 20
10 INIT = 0
IF (TOUT .EQ. T) RETURN
C-----------------------------------------------------------------------
C Block B.
C The next code block is executed for the initial call (ISTATE = 0 or 1)
C or for a continuation call with parameter changes (ISTATE = 3).
C It contains checking of all inputs and various initializations.
C
C First check legality of the non-optional inputs NEQ, ITOL, IOPT,
C MF, ML, and MU.
C-----------------------------------------------------------------------
20 IF (NEQ(1) .LE. 0) GO TO 604
IF (ISTATE .LE. 1) GO TO 25
IF (NEQ(1) .GT. N) GO TO 605
25 N = NEQ(1)
IF (ITOL .LT. 1 .OR. ITOL .GT. 4) GO TO 606
IF (IOPT .LT. 0 .OR. IOPT .GT. 1) GO TO 607
METH = MF/10
MITER = MF - 10*METH
IF (METH .LT. 1 .OR. METH .GT. 2) GO TO 608
IF (MITER .LE. 0 .OR. MITER .GT. 5) GO TO 608
IF (MITER .EQ. 3) GO TO 608
IF (MITER .LT. 3) GO TO 30
ML = IWORK(1)
MU = IWORK(2)
IF (ML .LT. 0 .OR. ML .GE. N) GO TO 609
IF (MU .LT. 0 .OR. MU .GE. N) GO TO 610
30 CONTINUE
C Next process and check the optional inputs. --------------------------
IF (IOPT .EQ. 1) GO TO 40
MAXORD = MORD(METH)
MXSTEP = MXSTP0
MXHNIL = MXHNL0
IF (ISTATE .LE. 1) H0 = 0.0D0
HMXI = 0.0D0
HMIN = 0.0D0
GO TO 60
40 MAXORD = IWORK(5)
IF (MAXORD .LT. 0) GO TO 611
IF (MAXORD .EQ. 0) MAXORD = 100
MAXORD = MIN(MAXORD,MORD(METH))
MXSTEP = IWORK(6)
IF (MXSTEP .LT. 0) GO TO 612
IF (MXSTEP .EQ. 0) MXSTEP = MXSTP0
MXHNIL = IWORK(7)
IF (MXHNIL .LT. 0) GO TO 613
IF (MXHNIL .EQ. 0) MXHNIL = MXHNL0
IF (ISTATE .GT. 1) GO TO 50
H0 = RWORK(5)
IF ((TOUT - T)*H0 .LT. 0.0D0) GO TO 614
50 HMAX = RWORK(6)
IF (HMAX .LT. 0.0D0) GO TO 615
HMXI = 0.0D0
IF (HMAX .GT. 0.0D0) HMXI = 1.0D0/HMAX
HMIN = RWORK(7)
IF (HMIN .LT. 0.0D0) GO TO 616
C-----------------------------------------------------------------------
C Set work array pointers and check lengths LRW and LIW.
C Pointers to segments of RWORK and IWORK are named by prefixing L to
C the name of the segment. E.g., the segment YH starts at RWORK(LYH).
C Segments of RWORK (in order) are denoted YH, WM, EWT, SAVR, ACOR.
C-----------------------------------------------------------------------
60 LYH = 21
IF (ISTATE .LE. 1) NYH = N
LWM = LYH + (MAXORD + 1)*NYH
IF (MITER .LE. 2) LENWM = N*N + 2
IF (MITER .GE. 4) LENWM = (2*ML + MU + 1)*N + 2
LEWT = LWM + LENWM
LSAVF = LEWT + N
LACOR = LSAVF + N
LENRW = LACOR + N - 1
IWORK(17) = LENRW
LIWM = 1
LENIW = 20 + N
IWORK(18) = LENIW
IF (LENRW .GT. LRW) GO TO 617
IF (LENIW .GT. LIW) GO TO 618
C Check RTOL and ATOL for legality. ------------------------------------
RTOLI = RTOL(1)
ATOLI = ATOL(1)
DO 70 I = 1,N
IF (ITOL .GE. 3) RTOLI = RTOL(I)
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
IF (RTOLI .LT. 0.0D0) GO TO 619
IF (ATOLI .LT. 0.0D0) GO TO 620
70 CONTINUE
IF (ISTATE .LE. 1) GO TO 100
C If ISTATE = 3, set flag to signal parameter changes to DSTODI. -------
JSTART = -1
IF (NQ .LE. MAXORD) GO TO 90
C MAXORD was reduced below NQ. Copy YH(*,MAXORD+2) into YDOTI.---------
DO 80 I = 1,N
80 YDOTI(I) = RWORK(I+LWM-1)
C Reload WM(1) = RWORK(lWM), since lWM may have changed. ---------------
90 RWORK(LWM) = SQRT(UROUND)
IF (N .EQ. NYH) GO TO 200
C NEQ was reduced. Zero part of YH to avoid undefined references. -----
I1 = LYH + L*NYH
I2 = LYH + (MAXORD + 1)*NYH - 1
IF (I1 .GT. I2) GO TO 200
DO 95 I = I1,I2
95 RWORK(I) = 0.0D0
GO TO 200
C-----------------------------------------------------------------------
C Block C.
C The next block is for the initial call only (ISTATE = 0 or 1).
C It contains all remaining initializations, the call to DAINVG
C (if ISTATE = 1), and the calculation of the initial step size.
C The error weights in EWT are inverted after being loaded.
C-----------------------------------------------------------------------
100 UROUND = DUMACH()
TN = T
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 105
TCRIT = RWORK(1)
IF ((TCRIT - TOUT)*(TOUT - T) .LT. 0.0D0) GO TO 625
IF (H0 .NE. 0.0D0 .AND. (T + H0 - TCRIT)*H0 .GT. 0.0D0)
1 H0 = TCRIT - T
105 JSTART = 0
RWORK(LWM) = SQRT(UROUND)
NHNIL = 0
NST = 0
NFE = 0
NJE = 0
NSLAST = 0
HU = 0.0D0
NQU = 0
CCMAX = 0.3D0
MAXCOR = 3
MSBP = 20
MXNCF = 10
C Compute initial dy/dt, if necessary, and load it and initial Y into YH
LYD0 = LYH + NYH
LP = LWM + 1
IF (ISTATE .EQ. 1) GO TO 120
C DLSODI must compute initial dy/dt (LYD0 points to YH(*,2)). ----------
CALL DAINVG( RES, ADDA, NEQ, T, Y, RWORK(LYD0), MITER,
1 ML, MU, RWORK(LP), IWORK(21), IER )
NFE = NFE + 1
IF (IER .LT. 0) GO TO 560
IF (IER .GT. 0) GO TO 565
DO 115 I = 1,N
115 RWORK(I+LYH-1) = Y(I)
GO TO 130
C Initial dy/dt was supplied. Load into YH (LYD0 points to YH(*,2).). -
120 DO 125 I = 1,N
RWORK(I+LYH-1) = Y(I)
125 RWORK(I+LYD0-1) = YDOTI(I)
C Load and invert the EWT array. (H is temporarily set to 1.0.) -------
130 CONTINUE
NQ = 1
H = 1.0D0
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 135 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 621
135 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
C-----------------------------------------------------------------------
C The coding below computes the step size, H0, to be attempted on the
C first step, unless the user has supplied a value for this.
C First check that TOUT - T differs significantly from zero.
C A scalar tolerance quantity TOL is computed, as MAX(RTOL(i))
C if this is positive, or MAX(ATOL(i)/ABS(Y(i))) otherwise, adjusted
C so as to be between 100*UROUND and 1.0E-3.
C Then the computed value H0 is given by..
C NEQ
C H0**2 = TOL / ( w0**-2 + (1/NEQ) * Sum ( YDOT(i)/ywt(i) )**2 )
C 1
C where w0 = MAX ( ABS(T), ABS(TOUT) ),
C YDOT(i) = i-th component of initial value of dy/dt,
C ywt(i) = EWT(i)/TOL (a weight for y(i)).
C The sign of H0 is inferred from the initial values of TOUT and T.
C-----------------------------------------------------------------------
IF (H0 .NE. 0.0D0) GO TO 180
TDIST = ABS(TOUT - T)
W0 = MAX(ABS(T),ABS(TOUT))
IF (TDIST .LT. 2.0D0*UROUND*W0) GO TO 622
TOL = RTOL(1)
IF (ITOL .LE. 2) GO TO 145
DO 140 I = 1,N
140 TOL = MAX(TOL,RTOL(I))
145 IF (TOL .GT. 0.0D0) GO TO 160
ATOLI = ATOL(1)
DO 150 I = 1,N
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
AYI = ABS(Y(I))
IF (AYI .NE. 0.0D0) TOL = MAX(TOL,ATOLI/AYI)
150 CONTINUE
160 TOL = MAX(TOL,100.0D0*UROUND)
TOL = MIN(TOL,0.001D0)
SUM = DVNORM (N, RWORK(LYD0), RWORK(LEWT))
SUM = 1.0D0/(TOL*W0*W0) + TOL*SUM**2
H0 = 1.0D0/SQRT(SUM)
H0 = MIN(H0,TDIST)
H0 = SIGN(H0,TOUT-T)
C Adjust H0 if necessary to meet HMAX bound. ---------------------------
180 RH = ABS(H0)*HMXI
IF (RH .GT. 1.0D0) H0 = H0/RH
C Load H with H0 and scale YH(*,2) by H0. ------------------------------
H = H0
DO 190 I = 1,N
190 RWORK(I+LYD0-1) = H0*RWORK(I+LYD0-1)
GO TO 270
C-----------------------------------------------------------------------
C Block D.
C The next code block is for continuation calls only (ISTATE = 2 or 3)
C and is to check stop conditions before taking a step.
C-----------------------------------------------------------------------
200 NSLAST = NST
GO TO (210, 250, 220, 230, 240), ITASK
210 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
220 TP = TN - HU*(1.0D0 + 100.0D0*UROUND)
IF ((TP - TOUT)*H .GT. 0.0D0) GO TO 623
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
GO TO 400
230 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
IF ((TCRIT - TOUT)*H .LT. 0.0D0) GO TO 625
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 245
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
240 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
245 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
IF (ISTATE .EQ. 2) JSTART = -2
C-----------------------------------------------------------------------
C Block E.
C The next block is normally executed for all calls and contains
C the call to the one-step core integrator DSTODI.
C
C This is a looping point for the integration steps.
C
C First check for too many steps being taken, update EWT (if not at
C start of problem), check for too much accuracy being requested, and
C check for H below the roundoff level in T.
C-----------------------------------------------------------------------
250 CONTINUE
IF ((NST-NSLAST) .GE. MXSTEP) GO TO 500
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 260 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 510
260 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
270 TOLSF = UROUND*DVNORM (N, RWORK(LYH), RWORK(LEWT))
IF (TOLSF .LE. 1.0D0) GO TO 280
TOLSF = TOLSF*2.0D0
IF (NST .EQ. 0) GO TO 626
GO TO 520
280 IF ((TN + H) .NE. TN) GO TO 290
NHNIL = NHNIL + 1
IF (NHNIL .GT. MXHNIL) GO TO 290
MSG = 'DLSODI- Warning..Internal T (=R1) and H (=R2) are'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' such that in the machine, T + H = T on the next step '
CALL XERRWD (MSG, 60, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' (H = step size). Solver will continue anyway.'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 2, TN, H)
IF (NHNIL .LT. MXHNIL) GO TO 290
MSG = 'DLSODI- Above warning has been issued I1 times. '
CALL XERRWD (MSG, 50, 102, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' It will not be issued again for this problem.'
CALL XERRWD (MSG, 50, 102, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
290 CONTINUE
C-----------------------------------------------------------------------
C CALL DSTODI(NEQ,Y,YH,NYH,YH1,EWT,SAVF,SAVR,ACOR,WM,IWM,RES,
C ADDA,JAC,DPREPJI,DSOLSY)
C Note: SAVF in DSTODI occupies the same space as YDOTI in DLSODI.
C-----------------------------------------------------------------------
CALL DSTODI (NEQ, Y, RWORK(LYH), NYH, RWORK(LYH), RWORK(LEWT),
1 YDOTI, RWORK(LSAVF), RWORK(LACOR), RWORK(LWM),
2 IWORK(LIWM), RES, ADDA, JAC, DPREPJI, DSOLSY )
KGO = 1 - KFLAG
GO TO (300, 530, 540, 400, 550), KGO
C
C KGO = 1:success; 2:error test failure; 3:convergence failure;
C 4:RES ordered return. 5:RES returned error.
C-----------------------------------------------------------------------
C Block F.
C The following block handles the case of a successful return from the
C core integrator (KFLAG = 0). Test for stop conditions.
C-----------------------------------------------------------------------
300 INIT = 1
GO TO (310, 400, 330, 340, 350), ITASK
C ITASK = 1. If TOUT has been reached, interpolate. -------------------
310 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
C ITASK = 3. Jump to exit if TOUT was reached. ------------------------
330 IF ((TN - TOUT)*H .GE. 0.0D0) GO TO 400
GO TO 250
C ITASK = 4. see if TOUT or TCRIT was reached. adjust h if necessary.
340 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 345
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
345 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
JSTART = -2
GO TO 250
C ITASK = 5. See if TCRIT was reached and jump to exit. ---------------
350 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
C-----------------------------------------------------------------------
C Block G.
C The following block handles all successful returns from DLSODI.
C if ITASK .ne. 1, Y is loaded from YH and T is set accordingly.
C ISTATE is set to 2, and the optional outputs are loaded into the
C work arrays before returning.
C-----------------------------------------------------------------------
400 DO 410 I = 1,N
410 Y(I) = RWORK(I+LYH-1)
T = TN
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 420
IF (IHIT) T = TCRIT
420 ISTATE = 2
IF (KFLAG .EQ. -3) ISTATE = 3
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
RETURN
C-----------------------------------------------------------------------
C Block H.
C The following block handles all unsuccessful returns other than
C those for illegal input. First the error message routine is called.
C If there was an error test or convergence test failure, IMXER is set.
C Then Y is loaded from YH and T is set to TN.
C The optional outputs are loaded into the work arrays before returning.
C-----------------------------------------------------------------------
C The maximum number of steps was taken before reaching TOUT. ----------
500 MSG = 'DLSODI- At current T (=R1), MXSTEP (=I1) steps '
CALL XERRWD (MSG, 50, 201, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' taken on this call before reaching TOUT '
CALL XERRWD (MSG, 50, 201, 0, 1, MXSTEP, 0, 1, TN, 0.0D0)
ISTATE = -1
GO TO 580
C EWT(i) .le. 0.0 for some i (not at start of problem). ----------------
510 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODI- At T (=R1), EWT(I1) has become R2 .le. 0.'
CALL XERRWD (MSG, 50, 202, 0, 1, I, 0, 2, TN, EWTI)
ISTATE = -6
GO TO 590
C Too much accuracy requested for machine precision. -------------------
520 MSG = 'DLSODI- At T (=R1), too much accuracy requested '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' for precision of machine.. See TOLSF (=R2) '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 2, TN, TOLSF)
RWORK(14) = TOLSF
ISTATE = -2
GO TO 590
C KFLAG = -1. Error test failed repeatedly or with ABS(H) = HMIN. -----
530 MSG = 'DLSODI- At T(=R1) and step size H(=R2), the error'
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' test failed repeatedly or with ABS(H) = HMIN'
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 2, TN, H)
ISTATE = -4
GO TO 570
C KFLAG = -2. Convergence failed repeatedly or with ABS(H) = HMIN. ----
540 MSG = 'DLSODI- At T (=R1) and step size H (=R2), the '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' corrector convergence failed repeatedly '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' or with ABS(H) = HMIN '
CALL XERRWD (MSG, 30, 205, 0, 0, 0, 0, 2, TN, H)
ISTATE = -5
GO TO 570
C IRES = 3 returned by RES, despite retries by DSTODI. -----------------
550 MSG = 'DLSODI- At T (=R1) residual routine returned '
CALL XERRWD (MSG, 50, 206, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' error IRES = 3 repeatedly. '
CALL XERRWD (MSG, 40, 206, 0, 0, 0, 0, 1, TN, 0.0D0)
ISTATE = -7
GO TO 590
C DAINVG failed because matrix A was singular. -------------------------
560 IER = -IER
MSG='DLSODI- Attempt to initialize dy/dt failed: Matrix A is '
CALL XERRWD (MSG, 60, 207, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' singular. DGEFA or DGBFA returned INFO = I1'
CALL XERRWD (MSG, 50, 207, 0, 1, IER, 0, 0, 0.0D0, 0.0D0)
ISTATE = -8
RETURN
C DAINVG failed because RES set IRES to 2 or 3. ------------------------
565 MSG = 'DLSODI- Attempt to initialize dy/dt failed '
CALL XERRWD (MSG, 50, 208, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' because residual routine set its error flag '
CALL XERRWD (MSG, 50, 208, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' to IRES = (I1)'
CALL XERRWD (MSG, 20, 208, 0, 1, IER, 0, 0, 0.0D0, 0.0D0)
ISTATE = -8
RETURN
C Compute IMXER if relevant. -------------------------------------------
570 BIG = 0.0D0
IMXER = 1
DO 575 I = 1,N
SIZE = ABS(RWORK(I+LACOR-1)*RWORK(I+LEWT-1))
IF (BIG .GE. SIZE) GO TO 575
BIG = SIZE
IMXER = I
575 CONTINUE
IWORK(16) = IMXER
C Compute residual if relevant. ----------------------------------------
580 LYD0 = LYH + NYH
DO 585 I = 1,N
RWORK(I+LSAVF-1) = RWORK(I+LYD0-1)/H
585 Y(I) = RWORK(I+LYH-1)
IRES = 1
CALL RES (NEQ, TN, Y, RWORK(LSAVF), YDOTI, IRES )
NFE = NFE + 1
IF (IRES .LE. 1) GO TO 595
MSG = 'DLSODI- Residual routine set its flag IRES '
CALL XERRWD (MSG, 50, 210, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' to (I1) when called for final output. '
CALL XERRWD (MSG, 50, 210, 0, 1, IRES, 0, 0, 0.0D0, 0.0D0)
GO TO 595
C Set Y vector, T, and optional outputs. -------------------------------
590 DO 592 I = 1,N
592 Y(I) = RWORK(I+LYH-1)
595 T = TN
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
RETURN
C-----------------------------------------------------------------------
C Block I.
C The following block handles all error returns due to illegal input
C (ISTATE = -3), as detected before calling the core integrator.
C First the error message routine is called. If the illegal input
C is a negative ISTATE, the run is aborted (apparent infinite loop).
C-----------------------------------------------------------------------
601 MSG = 'DLSODI- ISTATE (=I1) illegal.'
CALL XERRWD (MSG, 30, 1, 0, 1, ISTATE, 0, 0, 0.0D0, 0.0D0)
IF (ISTATE .LT. 0) GO TO 800
GO TO 700
602 MSG = 'DLSODI- ITASK (=I1) illegal. '
CALL XERRWD (MSG, 30, 2, 0, 1, ITASK, 0, 0, 0.0D0, 0.0D0)
GO TO 700
603 MSG = 'DLSODI- ISTATE .gt. 1 but DLSODI not initialized.'
CALL XERRWD (MSG, 50, 3, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
GO TO 700
604 MSG = 'DLSODI- NEQ (=I1) .lt. 1 '
CALL XERRWD (MSG, 30, 4, 0, 1, NEQ(1), 0, 0, 0.0D0, 0.0D0)
GO TO 700
605 MSG = 'DLSODI- ISTATE = 3 and NEQ increased (I1 to I2). '
CALL XERRWD (MSG, 50, 5, 0, 2, N, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
606 MSG = 'DLSODI- ITOL (=I1) illegal. '
CALL XERRWD (MSG, 30, 6, 0, 1, ITOL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
607 MSG = 'DLSODI- IOPT (=I1) illegal. '
CALL XERRWD (MSG, 30, 7, 0, 1, IOPT, 0, 0, 0.0D0, 0.0D0)
GO TO 700
608 MSG = 'DLSODI- MF (=I1) illegal. '
CALL XERRWD (MSG, 30, 8, 0, 1, MF, 0, 0, 0.0D0, 0.0D0)
GO TO 700
609 MSG = 'DLSODI- ML(=I1) illegal: .lt. 0 or .ge. NEQ(=I2) '
CALL XERRWD (MSG, 50, 9, 0, 2, ML, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
610 MSG = 'DLSODI- MU(=I1) illegal: .lt. 0 or .ge. NEQ(=I2) '
CALL XERRWD (MSG, 50, 10, 0, 2, MU, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
611 MSG = 'DLSODI- MAXORD (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 11, 0, 1, MAXORD, 0, 0, 0.0D0, 0.0D0)
GO TO 700
612 MSG = 'DLSODI- MXSTEP (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 12, 0, 1, MXSTEP, 0, 0, 0.0D0, 0.0D0)
GO TO 700
613 MSG = 'DLSODI- MXHNIL (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 13, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
614 MSG = 'DLSODI- TOUT (=R1) behind T (=R2) '
CALL XERRWD (MSG, 40, 14, 0, 0, 0, 0, 2, TOUT, T)
MSG = ' Integration direction is given by H0 (=R1) '
CALL XERRWD (MSG, 50, 14, 0, 0, 0, 0, 1, H0, 0.0D0)
GO TO 700
615 MSG = 'DLSODI- HMAX (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 15, 0, 0, 0, 0, 1, HMAX, 0.0D0)
GO TO 700
616 MSG = 'DLSODI- HMIN (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 16, 0, 0, 0, 0, 1, HMIN, 0.0D0)
GO TO 700
617 MSG='DLSODI- RWORK length needed, LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 17, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
618 MSG='DLSODI- IWORK length needed, LENIW (=I1), exceeds LIW (=I2)'
CALL XERRWD (MSG, 60, 18, 0, 2, LENIW, LIW, 0, 0.0D0, 0.0D0)
GO TO 700
619 MSG = 'DLSODI- RTOL(=I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 19, 0, 1, I, 0, 1, RTOLI, 0.0D0)
GO TO 700
620 MSG = 'DLSODI- ATOL(=I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 20, 0, 1, I, 0, 1, ATOLI, 0.0D0)
GO TO 700
621 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODI- EWT(I1) is R1 .le. 0.0 '
CALL XERRWD (MSG, 40, 21, 0, 1, I, 0, 1, EWTI, 0.0D0)
GO TO 700
622 MSG='DLSODI- TOUT(=R1) too close to T(=R2) to start integration.'
CALL XERRWD (MSG, 60, 22, 0, 0, 0, 0, 2, TOUT, T)
GO TO 700
623 MSG='DLSODI- ITASK = I1 and TOUT (=R1) behind TCUR - HU (= R2) '
CALL XERRWD (MSG, 60, 23, 0, 1, ITASK, 0, 2, TOUT, TP)
GO TO 700
624 MSG='DLSODI- ITASK = 4 or 5 and TCRIT (=R1) behind TCUR (=R2) '
CALL XERRWD (MSG, 60, 24, 0, 0, 0, 0, 2, TCRIT, TN)
GO TO 700
625 MSG='DLSODI- ITASK = 4 or 5 and TCRIT (=R1) behind TOUT (=R2) '
CALL XERRWD (MSG, 60, 25, 0, 0, 0, 0, 2, TCRIT, TOUT)
GO TO 700
626 MSG = 'DLSODI- At start of problem, too much accuracy '
CALL XERRWD (MSG, 50, 26, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' requested for precision of machine.. See TOLSF (=R1) '
CALL XERRWD (MSG, 60, 26, 0, 0, 0, 0, 1, TOLSF, 0.0D0)
RWORK(14) = TOLSF
GO TO 700
627 MSG = 'DLSODI- Trouble in DINTDY. ITASK = I1, TOUT = R1'
CALL XERRWD (MSG, 50, 27, 0, 1, ITASK, 0, 1, TOUT, 0.0D0)
C
700 ISTATE = -3
RETURN
C
800 MSG = 'DLSODI- Run aborted.. apparent infinite loop. '
CALL XERRWD (MSG, 50, 303, 2, 0, 0, 0, 0, 0.0D0, 0.0D0)
RETURN
C----------------------- End of Subroutine DLSODI ----------------------
END
*DECK DLSOIBT
SUBROUTINE DLSOIBT (RES, ADDA, JAC, NEQ, Y, YDOTI, T, TOUT, ITOL,
1 RTOL, ATOL, ITASK, ISTATE, IOPT, RWORK, LRW, IWORK, LIW, MF )
EXTERNAL RES, ADDA, JAC
INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK, LIW, MF
DOUBLE PRECISION Y, YDOTI, T, TOUT, RTOL, ATOL, RWORK
DIMENSION NEQ(*), Y(*), YDOTI(*), RTOL(*), ATOL(*), RWORK(LRW),
1 IWORK(LIW)
C-----------------------------------------------------------------------
C This is the 18 November 2003 version of
C DLSOIBT: Livermore Solver for Ordinary differential equations given
C in Implicit form, with Block-Tridiagonal Jacobian treatment.
C
C This version is in double precision.
C
C DLSOIBT solves the initial value problem for linearly implicit
C systems of first order ODEs,
C A(t,y) * dy/dt = g(t,y) , where A(t,y) is a square matrix,
C or, in component form,
C ( a * ( dy / dt )) + ... + ( a * ( dy / dt )) =
C i,1 1 i,NEQ NEQ
C
C = g ( t, y , y ,..., y ) ( i = 1,...,NEQ )
C i 1 2 NEQ
C
C If A is singular, this is a differential-algebraic system.
C
C DLSOIBT is a variant version of the DLSODI package, for the case where
C the matrices A, dg/dy, and d(A*s)/dy are all block-tridiagonal.
C-----------------------------------------------------------------------
C Reference:
C Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE
C Solvers, in Scientific Computing, R. S. Stepleman et al. (Eds.),
C North-Holland, Amsterdam, 1983, pp. 55-64.
C-----------------------------------------------------------------------
C Authors: Alan C. Hindmarsh and Jeffrey F. Painter
C Center for Applied Scientific Computing, L-561
C Lawrence Livermore National Laboratory
C Livermore, CA 94551
C and
C Charles S. Kenney
C formerly at: Naval Weapons Center
C China Lake, CA 93555
C-----------------------------------------------------------------------
C Summary of Usage.
C
C Communication between the user and the DLSOIBT package, for normal
C situations, is summarized here. This summary describes only a subset
C of the full set of options available. See the full description for
C details, including optional communication, nonstandard options,
C and instructions for special situations. See also the example
C problem (with program and output) following this summary.
C
C A. First, provide a subroutine of the form:
C SUBROUTINE RES (NEQ, T, Y, S, R, IRES)
C DOUBLE PRECISION T, Y(*), S(*), R(*)
C which computes the residual function
C r = g(t,y) - A(t,y) * s ,
C as a function of t and the vectors y and s. (s is an internally
C generated approximation to dy/dt.) The arrays Y and S are inputs
C to the RES routine and should not be altered. The residual
C vector is to be stored in the array R. The argument IRES should be
C ignored for casual use of DLSOIBT. (For uses of IRES, see the
C paragraph on RES in the full description below.)
C
C B. Next, identify the block structure of the matrices A = A(t,y) and
C dr/dy. DLSOIBT must deal internally with a linear combination, P, of
C these two matrices. The matrix P (hence both A and dr/dy) must have
C a block-tridiagonal form with fixed structure parameters
C MB = block size, MB .ge. 1, and
C NB = number of blocks in each direction, NB .ge. 4,
C with MB*NB = NEQ. In each of the NB block-rows of the matrix P
C (each consisting of MB consecutive rows), the nonzero elements are
C to lie in three consecutive MB by MB blocks. In block-rows
C 2 through NB - 1, these are centered about the main diagonal.
C in block-rows 1 and NB, they are the diagonal blocks and the two
C blocks adjacent to the diagonal block. (Thus block positions (1,3)
C and (NB,NB-2) can be nonzero.)
C Alternatively, P (hence A and dr/dy) may be only approximately
C equal to matrices with this form, and DLSOIBT should still succeed.
C The block-tridiagonal matrix P is described by three arrays,
C each of size MB by MB by NB:
C PA = array of diagonal blocks,
C PB = array of superdiagonal (and one subdiagonal) blocks, and
C PC = array of subdiagonal (and one superdiagonal) blocks.
C Specifically, the three MB by MB blocks in the k-th block-row of P
C are stored in (reading across):
C PC(*,*,k) = block to the left of the diagonal block,
C PA(*,*,k) = diagonal block, and
C PB(*,*,k) = block to the right of the diagonal block,
C except for k = 1, where the three blocks (reading across) are
C PA(*,*,1) (= diagonal block), PB(*,*,1), and PC(*,*,1),
C and k = NB, where they are
C PB(*,*,NB), PC(*,*,NB), and PA(*,*,NB) (= diagonal block).
C (Each asterisk * stands for an index that ranges from 1 to MB.)
C
C C. You must also provide a subroutine of the form:
C SUBROUTINE ADDA (NEQ, T, Y, MB, NB, PA, PB, PC)
C DOUBLE PRECISION T, Y(*), PA(MB,MB,NB), PB(MB,MB,NB), PC(MB,MB,NB)
C which adds the nonzero blocks of the matrix A = A(t,y) to the
C contents of the arrays PA, PB, and PC, following the structure
C description in Paragraph B above.
C T and the Y array are input and should not be altered.
C Thus the affect of ADDA should be the following:
C DO 30 K = 1,NB
C DO 20 J = 1,MB
C DO 10 I = 1,MB
C PA(I,J,K) = PA(I,J,K) +
C ( (I,J) element of K-th diagonal block of A)
C PB(I,J,K) = PB(I,J,K) +
C ( (I,J) element of block in block position (K,K+1) of A,
C or in block position (NB,NB-2) if K = NB)
C PC(I,J,K) = PC(I,J,K) +
C ( (I,J) element of block in block position (K,K-1) of A,
C or in block position (1,3) if K = 1)
C 10 CONTINUE
C 20 CONTINUE
C 30 CONTINUE
C
C D. For the sake of efficiency, you are encouraged to supply the
C Jacobian matrix dr/dy in closed form, where r = g(t,y) - A(t,y)*s
C (s = a fixed vector) as above. If dr/dy is being supplied,
C use MF = 21, and provide a subroutine of the form:
C SUBROUTINE JAC (NEQ, T, Y, S, MB, NB, PA, PB, PC)
C DOUBLE PRECISION T, Y(*), S(*), PA(MB,MB,NB), PB(MB,MB,NB),
C 1 PC(MB,MB,NB)
C which computes dr/dy as a function of t, y, and s. Here T, Y, and
C S are inputs, and the routine is to load dr/dy into PA, PB, PC,
C according to the structure description in Paragraph B above.
C That is, load the diagonal blocks into PA, the superdiagonal blocks
C (and block (NB,NB-2) ) into PB, and the subdiagonal blocks (and
C block (1,3) ) into PC. The blocks in block-row k of dr/dy are to
C be loaded into PA(*,*,k), PB(*,*,k), and PC(*,*,k).
C Only nonzero elements need be loaded, and the indexing
C of PA, PB, and PC is the same as in the ADDA routine.
C Note that if A is independent of Y (or this dependence
C is weak enough to be ignored) then JAC is to compute dg/dy.
C If it is not feasible to provide a JAC routine, use
C MF = 22, and DLSOIBT will compute an approximate Jacobian
C internally by difference quotients.
C
C E. Next decide whether or not to provide the initial value of the
C derivative vector dy/dt. If the initial value of A(t,y) is
C nonsingular (and not too ill-conditioned), you may let DLSOIBT compute
C this vector (ISTATE = 0). (DLSOIBT will solve the system A*s = g for
C s, with initial values of A and g.) If A(t,y) is initially
C singular, then the system is a differential-algebraic system, and
C you must make use of the particular form of the system to compute the
C initial values of y and dy/dt. In that case, use ISTATE = 1 and
C load the initial value of dy/dt into the array YDOTI.
C The input array YDOTI and the initial Y array must be consistent with
C the equations A*dy/dt = g. This implies that the initial residual
C r = g(t,y) - A(t,y)*YDOTI must be approximately zero.
C
C F. Write a main program which calls Subroutine DLSOIBT once for
C each point at which answers are desired. This should also provide
C for possible use of logical unit 6 for output of error messages by
C DLSOIBT. on the first call to DLSOIBT, supply arguments as follows:
C RES = name of user subroutine for residual function r.
C ADDA = name of user subroutine for computing and adding A(t,y).
C JAC = name of user subroutine for Jacobian matrix dr/dy
C (MF = 21). If not used, pass a dummy name.
C Note: the names for the RES and ADDA routines and (if used) the
C JAC routine must be declared External in the calling program.
C NEQ = number of scalar equations in the system.
C Y = array of initial values, of length NEQ.
C YDOTI = array of length NEQ (containing initial dy/dt if ISTATE = 1).
C T = the initial value of the independent variable.
C TOUT = first point where output is desired (.ne. T).
C ITOL = 1 or 2 according as ATOL (below) is a scalar or array.
C RTOL = relative tolerance parameter (scalar).
C ATOL = absolute tolerance parameter (scalar or array).
C the estimated local error in y(i) will be controlled so as
C to be roughly less (in magnitude) than
C EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
C EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
C Thus the local error test passes if, in each component,
C either the absolute error is less than ATOL (or ATOL(i)),
C or the relative error is less than RTOL.
C Use RTOL = 0.0 for pure absolute error control, and
C use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative error
C control. Caution: Actual (global) errors may exceed these
C local tolerances, so choose them conservatively.
C ITASK = 1 for normal computation of output values of y at t = TOUT.
C ISTATE = integer flag (input and output). Set ISTATE = 1 if the
C initial dy/dt is supplied, and 0 otherwise.
C IOPT = 0 to indicate no optional inputs used.
C RWORK = real work array of length at least:
C 22 + 9*NEQ + 3*MB*MB*NB for MF = 21 or 22.
C LRW = declared length of RWORK (in user's dimension).
C IWORK = integer work array of length at least 20 + NEQ.
C Input in IWORK(1) the block size MB and in IWORK(2) the
C number NB of blocks in each direction along the matrix A.
C These must satisfy MB .ge. 1, NB .ge. 4, and MB*NB = NEQ.
C LIW = declared length of IWORK (in user's dimension).
C MF = method flag. Standard values are:
C 21 for a user-supplied Jacobian.
C 22 for an internally generated Jacobian.
C For other choices of MF, see the paragraph on MF in
C the full description below.
C Note that the main program must declare arrays Y, YDOTI, RWORK, IWORK,
C and possibly ATOL.
C
C G. The output from the first call (or any call) is:
C Y = array of computed values of y(t) vector.
C T = corresponding value of independent variable (normally TOUT).
C ISTATE = 2 if DLSOIBT was successful, negative otherwise.
C -1 means excess work done on this call (check all inputs).
C -2 means excess accuracy requested (tolerances too small).
C -3 means illegal input detected (see printed message).
C -4 means repeated error test failures (check all inputs).
C -5 means repeated convergence failures (perhaps bad Jacobian
C supplied or wrong choice of tolerances).
C -6 means error weight became zero during problem. (Solution
C component i vanished, and ATOL or ATOL(i) = 0.)
C -7 cannot occur in casual use.
C -8 means DLSOIBT was unable to compute the initial dy/dt.
C In casual use, this means A(t,y) is initially singular.
C Supply YDOTI and use ISTATE = 1 on the first call.
C
C If DLSOIBT returns ISTATE = -1, -4, or -5, then the output of
C DLSOIBT also includes YDOTI = array containing residual vector
C r = g - A * dy/dt evaluated at the current t, y, and dy/dt.
C
C H. To continue the integration after a successful return, simply
C reset TOUT and call DLSOIBT again. No other parameters need be reset.
C
C-----------------------------------------------------------------------
C Example Problem.
C
C The following is an example problem, with the coding needed
C for its solution by DLSOIBT. The problem comes from the partial
C differential equation (the Burgers equation)
C du/dt = - u * du/dx + eta * d**2 u/dx**2, eta = .05,
C on -1 .le. x .le. 1. The boundary conditions are
C du/dx = 0 at x = -1 and at x = 1.
C The initial profile is a square wave,
C u = 1 in ABS(x) .lt. .5, u = .5 at ABS(x) = .5, u = 0 elsewhere.
C The PDE is discretized in x by a simplified Galerkin method,
C using piecewise linear basis functions, on a grid of 40 intervals.
C The equations at x = -1 and 1 use a 3-point difference approximation
C for the right-hand side. The result is a system A * dy/dt = g(y),
C of size NEQ = 41, where y(i) is the approximation to u at x = x(i),
C with x(i) = -1 + (i-1)*delx, delx = 2/(NEQ-1) = .05. The individual
C equations in the system are
C dy(1)/dt = ( y(3) - 2*y(2) + y(1) ) * eta / delx**2,
C dy(NEQ)/dt = ( y(NEQ-2) - 2*y(NEQ-1) + y(NEQ) ) * eta / delx**2,
C and for i = 2, 3, ..., NEQ-1,
C (1/6) dy(i-1)/dt + (4/6) dy(i)/dt + (1/6) dy(i+1)/dt
C = ( y(i-1)**2 - y(i+1)**2 ) / (4*delx)
C + ( y(i+1) - 2*y(i) + y(i-1) ) * eta / delx**2.
C The following coding solves the problem with MF = 21, with output
C of solution statistics at t = .1, .2, .3, and .4, and of the
C solution vector at t = .4. Here the block size is just MB = 1.
C
C EXTERNAL RESID, ADDABT, JACBT
C DOUBLE PRECISION ATOL, RTOL, RWORK, T, TOUT, Y, YDOTI
C DIMENSION Y(41), YDOTI(41), RWORK(514), IWORK(61)
C NEQ = 41
C DO 10 I = 1,NEQ
C 10 Y(I) = 0.0
C Y(11) = 0.5
C DO 20 I = 12,30
C 20 Y(I) = 1.0
C Y(31) = 0.5
C T = 0.0
C TOUT = 0.1
C ITOL = 1
C RTOL = 1.0D-4
C ATOL = 1.0D-5
C ITASK = 1
C ISTATE = 0
C IOPT = 0
C LRW = 514
C LIW = 61
C IWORK(1) = 1
C IWORK(2) = NEQ
C MF = 21
C DO 40 IO = 1,4
C CALL DLSOIBT (RESID, ADDABT, JACBT, NEQ, Y, YDOTI, T, TOUT,
C 1 ITOL,RTOL,ATOL, ITASK, ISTATE, IOPT, RWORK,LRW,IWORK,LIW, MF)
C WRITE (6,30) T, IWORK(11), IWORK(12), IWORK(13)
C 30 FORMAT(' At t =',F5.2,' No. steps =',I4,' No. r-s =',I4,
C 1 ' No. J-s =',I3)
C IF (ISTATE .NE. 2) GO TO 90
C TOUT = TOUT + 0.1
C 40 CONTINUE
C WRITE(6,50) (Y(I),I=1,NEQ)
C 50 FORMAT(/' Final solution values..'/9(5D12.4/))
C STOP
C 90 WRITE(6,95) ISTATE
C 95 FORMAT(///' Error halt.. ISTATE =',I3)
C STOP
C END
C
C SUBROUTINE RESID (N, T, Y, S, R, IRES)
C DOUBLE PRECISION T, Y, S, R, ETA, DELX, EODSQ
C DIMENSION Y(N), S(N), R(N)
C DATA ETA/0.05/, DELX/0.05/
C EODSQ = ETA/DELX**2
C R(1) = EODSQ*(Y(3) - 2.0*Y(2) + Y(1)) - S(1)
C NM1 = N - 1
C DO 10 I = 2,NM1
C R(I) = (Y(I-1)**2 - Y(I+1)**2)/(4.0*DELX)
C 1 + EODSQ*(Y(I+1) - 2.0*Y(I) + Y(I-1))
C 2 - (S(I-1) + 4.0*S(I) + S(I+1))/6.0
C 10 CONTINUE
C R(N) = EODSQ*(Y(N-2) - 2.0*Y(NM1) + Y(N)) - S(N)
C RETURN
C END
C
C SUBROUTINE ADDABT (N, T, Y, MB, NB, PA, PB, PC)
C DOUBLE PRECISION T, Y, PA, PB, PC
C DIMENSION Y(N), PA(MB,MB,NB), PB(MB,MB,NB), PC(MB,MB,NB)
C PA(1,1,1) = PA(1,1,1) + 1.0
C NM1 = N - 1
C DO 10 K = 2,NM1
C PA(1,1,K) = PA(1,1,K) + (4.0/6.0)
C PB(1,1,K) = PB(1,1,K) + (1.0/6.0)
C PC(1,1,K) = PC(1,1,K) + (1.0/6.0)
C 10 CONTINUE
C PA(1,1,N) = PA(1,1,N) + 1.0
C RETURN
C END
C
C SUBROUTINE JACBT (N, T, Y, S, MB, NB, PA, PB, PC)
C DOUBLE PRECISION T, Y, S, PA, PB, PC, ETA, DELX, EODSQ
C DIMENSION Y(N), S(N), PA(MB,MB,NB),PB(MB,MB,NB),PC(MB,MB,NB)
C DATA ETA/0.05/, DELX/0.05/
C EODSQ = ETA/DELX**2
C PA(1,1,1) = EODSQ
C PB(1,1,1) = -2.0*EODSQ
C PC(1,1,1) = EODSQ
C DO 10 K = 2,N
C PA(1,1,K) = -2.0*EODSQ
C PB(1,1,K) = -Y(K+1)*(0.5/DELX) + EODSQ
C PC(1,1,K) = Y(K-1)*(0.5/DELX) + EODSQ
C 10 CONTINUE
C PB(1,1,N) = EODSQ
C PC(1,1,N) = -2.0*EODSQ
C PA(1,1,N) = EODSQ
C RETURN
C END
C
C The output of this program (on a CDC-7600 in single precision)
C is as follows:
C
C At t = 0.10 No. steps = 35 No. r-s = 45 No. J-s = 9
C At t = 0.20 No. steps = 43 No. r-s = 54 No. J-s = 10
C At t = 0.30 No. steps = 48 No. r-s = 60 No. J-s = 11
C At t = 0.40 No. steps = 51 No. r-s = 64 No. J-s = 12
C
C Final solution values..
C 1.2747e-02 1.1997e-02 1.5560e-02 2.3767e-02 3.7224e-02
C 5.6646e-02 8.2645e-02 1.1557e-01 1.5541e-01 2.0177e-01
C 2.5397e-01 3.1104e-01 3.7189e-01 4.3530e-01 5.0000e-01
C 5.6472e-01 6.2816e-01 6.8903e-01 7.4612e-01 7.9829e-01
C 8.4460e-01 8.8438e-01 9.1727e-01 9.4330e-01 9.6281e-01
C 9.7632e-01 9.8426e-01 9.8648e-01 9.8162e-01 9.6617e-01
C 9.3374e-01 8.7535e-01 7.8236e-01 6.5321e-01 5.0003e-01
C 3.4709e-01 2.1876e-01 1.2771e-01 7.3671e-02 5.0642e-02
C 5.4496e-02
C
C-----------------------------------------------------------------------
C Full Description of User Interface to DLSOIBT.
C
C The user interface to DLSOIBT consists of the following parts.
C
C 1. The call sequence to Subroutine DLSOIBT, which is a driver
C routine for the solver. This includes descriptions of both
C the call sequence arguments and of user-supplied routines.
C Following these descriptions is a description of
C optional inputs available through the call sequence, and then
C a description of optional outputs (in the work arrays).
C
C 2. Descriptions of other routines in the DLSOIBT package that may be
C (optionally) called by the user. These provide the ability to
C alter error message handling, save and restore the internal
C Common, and obtain specified derivatives of the solution y(t).
C
C 3. Descriptions of Common blocks to be declared in overlay
C or similar environments, or to be saved when doing an interrupt
C of the problem and continued solution later.
C
C 4. Description of two routines in the DLSOIBT package, either of
C which the user may replace with his/her own version, if desired.
C These relate to the measurement of errors.
C
C-----------------------------------------------------------------------
C Part 1. Call Sequence.
C
C The call sequence parameters used for input only are
C RES, ADDA, JAC, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK,
C IOPT, LRW, LIW, MF,
C and those used for both input and output are
C Y, T, ISTATE, YDOTI.
C The work arrays RWORK and IWORK are also used for additional and
C optional inputs and optional outputs. (The term output here refers
C to the return from Subroutine DLSOIBT to the user's calling program.)
C
C The legality of input parameters will be thoroughly checked on the
C initial call for the problem, but not checked thereafter unless a
C change in input parameters is flagged by ISTATE = 3 on input.
C
C The descriptions of the call arguments are as follows.
C
C RES = the name of the user-supplied subroutine which supplies
C the residual vector for the ODE system, defined by
C r = g(t,y) - A(t,y) * s
C as a function of the scalar t and the vectors
C s and y (s approximates dy/dt). This subroutine
C is to have the form
C SUBROUTINE RES (NEQ, T, Y, S, R, IRES)
C DOUBLE PRECISION T, Y(*), S(*), R(*)
C where NEQ, T, Y, S, and IRES are input, and R and
C IRES are output. Y, S, and R are arrays of length NEQ.
C On input, IRES indicates how DLSOIBT will use the
C returned array R, as follows:
C IRES = 1 means that DLSOIBT needs the full residual,
C r = g - A*s, exactly.
C IRES = -1 means that DLSOIBT is using R only to compute
C the Jacobian dr/dy by difference quotients.
C The RES routine can ignore IRES, or it can omit some terms
C if IRES = -1. If A does not depend on y, then RES can
C just return R = g when IRES = -1. If g - A*s contains other
C additive terms that are independent of y, these can also be
C dropped, if done consistently, when IRES = -1.
C The subroutine should set the flag IRES if it
C encounters a halt condition or illegal input.
C Otherwise, it should not reset IRES. On output,
C IRES = 1 or -1 represents a normal return, and
C DLSOIBT continues integrating the ODE. Leave IRES
C unchanged from its input value.
C IRES = 2 tells DLSOIBT to immediately return control
C to the calling program, with ISTATE = 3. This lets
C the calling program change parameters of the problem
C if necessary.
C IRES = 3 represents an error condition (for example, an
C illegal value of y). DLSOIBT tries to integrate the system
C without getting IRES = 3 from RES. If it cannot, DLSOIBT
C returns with ISTATE = -7 or -1.
C On an DLSOIBT return with ISTATE = 3, -1, or -7, the
C values of T and Y returned correspond to the last point
C reached successfully without getting the flag IRES = 2 or 3.
C The flag values IRES = 2 and 3 should not be used to
C handle switches or root-stop conditions. This is better
C done by calling DLSOIBT in a one-step mode and checking the
C stopping function for a sign change at each step.
C If quantities computed in the RES routine are needed
C externally to DLSOIBT, an extra call to RES should be made
C for this purpose, for consistent and accurate results.
C To get the current dy/dt for the S argument, use DINTDY.
C RES must be declared External in the calling
C program. See note below for more about RES.
C
C ADDA = the name of the user-supplied subroutine which adds the
C matrix A = A(t,y) to another matrix, P, stored in
C block-tridiagonal form. This routine is to have the form
C SUBROUTINE ADDA (NEQ, T, Y, MB, NB, PA, PB, PC)
C DOUBLE PRECISION T, Y(*), PA(MB,MB,NB), PB(MB,MB,NB),
C 1 PC(MB,MB,NB)
C where NEQ, T, Y, MB, NB, and the arrays PA, PB, and PC
C are input, and the arrays PA, PB, and PC are output.
C Y is an array of length NEQ, and the arrays PA, PB, PC
C are all MB by MB by NB.
C Here a block-tridiagonal structure is assumed for A(t,y),
C and also for the matrix P to which A is added here,
C as described in Paragraph B of the Summary of Usage above.
C Thus the affect of ADDA should be the following:
C DO 30 K = 1,NB
C DO 20 J = 1,MB
C DO 10 I = 1,MB
C PA(I,J,K) = PA(I,J,K) +
C ( (I,J) element of K-th diagonal block of A)
C PB(I,J,K) = PB(I,J,K) +
C ( (I,J) element of block (K,K+1) of A,
C or block (NB,NB-2) if K = NB)
C PC(I,J,K) = PC(I,J,K) +
C ( (I,J) element of block (K,K-1) of A,
C or block (1,3) if K = 1)
C 10 CONTINUE
C 20 CONTINUE
C 30 CONTINUE
C ADDA must be declared External in the calling program.
C See note below for more information about ADDA.
C
C JAC = the name of the user-supplied subroutine which supplies
C the Jacobian matrix, dr/dy, where r = g - A*s. JAC is
C required if MITER = 1. Otherwise a dummy name can be
C passed. This subroutine is to have the form
C SUBROUTINE JAC (NEQ, T, Y, S, MB, NB, PA, PB, PC)
C DOUBLE PRECISION T, Y(*), S(*), PA(MB,MB,NB),
C 1 PB(MB,MB,NB), PC(MB,MB,NB)
C where NEQ, T, Y, S, MB, NB, and the arrays PA, PB, and PC
C are input, and the arrays PA, PB, and PC are output.
C Y and S are arrays of length NEQ, and the arrays PA, PB, PC
C are all MB by MB by NB.
C PA, PB, and PC are to be loaded with partial derivatives
C (elements of the Jacobian matrix) on output, in terms of the
C block-tridiagonal structure assumed, as described
C in Paragraph B of the Summary of Usage above.
C That is, load the diagonal blocks into PA, the
C superdiagonal blocks (and block (NB,NB-2) ) into PB, and
C the subdiagonal blocks (and block (1,3) ) into PC.
C The blocks in block-row k of dr/dy are to be loaded into
C PA(*,*,k), PB(*,*,k), and PC(*,*,k).
C Thus the affect of JAC should be the following:
C DO 30 K = 1,NB
C DO 20 J = 1,MB
C DO 10 I = 1,MB
C PA(I,J,K) = ( (I,J) element of
C K-th diagonal block of dr/dy)
C PB(I,J,K) = ( (I,J) element of block (K,K+1)
C of dr/dy, or block (NB,NB-2) if K = NB)
C PC(I,J,K) = ( (I,J) element of block (K,K-1)
C of dr/dy, or block (1,3) if K = 1)
C 10 CONTINUE
C 20 CONTINUE
C 30 CONTINUE
C PA, PB, and PC are preset to zero by the solver,
C so that only the nonzero elements need be loaded by JAC.
C Each call to JAC is preceded by a call to RES with the same
C arguments NEQ, T, Y, and S. Thus to gain some efficiency,
C intermediate quantities shared by both calculations may be
C saved in a user Common block by RES and not recomputed by JAC
C if desired. Also, JAC may alter the Y array, if desired.
C JAC need not provide dr/dy exactly. A crude
C approximation will do, so that DLSOIBT may be used when
C A and dr/dy are not really block-tridiagonal, but are close
C to matrices that are.
C JAC must be declared External in the calling program.
C See note below for more about JAC.
C
C Note on RES, ADDA, and JAC:
C These subroutines may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in the subroutines) and/or Y has length
C exceeding NEQ(1). However, these routines should not alter
C NEQ(1), Y(1),...,Y(NEQ) or any other input variables.
C See the descriptions of NEQ and Y below.
C
C NEQ = the size of the system (number of first order ordinary
C differential equations or scalar algebraic equations).
C Used only for input.
C NEQ may be decreased, but not increased, during the problem.
C If NEQ is decreased (with ISTATE = 3 on input), the
C remaining components of Y should be left undisturbed, if
C these are to be accessed in RES, ADDA, or JAC.
C
C Normally, NEQ is a scalar, and it is generally referred to
C as a scalar in this user interface description. However,
C NEQ may be an array, with NEQ(1) set to the system size.
C (The DLSOIBT package accesses only NEQ(1).) In either case,
C this parameter is passed as the NEQ argument in all calls
C to RES, ADDA, and JAC. Hence, if it is an array,
C locations NEQ(2),... may be used to store other integer data
C and pass it to RES, ADDA, or JAC. Each such subroutine
C must include NEQ in a Dimension statement in that case.
C
C Y = a real array for the vector of dependent variables, of
C length NEQ or more. Used for both input and output on the
C first call (ISTATE = 0 or 1), and only for output on other
C calls. On the first call, Y must contain the vector of
C initial values. On output, Y contains the computed solution
C vector, evaluated at t. If desired, the Y array may be used
C for other purposes between calls to the solver.
C
C This array is passed as the Y argument in all calls to RES,
C ADDA, and JAC. Hence its length may exceed NEQ,
C and locations Y(NEQ+1),... may be used to store other real
C data and pass it to RES, ADDA, or JAC. (The DLSOIBT
C package accesses only Y(1),...,Y(NEQ). )
C
C YDOTI = a real array for the initial value of the vector
C dy/dt and for work space, of dimension at least NEQ.
C
C On input:
C If ISTATE = 0 then DLSOIBT will compute the initial value
C of dy/dt, if A is nonsingular. Thus YDOTI will
C serve only as work space and may have any value.
C If ISTATE = 1 then YDOTI must contain the initial value
C of dy/dt.
C If ISTATE = 2 or 3 (continuation calls) then YDOTI
C may have any value.
C Note: If the initial value of A is singular, then
C DLSOIBT cannot compute the initial value of dy/dt, so
C it must be provided in YDOTI, with ISTATE = 1.
C
C On output, when DLSOIBT terminates abnormally with ISTATE =
C -1, -4, or -5, YDOTI will contain the residual
C r = g(t,y) - A(t,y)*(dy/dt). If r is large, t is near
C its initial value, and YDOTI is supplied with ISTATE = 1,
C there may have been an incorrect input value of
C YDOTI = dy/dt, or the problem (as given to DLSOIBT)
C may not have a solution.
C
C If desired, the YDOTI array may be used for other
C purposes between calls to the solver.
C
C T = the independent variable. On input, T is used only on the
C first call, as the initial point of the integration.
C On output, after each call, T is the value at which a
C computed solution y is evaluated (usually the same as TOUT).
C On an error return, T is the farthest point reached.
C
C TOUT = the next value of t at which a computed solution is desired.
C Used only for input.
C
C When starting the problem (ISTATE = 0 or 1), TOUT may be
C equal to T for one call, then should .ne. T for the next
C call. For the initial T, an input value of TOUT .ne. T is
C used in order to determine the direction of the integration
C (i.e. the algebraic sign of the step sizes) and the rough
C scale of the problem. Integration in either direction
C (forward or backward in t) is permitted.
C
C If ITASK = 2 or 5 (one-step modes), TOUT is ignored after
C the first call (i.e. the first call with TOUT .ne. T).
C Otherwise, TOUT is required on every call.
C
C If ITASK = 1, 3, or 4, the values of TOUT need not be
C monotone, but a value of TOUT which backs up is limited
C to the current internal T interval, whose endpoints are
C TCUR - HU and TCUR (see optional outputs, below, for
C TCUR and HU).
C
C ITOL = an indicator for the type of error control. See
C description below under ATOL. Used only for input.
C
C RTOL = a relative error tolerance parameter, either a scalar or
C an array of length NEQ. See description below under ATOL.
C Input only.
C
C ATOL = an absolute error tolerance parameter, either a scalar or
C an array of length NEQ. Input only.
C
C The input parameters ITOL, RTOL, and ATOL determine
C the error control performed by the solver. The solver will
C control the vector E = (E(i)) of estimated local errors
C in y, according to an inequality of the form
C RMS-norm of ( E(i)/EWT(i) ) .le. 1,
C where EWT(i) = RTOL(i)*ABS(Y(i)) + ATOL(i),
C and the RMS-norm (root-mean-square norm) here is
C RMS-norm(v) = SQRT(sum v(i)**2 / NEQ). Here EWT = (EWT(i))
C is a vector of weights which must always be positive, and
C the values of RTOL and ATOL should all be non-negative.
C The following table gives the types (scalar/array) of
C RTOL and ATOL, and the corresponding form of EWT(i).
C
C ITOL RTOL ATOL EWT(i)
C 1 scalar scalar RTOL*ABS(Y(i)) + ATOL
C 2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
C 3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
C 4 array scalar RTOL(i)*ABS(Y(i)) + ATOL(i)
C
C When either of these parameters is a scalar, it need not
C be dimensioned in the user's calling program.
C
C If none of the above choices (with ITOL, RTOL, and ATOL
C fixed throughout the problem) is suitable, more general
C error controls can be obtained by substituting
C user-supplied routines for the setting of EWT and/or for
C the norm calculation. See Part 4 below.
C
C If global errors are to be estimated by making a repeated
C run on the same problem with smaller tolerances, then all
C components of RTOL and ATOL (i.e. of EWT) should be scaled
C down uniformly.
C
C ITASK = an index specifying the task to be performed.
C Input only. ITASK has the following values and meanings.
C 1 means normal computation of output values of y(t) at
C t = TOUT (by overshooting and interpolating).
C 2 means take one step only and return.
C 3 means stop at the first internal mesh point at or
C beyond t = TOUT and return.
C 4 means normal computation of output values of y(t) at
C t = TOUT but without overshooting t = TCRIT.
C TCRIT must be input as RWORK(1). TCRIT may be equal to
C or beyond TOUT, but not behind it in the direction of
C integration. This option is useful if the problem
C has a singularity at or beyond t = TCRIT.
C 5 means take one step, without passing TCRIT, and return.
C TCRIT must be input as RWORK(1).
C
C Note: If ITASK = 4 or 5 and the solver reaches TCRIT
C (within roundoff), it will return T = TCRIT (exactly) to
C indicate this (unless ITASK = 4 and TOUT comes before TCRIT,
C in which case answers at t = TOUT are returned first).
C
C ISTATE = an index used for input and output to specify the
C state of the calculation.
C
C On input, the values of ISTATE are as follows.
C 0 means this is the first call for the problem, and
C DLSOIBT is to compute the initial value of dy/dt
C (while doing other initializations). See note below.
C 1 means this is the first call for the problem, and
C the initial value of dy/dt has been supplied in
C YDOTI (DLSOIBT will do other initializations).
C See note below.
C 2 means this is not the first call, and the calculation
C is to continue normally, with no change in any input
C parameters except possibly TOUT and ITASK.
C (If ITOL, RTOL, and/or ATOL are changed between calls
C with ISTATE = 2, the new values will be used but not
C tested for legality.)
C 3 means this is not the first call, and the
C calculation is to continue normally, but with
C a change in input parameters other than
C TOUT and ITASK. Changes are allowed in
C NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, MF, MB, NB,
C and any of the optional inputs except H0.
C (See IWORK description for MB and NB.)
C Note: A preliminary call with TOUT = T is not counted
C as a first call here, as no initialization or checking of
C input is done. (Such a call is sometimes useful for the
C purpose of outputting the initial conditions.)
C Thus the first call for which TOUT .ne. T requires
C ISTATE = 0 or 1 on input.
C
C On output, ISTATE has the following values and meanings.
C 0 or 1 means nothing was done; TOUT = t and
C ISTATE = 0 or 1 on input.
C 2 means that the integration was performed successfully.
C 3 means that the user-supplied Subroutine RES signalled
C DLSOIBT to halt the integration and return (IRES = 2).
C Integration as far as T was achieved with no occurrence
C of IRES = 2, but this flag was set on attempting the
C next step.
C -1 means an excessive amount of work (more than MXSTEP
C steps) was done on this call, before completing the
C requested task, but the integration was otherwise
C successful as far as T. (MXSTEP is an optional input
C and is normally 500.) To continue, the user may
C simply reset ISTATE to a value .gt. 1 and call again
C (the excess work step counter will be reset to 0).
C In addition, the user may increase MXSTEP to avoid
C this error return (see below on optional inputs).
C -2 means too much accuracy was requested for the precision
C of the machine being used. This was detected before
C completing the requested task, but the integration
C was successful as far as T. To continue, the tolerance
C parameters must be reset, and ISTATE must be set
C to 3. The optional output TOLSF may be used for this
C purpose. (Note: If this condition is detected before
C taking any steps, then an illegal input return
C (ISTATE = -3) occurs instead.)
C -3 means illegal input was detected, before taking any
C integration steps. See written message for details.
C Note: If the solver detects an infinite loop of calls
C to the solver with illegal input, it will cause
C the run to stop.
C -4 means there were repeated error test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C The problem may have a singularity, or the input
C may be inappropriate.
C -5 means there were repeated convergence test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C This may be caused by an inaccurate Jacobian matrix.
C -6 means EWT(i) became zero for some i during the
C integration. Pure relative error control (ATOL(i) = 0.0)
C was requested on a variable which has now vanished.
C The integration was successful as far as T.
C -7 means that the user-supplied Subroutine RES set
C its error flag (IRES = 3) despite repeated tries by
C DLSOIBT to avoid that condition.
C -8 means that ISTATE was 0 on input but DLSOIBT was unable
C to compute the initial value of dy/dt. See the
C printed message for details.
C
C Note: Since the normal output value of ISTATE is 2,
C it does not need to be reset for normal continuation.
C Similarly, ISTATE (= 3) need not be reset if RES told
C DLSOIBT to return because the calling program must change
C the parameters of the problem.
C Also, since a negative input value of ISTATE will be
C regarded as illegal, a negative output value requires the
C user to change it, and possibly other inputs, before
C calling the solver again.
C
C IOPT = an integer flag to specify whether or not any optional
C inputs are being used on this call. Input only.
C The optional inputs are listed separately below.
C IOPT = 0 means no optional inputs are being used.
C Default values will be used in all cases.
C IOPT = 1 means one or more optional inputs are being used.
C
C RWORK = a real working array (double precision).
C The length of RWORK must be at least
C 20 + NYH*(MAXORD + 1) + 3*NEQ + LENWM where
C NYH = the initial value of NEQ,
C MAXORD = 12 (if METH = 1) or 5 (if METH = 2) (unless a
C smaller value is given as an optional input),
C LENWM = 3*MB*MB*NB + 2.
C (See MF description for the definition of METH.)
C Thus if MAXORD has its default value and NEQ is constant,
C this length is
C 22 + 16*NEQ + 3*MB*MB*NB for MF = 11 or 12,
C 22 + 9*NEQ + 3*MB*MB*NB for MF = 21 or 22.
C The first 20 words of RWORK are reserved for conditional
C and optional inputs and optional outputs.
C
C The following word in RWORK is a conditional input:
C RWORK(1) = TCRIT = critical value of t which the solver
C is not to overshoot. Required if ITASK is
C 4 or 5, and ignored otherwise. (See ITASK.)
C
C LRW = the length of the array RWORK, as declared by the user.
C (This will be checked by the solver.)
C
C IWORK = an integer work array. The length of IWORK must be at least
C 20 + NEQ . The first few words of IWORK are used for
C additional and optional inputs and optional outputs.
C
C The following 2 words in IWORK are additional required
C inputs to DLSOIBT:
C IWORK(1) = MB = block size
C IWORK(2) = NB = number of blocks in the main diagonal
C These must satisfy MB .ge. 1, NB .ge. 4, and MB*NB = NEQ.
C
C LIW = the length of the array IWORK, as declared by the user.
C (This will be checked by the solver.)
C
C Note: The work arrays must not be altered between calls to DLSOIBT
C for the same problem, except possibly for the additional and
C optional inputs, and except for the last 3*NEQ words of RWORK.
C The latter space is used for internal scratch space, and so is
C available for use by the user outside DLSOIBT between calls, if
C desired (but not for use by RES, ADDA, or JAC).
C
C MF = the method flag. used only for input. The legal values of
C MF are 11, 12, 21, and 22.
C MF has decimal digits METH and MITER: MF = 10*METH + MITER.
C METH indicates the basic linear multistep method:
C METH = 1 means the implicit Adams method.
C METH = 2 means the method based on Backward
C Differentiation Formulas (BDFS).
C The BDF method is strongly preferred for stiff
C problems, while the Adams method is preferred when the
C problem is not stiff. If the matrix A(t,y) is
C nonsingular, stiffness here can be taken to mean that of
C the explicit ODE system dy/dt = A-inverse * g. If A is
C singular, the concept of stiffness is not well defined.
C If you do not know whether the problem is stiff, we
C recommend using METH = 2. If it is stiff, the advantage
C of METH = 2 over METH = 1 will be great, while if it is
C not stiff, the advantage of METH = 1 will be slight.
C If maximum efficiency is important, some experimentation
C with METH may be necessary.
C MITER indicates the corrector iteration method:
C MITER = 1 means chord iteration with a user-supplied
C block-tridiagonal Jacobian.
C MITER = 2 means chord iteration with an internally
C generated (difference quotient) block-
C tridiagonal Jacobian approximation, using
C 3*MB+1 extra calls to RES per dr/dy evaluation.
C If MITER = 1, the user must supply a Subroutine JAC
C (the name is arbitrary) as described above under JAC.
C For MITER = 2, a dummy argument can be used.
C-----------------------------------------------------------------------
C Optional Inputs.
C
C The following is a list of the optional inputs provided for in the
C call sequence. (See also Part 2.) For each such input variable,
C this table lists its name as used in this documentation, its
C location in the call sequence, its meaning, and the default value.
C The use of any of these inputs requires IOPT = 1, and in that
C case all of these inputs are examined. A value of zero for any
C of these optional inputs will cause the default value to be used.
C Thus to use a subset of the optional inputs, simply preload
C locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively, and
C then set those of interest to nonzero values.
C
C Name Location Meaning and Default Value
C
C H0 RWORK(5) the step size to be attempted on the first step.
C The default value is determined by the solver.
C
C HMAX RWORK(6) the maximum absolute step size allowed.
C The default value is infinite.
C
C HMIN RWORK(7) the minimum absolute step size allowed.
C The default value is 0. (This lower bound is not
C enforced on the final step before reaching TCRIT
C when ITASK = 4 or 5.)
C
C MAXORD IWORK(5) the maximum order to be allowed. The default
C value is 12 if METH = 1, and 5 if METH = 2.
C If MAXORD exceeds the default value, it will
C be reduced to the default value.
C If MAXORD is changed during the problem, it may
C cause the current order to be reduced.
C
C MXSTEP IWORK(6) maximum number of (internally defined) steps
C allowed during one call to the solver.
C The default value is 500.
C
C MXHNIL IWORK(7) maximum number of messages printed (per problem)
C warning that T + H = T on a step (H = step size).
C This must be positive to result in a non-default
C value. The default value is 10.
C-----------------------------------------------------------------------
C Optional Outputs.
C
C As optional additional output from DLSOIBT, the variables listed
C below are quantities related to the performance of DLSOIBT
C which are available to the user. These are communicated by way of
C the work arrays, but also have internal mnemonic names as shown.
C Except where stated otherwise, all of these outputs are defined
C on any successful return from DLSOIBT, and on any return with
C ISTATE = -1, -2, -4, -5, -6, or -7. On a return with -3 (illegal
C input) or -8, they will be unchanged from their existing values
C (if any), except possibly for TOLSF, LENRW, and LENIW.
C On any error return, outputs relevant to the error will be defined,
C as noted below.
C
C Name Location Meaning
C
C HU RWORK(11) the step size in t last used (successfully).
C
C HCUR RWORK(12) the step size to be attempted on the next step.
C
C TCUR RWORK(13) the current value of the independent variable
C which the solver has actually reached, i.e. the
C current internal mesh point in t. On output, TCUR
C will always be at least as far as the argument
C T, but may be farther (if interpolation was done).
C
C TOLSF RWORK(14) a tolerance scale factor, greater than 1.0,
C computed when a request for too much accuracy was
C detected (ISTATE = -3 if detected at the start of
C the problem, ISTATE = -2 otherwise). If ITOL is
C left unaltered but RTOL and ATOL are uniformly
C scaled up by a factor of TOLSF for the next call,
C then the solver is deemed likely to succeed.
C (The user may also ignore TOLSF and alter the
C tolerance parameters in any other way appropriate.)
C
C NST IWORK(11) the number of steps taken for the problem so far.
C
C NRE IWORK(12) the number of residual evaluations (RES calls)
C for the problem so far.
C
C NJE IWORK(13) the number of Jacobian evaluations (each involving
C an evaluation of a and dr/dy) for the problem so
C far. This equals the number of calls to ADDA and
C (if MITER = 1) to JAC, and the number of matrix
C LU decompositions.
C
C NQU IWORK(14) the method order last used (successfully).
C
C NQCUR IWORK(15) the order to be attempted on the next step.
C
C IMXER IWORK(16) the index of the component of largest magnitude in
C the weighted local error vector ( E(i)/EWT(i) ),
C on an error return with ISTATE = -4 or -5.
C
C LENRW IWORK(17) the length of RWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C LENIW IWORK(18) the length of IWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C
C The following two arrays are segments of the RWORK array which
C may also be of interest to the user as optional outputs.
C For each array, the table below gives its internal name,
C its base address in RWORK, and its description.
C
C Name Base Address Description
C
C YH 21 the Nordsieck history array, of size NYH by
C (NQCUR + 1), where NYH is the initial value
C of NEQ. For j = 0,1,...,NQCUR, column j+1
C of YH contains HCUR**j/factorial(j) times
C the j-th derivative of the interpolating
C polynomial currently representing the solution,
C evaluated at t = TCUR.
C
C ACOR LENRW-NEQ+1 array of size NEQ used for the accumulated
C corrections on each step, scaled on output to
C represent the estimated local error in y on
C the last step. This is the vector E in the
C description of the error control. It is
C defined only on a return from DLSOIBT with
C ISTATE = 2.
C
C-----------------------------------------------------------------------
C Part 2. Other Routines Callable.
C
C The following are optional calls which the user may make to
C gain additional capabilities in conjunction with DLSOIBT.
C (The routines XSETUN and XSETF are designed to conform to the
C SLATEC error handling package.)
C
C Form of Call Function
C CALL XSETUN(LUN) Set the logical unit number, LUN, for
C output of messages from DLSOIBT, if
C the default is not desired.
C The default value of LUN is 6.
C
C CALL XSETF(MFLAG) Set a flag to control the printing of
C messages by DLSOIBT.
C MFLAG = 0 means do not print. (Danger:
C This risks losing valuable information.)
C MFLAG = 1 means print (the default).
C
C Either of the above calls may be made at
C any time and will take effect immediately.
C
C CALL DSRCOM(RSAV,ISAV,JOB) saves and restores the contents of
C the internal Common blocks used by
C DLSOIBT (see Part 3 below).
C RSAV must be a real array of length 218
C or more, and ISAV must be an integer
C array of length 37 or more.
C JOB=1 means save Common into RSAV/ISAV.
C JOB=2 means restore Common from RSAV/ISAV.
C DSRCOM is useful if one is
C interrupting a run and restarting
C later, or alternating between two or
C more problems solved with DLSOIBT.
C
C CALL DINTDY(,,,,,) Provide derivatives of y, of various
C (see below) orders, at a specified point t, if
C desired. It may be called only after
C a successful return from DLSOIBT.
C
C The detailed instructions for using DINTDY are as follows.
C The form of the call is:
C
C CALL DINTDY (T, K, RWORK(21), NYH, DKY, IFLAG)
C
C The input parameters are:
C
C T = value of independent variable where answers are desired
C (normally the same as the t last returned by DLSOIBT).
C For valid results, T must lie between TCUR - HU and TCUR.
C (See optional outputs for TCUR and HU.)
C K = integer order of the derivative desired. K must satisfy
C 0 .le. K .le. NQCUR, where NQCUR is the current order
C (see optional outputs). The capability corresponding
C to K = 0, i.e. computing y(t), is already provided
C by DLSOIBT directly. Since NQCUR .ge. 1, the first
C derivative dy/dt is always available with DINTDY.
C RWORK(21) = the base address of the history array YH.
C NYH = column length of YH, equal to the initial value of NEQ.
C
C The output parameters are:
C
C DKY = a real array of length NEQ containing the computed value
C of the K-th derivative of y(t).
C IFLAG = integer flag, returned as 0 if K and T were legal,
C -1 if K was illegal, and -2 if T was illegal.
C On an error return, a message is also written.
C-----------------------------------------------------------------------
C Part 3. Common Blocks.
C
C If DLSOIBT is to be used in an overlay situation, the user
C must declare, in the primary overlay, the variables in:
C (1) the call sequence to DLSOIBT, and
C (2) the internal Common block
C /DLS001/ of length 255 (218 double precision words
C followed by 37 integer words),
C
C If DLSOIBT is used on a system in which the contents of internal
C Common blocks are not preserved between calls, the user should
C declare the above Common block in the calling program to insure
C that their contents are preserved.
C
C If the solution of a given problem by DLSOIBT is to be interrupted
C and then later continued, such as when restarting an interrupted run
C or alternating between two or more problems, the user should save,
C following the return from the last DLSOIBT call prior to the
C interruption, the contents of the call sequence variables and the
C internal Common blocks, and later restore these values before the
C next DLSOIBT call for that problem. To save and restore the Common
C blocks, use Subroutine DSRCOM (see Part 2 above).
C
C-----------------------------------------------------------------------
C Part 4. Optionally Replaceable Solver Routines.
C
C Below are descriptions of two routines in the DLSOIBT package which
C relate to the measurement of errors. Either routine can be
C replaced by a user-supplied version, if desired. However, since such
C a replacement may have a major impact on performance, it should be
C done only when absolutely necessary, and only with great caution.
C (Note: The means by which the package version of a routine is
C superseded by the user's version may be system-dependent.)
C
C (a) DEWSET.
C The following subroutine is called just before each internal
C integration step, and sets the array of error weights, EWT, as
C described under ITOL/RTOL/ATOL above:
C SUBROUTINE DEWSET (NEQ, ITOL, RTOL, ATOL, YCUR, EWT)
C where NEQ, ITOL, RTOL, and ATOL are as in the DLSOIBT call sequence,
C YCUR contains the current dependent variable vector, and
C EWT is the array of weights set by DEWSET.
C
C If the user supplies this subroutine, it must return in EWT(i)
C (i = 1,...,NEQ) a positive quantity suitable for comparing errors
C in y(i) to. The EWT array returned by DEWSET is passed to the DVNORM
C routine (see below), and also used by DLSOIBT in the computation
C of the optional output IMXER, the diagonal Jacobian approximation,
C and the increments for difference quotient Jacobians.
C
C In the user-supplied version of DEWSET, it may be desirable to use
C the current values of derivatives of y. Derivatives up to order NQ
C are available from the history array YH, described above under
C optional outputs. In DEWSET, YH is identical to the YCUR array,
C extended to NQ + 1 columns with a column length of NYH and scale
C factors of H**j/factorial(j). On the first call for the problem,
C given by NST = 0, NQ is 1 and H is temporarily set to 1.0.
C NYH is the initial value of NEQ. The quantities NQ, H, and NST
C can be obtained by including in DEWSET the statements:
C DOUBLE PRECISION RLS
C COMMON /DLS001/ RLS(218),ILS(37)
C NQ = ILS(33)
C NST = ILS(34)
C H = RLS(212)
C Thus, for example, the current value of dy/dt can be obtained as
C YCUR(NYH+i)/H (i=1,...,NEQ) (and the division by H is
C unnecessary when NST = 0).
C
C (b) DVNORM.
C The following is a real function routine which computes the weighted
C root-mean-square norm of a vector v:
C D = DVNORM (N, V, W)
C where:
C N = the length of the vector,
C V = real array of length N containing the vector,
C W = real array of length N containing weights,
C D = SQRT( (1/N) * sum(V(i)*W(i))**2 ).
C DVNORM is called with N = NEQ and with W(i) = 1.0/EWT(i), where
C EWT is as set by Subroutine DEWSET.
C
C If the user supplies this function, it should return a non-negative
C value of DVNORM suitable for use in the error control in DLSOIBT.
C None of the arguments should be altered by DVNORM.
C For example, a user-supplied DVNORM routine might:
C -substitute a max-norm of (V(i)*W(i)) for the RMS-norm, or
C -ignore some components of V in the norm, with the effect of
C suppressing the error control on those components of y.
C-----------------------------------------------------------------------
C
C***REVISION HISTORY (YYYYMMDD)
C 19840625 DATE WRITTEN
C 19870330 Major update: corrected comments throughout;
C removed TRET from Common; rewrote EWSET with 4 loops;
C fixed t test in INTDY; added Cray directives in STODI;
C in STODI, fixed DELP init. and logic around PJAC call;
C combined routines to save/restore Common;
C passed LEVEL = 0 in error message calls (except run abort).
C 20010425 Major update: convert source lines to upper case;
C added *DECK lines; changed from 1 to * in dummy dimensions;
C changed names R1MACH/D1MACH to RUMACH/DUMACH;
C renamed routines for uniqueness across single/double prec.;
C converted intrinsic names to generic form;
C removed ILLIN and NTREP (data loaded) from Common;
C removed all 'own' variables from Common;
C changed error messages to quoted strings;
C replaced XERRWV/XERRWD with 1993 revised version;
C converted prologues, comments, error messages to mixed case;
C converted arithmetic IF statements to logical IF statements;
C numerous corrections to prologues and internal comments.
C 20010507 Converted single precision source to double precision.
C 20020502 Corrected declarations in descriptions of user routines.
C 20031105 Restored 'own' variables to Common block, to enable
C interrupt/restart feature.
C 20031112 Added SAVE statements for data-loaded constants.
C 20031117 Changed internal names NRE, LSAVR to NFE, LSAVF resp.
C
C-----------------------------------------------------------------------
C Other routines in the DLSOIBT package.
C
C In addition to Subroutine DLSOIBT, the DLSOIBT package includes the
C following subroutines and function routines:
C DAIGBT computes the initial value of the vector
C dy/dt = A-inverse * g
C DINTDY computes an interpolated value of the y vector at t = TOUT.
C DSTODI is the core integrator, which does one step of the
C integration and the associated error control.
C DCFODE sets all method coefficients and test constants.
C DEWSET sets the error weight vector EWT before each step.
C DVNORM computes the weighted RMS-norm of a vector.
C DSRCOM is a user-callable routine to save and restore
C the contents of the internal Common blocks.
C DPJIBT computes and preprocesses the Jacobian matrix
C and the Newton iteration matrix P.
C DSLSBT manages solution of linear system in chord iteration.
C DDECBT and DSOLBT are routines for solving block-tridiagonal
C systems of linear algebraic equations.
C DGEFA and DGESL are routines from LINPACK for solving full
C systems of linear algebraic equations.
C DDOT is one of the basic linear algebra modules (BLAS).
C DUMACH computes the unit roundoff in a machine-independent manner.
C XERRWD, XSETUN, XSETF, IXSAV, and IUMACH handle the printing of all
C error messages and warnings. XERRWD is machine-dependent.
C Note: DVNORM, DDOT, DUMACH, IXSAV, and IUMACH are function routines.
C All the others are subroutines.
C
C-----------------------------------------------------------------------
EXTERNAL DPJIBT, DSLSBT
DOUBLE PRECISION DUMACH, DVNORM
INTEGER INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER I, I1, I2, IER, IFLAG, IMXER, IRES, KGO,
1 LENIW, LENRW, LENWM, LP, LYD0, MB, MORD, MXHNL0, MXSTP0, NB
DOUBLE PRECISION ROWNS,
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
DOUBLE PRECISION ATOLI, AYI, BIG, EWTI, H0, HMAX, HMX, RH, RTOLI,
1 TCRIT, TDIST, TNEXT, TOL, TOLSF, TP, SIZE, SUM, W0
DIMENSION MORD(2)
LOGICAL IHIT
CHARACTER*60 MSG
SAVE MORD, MXSTP0, MXHNL0
C-----------------------------------------------------------------------
C The following internal Common block contains
C (a) variables which are local to any subroutine but whose values must
C be preserved between calls to the routine ("own" variables), and
C (b) variables which are communicated between subroutines.
C The block DLS001 is declared in subroutines DLSOIBT, DINTDY, DSTODI,
C DPJIBT, and DSLSBT.
C Groups of variables are replaced by dummy arrays in the Common
C declarations in routines where those variables are not used.
C-----------------------------------------------------------------------
COMMON /DLS001/ ROWNS(209),
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
2 INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS(6),
3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
DATA MORD(1),MORD(2)/12,5/, MXSTP0/500/, MXHNL0/10/
C-----------------------------------------------------------------------
C Block A.
C This code block is executed on every call.
C It tests ISTATE and ITASK for legality and branches appropriately.
C If ISTATE .gt. 1 but the flag INIT shows that initialization has
C not yet been done, an error return occurs.
C If ISTATE = 0 or 1 and TOUT = T, return immediately.
C-----------------------------------------------------------------------
IF (ISTATE .LT. 0 .OR. ISTATE .GT. 3) GO TO 601
IF (ITASK .LT. 1 .OR. ITASK .GT. 5) GO TO 602
IF (ISTATE .LE. 1) GO TO 10
IF (INIT .EQ. 0) GO TO 603
IF (ISTATE .EQ. 2) GO TO 200
GO TO 20
10 INIT = 0
IF (TOUT .EQ. T) RETURN
C-----------------------------------------------------------------------
C Block B.
C The next code block is executed for the initial call (ISTATE = 0 or 1)
C or for a continuation call with parameter changes (ISTATE = 3).
C It contains checking of all inputs and various initializations.
C
C First check legality of the non-optional inputs NEQ, ITOL, IOPT,
C MF, MB, and NB.
C-----------------------------------------------------------------------
20 IF (NEQ(1) .LE. 0) GO TO 604
IF (ISTATE .LE. 1) GO TO 25
IF (NEQ(1) .GT. N) GO TO 605
25 N = NEQ(1)
IF (ITOL .LT. 1 .OR. ITOL .GT. 4) GO TO 606
IF (IOPT .LT. 0 .OR. IOPT .GT. 1) GO TO 607
METH = MF/10
MITER = MF - 10*METH
IF (METH .LT. 1 .OR. METH .GT. 2) GO TO 608
IF (MITER .LT. 1 .OR. MITER .GT. 2) GO TO 608
MB = IWORK(1)
NB = IWORK(2)
IF (MB .LT. 1 .OR. MB .GT. N) GO TO 609
IF (NB .LT. 4) GO TO 610
IF (MB*NB .NE. N) GO TO 609
C Next process and check the optional inputs. --------------------------
IF (IOPT .EQ. 1) GO TO 40
MAXORD = MORD(METH)
MXSTEP = MXSTP0
MXHNIL = MXHNL0
IF (ISTATE .LE. 1) H0 = 0.0D0
HMXI = 0.0D0
HMIN = 0.0D0
GO TO 60
40 MAXORD = IWORK(5)
IF (MAXORD .LT. 0) GO TO 611
IF (MAXORD .EQ. 0) MAXORD = 100
MAXORD = MIN(MAXORD,MORD(METH))
MXSTEP = IWORK(6)
IF (MXSTEP .LT. 0) GO TO 612
IF (MXSTEP .EQ. 0) MXSTEP = MXSTP0
MXHNIL = IWORK(7)
IF (MXHNIL .LT. 0) GO TO 613
IF (MXHNIL .EQ. 0) MXHNIL = MXHNL0
IF (ISTATE .GT. 1) GO TO 50
H0 = RWORK(5)
IF ((TOUT - T)*H0 .LT. 0.0D0) GO TO 614
50 HMAX = RWORK(6)
IF (HMAX .LT. 0.0D0) GO TO 615
HMXI = 0.0D0
IF (HMAX .GT. 0.0D0) HMXI = 1.0D0/HMAX
HMIN = RWORK(7)
IF (HMIN .LT. 0.0D0) GO TO 616
C-----------------------------------------------------------------------
C Set work array pointers and check lengths LRW and LIW.
C Pointers to segments of RWORK and IWORK are named by prefixing L to
C the name of the segment. E.g., the segment YH starts at RWORK(LYH).
C Segments of RWORK (in order) are denoted YH, WM, EWT, SAVR, ACOR.
C-----------------------------------------------------------------------
60 LYH = 21
IF (ISTATE .LE. 1) NYH = N
LWM = LYH + (MAXORD + 1)*NYH
LENWM = 3*MB*MB*NB + 2
LEWT = LWM + LENWM
LSAVF = LEWT + N
LACOR = LSAVF + N
LENRW = LACOR + N - 1
IWORK(17) = LENRW
LIWM = 1
LENIW = 20 + N
IWORK(18) = LENIW
IF (LENRW .GT. LRW) GO TO 617
IF (LENIW .GT. LIW) GO TO 618
C Check RTOL and ATOL for legality. ------------------------------------
RTOLI = RTOL(1)
ATOLI = ATOL(1)
DO 70 I = 1,N
IF (ITOL .GE. 3) RTOLI = RTOL(I)
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
IF (RTOLI .LT. 0.0D0) GO TO 619
IF (ATOLI .LT. 0.0D0) GO TO 620
70 CONTINUE
IF (ISTATE .LE. 1) GO TO 100
C If ISTATE = 3, set flag to signal parameter changes to DSTODI. -------
JSTART = -1
IF (NQ .LE. MAXORD) GO TO 90
C MAXORD was reduced below NQ. Copy YH(*,MAXORD+2) into YDOTI.---------
DO 80 I = 1,N
80 YDOTI(I) = RWORK(I+LWM-1)
C Reload WM(1) = RWORK(lWM), since lWM may have changed. ---------------
90 RWORK(LWM) = SQRT(UROUND)
IF (N .EQ. NYH) GO TO 200
C NEQ was reduced. Zero part of YH to avoid undefined references. -----
I1 = LYH + L*NYH
I2 = LYH + (MAXORD + 1)*NYH - 1
IF (I1 .GT. I2) GO TO 200
DO 95 I = I1,I2
95 RWORK(I) = 0.0D0
GO TO 200
C-----------------------------------------------------------------------
C Block C.
C The next block is for the initial call only (ISTATE = 0 or 1).
C It contains all remaining initializations, the call to DAIGBT
C (if ISTATE = 1), and the calculation of the initial step size.
C The error weights in EWT are inverted after being loaded.
C-----------------------------------------------------------------------
100 UROUND = DUMACH()
TN = T
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 105
TCRIT = RWORK(1)
IF ((TCRIT - TOUT)*(TOUT - T) .LT. 0.0D0) GO TO 625
IF (H0 .NE. 0.0D0 .AND. (T + H0 - TCRIT)*H0 .GT. 0.0D0)
1 H0 = TCRIT - T
105 JSTART = 0
RWORK(LWM) = SQRT(UROUND)
NHNIL = 0
NST = 0
NFE = 0
NJE = 0
NSLAST = 0
HU = 0.0D0
NQU = 0
CCMAX = 0.3D0
MAXCOR = 3
MSBP = 20
MXNCF = 10
C Compute initial dy/dt, if necessary, and load it and initial Y into YH
LYD0 = LYH + NYH
LP = LWM + 1
IF ( ISTATE .EQ. 1 ) GO TO 120
C DLSOIBT must compute initial dy/dt (LYD0 points to YH(*,2)). ---------
CALL DAIGBT( RES, ADDA, NEQ, T, Y, RWORK(LYD0),
1 MB, NB, RWORK(LP), IWORK(21), IER )
NFE = NFE + 1
IF (IER .LT. 0) GO TO 560
IF (IER .GT. 0) GO TO 565
DO 115 I = 1,N
115 RWORK(I+LYH-1) = Y(I)
GO TO 130
C Initial dy/dt was supplied. Load into YH (LYD0 points to YH(*,2).). -
120 DO 125 I = 1,N
RWORK(I+LYH-1) = Y(I)
125 RWORK(I+LYD0-1) = YDOTI(I)
C Load and invert the EWT array. (H is temporarily set to 1.0.) -------
130 CONTINUE
NQ = 1
H = 1.0D0
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 135 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 621
135 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
C-----------------------------------------------------------------------
C The coding below computes the step size, H0, to be attempted on the
C first step, unless the user has supplied a value for this.
C First check that TOUT - T differs significantly from zero.
C A scalar tolerance quantity TOL is computed, as MAX(RTOL(i))
C if this is positive, or MAX(ATOL(i)/ABS(Y(i))) otherwise, adjusted
C so as to be between 100*UROUND and 1.0E-3.
C Then the computed value H0 is given by..
C NEQ
C H0**2 = TOL / ( w0**-2 + (1/NEQ) * Sum ( YDOT(i)/ywt(i) )**2 )
C 1
C where w0 = MAX ( ABS(T), ABS(TOUT) ),
C YDOT(i) = i-th component of initial value of dy/dt,
C ywt(i) = EWT(i)/TOL (a weight for y(i)).
C The sign of H0 is inferred from the initial values of TOUT and T.
C-----------------------------------------------------------------------
IF (H0 .NE. 0.0D0) GO TO 180
TDIST = ABS(TOUT - T)
W0 = MAX(ABS(T),ABS(TOUT))
IF (TDIST .LT. 2.0D0*UROUND*W0) GO TO 622
TOL = RTOL(1)
IF (ITOL .LE. 2) GO TO 145
DO 140 I = 1,N
140 TOL = MAX(TOL,RTOL(I))
145 IF (TOL .GT. 0.0D0) GO TO 160
ATOLI = ATOL(1)
DO 150 I = 1,N
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
AYI = ABS(Y(I))
IF (AYI .NE. 0.0D0) TOL = MAX(TOL,ATOLI/AYI)
150 CONTINUE
160 TOL = MAX(TOL,100.0D0*UROUND)
TOL = MIN(TOL,0.001D0)
SUM = DVNORM (N, RWORK(LYD0), RWORK(LEWT))
SUM = 1.0D0/(TOL*W0*W0) + TOL*SUM**2
H0 = 1.0D0/SQRT(SUM)
H0 = MIN(H0,TDIST)
H0 = SIGN(H0,TOUT-T)
C Adjust H0 if necessary to meet HMAX bound. ---------------------------
180 RH = ABS(H0)*HMXI
IF (RH .GT. 1.0D0) H0 = H0/RH
C Load H with H0 and scale YH(*,2) by H0. ------------------------------
H = H0
DO 190 I = 1,N
190 RWORK(I+LYD0-1) = H0*RWORK(I+LYD0-1)
GO TO 270
C-----------------------------------------------------------------------
C Block D.
C The next code block is for continuation calls only (ISTATE = 2 or 3)
C and is to check stop conditions before taking a step.
C-----------------------------------------------------------------------
200 NSLAST = NST
GO TO (210, 250, 220, 230, 240), ITASK
210 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
220 TP = TN - HU*(1.0D0 + 100.0D0*UROUND)
IF ((TP - TOUT)*H .GT. 0.0D0) GO TO 623
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
GO TO 400
230 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
IF ((TCRIT - TOUT)*H .LT. 0.0D0) GO TO 625
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 245
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
240 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
245 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
IF (ISTATE .EQ. 2) JSTART = -2
C-----------------------------------------------------------------------
C Block E.
C The next block is normally executed for all calls and contains
C the call to the one-step core integrator DSTODI.
C
C This is a looping point for the integration steps.
C
C First check for too many steps being taken, update EWT (if not at
C start of problem), check for too much accuracy being requested, and
C check for H below the roundoff level in T.
C-----------------------------------------------------------------------
250 CONTINUE
IF ((NST-NSLAST) .GE. MXSTEP) GO TO 500
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 260 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 510
260 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
270 TOLSF = UROUND*DVNORM (N, RWORK(LYH), RWORK(LEWT))
IF (TOLSF .LE. 1.0D0) GO TO 280
TOLSF = TOLSF*2.0D0
IF (NST .EQ. 0) GO TO 626
GO TO 520
280 IF ((TN + H) .NE. TN) GO TO 290
NHNIL = NHNIL + 1
IF (NHNIL .GT. MXHNIL) GO TO 290
MSG = 'DLSOIBT- Warning..Internal T (=R1) and H (=R2) are'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' such that in the machine, T + H = T on the next step '
CALL XERRWD (MSG, 60, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' (H = step size). Solver will continue anyway.'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 2, TN, H)
IF (NHNIL .LT. MXHNIL) GO TO 290
MSG = 'DLSOIBT- Above warning has been issued I1 times. '
CALL XERRWD (MSG, 50, 102, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' It will not be issued again for this problem.'
CALL XERRWD (MSG, 50, 102, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
290 CONTINUE
C-----------------------------------------------------------------------
C CALL DSTODI(NEQ,Y,YH,NYH,YH1,EWT,SAVF,SAVR,ACOR,WM,IWM,RES,
C ADDA,JAC,DPJIBT,DSLSBT)
C Note: SAVF in DSTODI occupies the same space as YDOTI in DLSOIBT.
C-----------------------------------------------------------------------
CALL DSTODI (NEQ, Y, RWORK(LYH), NYH, RWORK(LYH), RWORK(LEWT),
1 YDOTI, RWORK(LSAVF), RWORK(LACOR), RWORK(LWM),
2 IWORK(LIWM), RES, ADDA, JAC, DPJIBT, DSLSBT )
KGO = 1 - KFLAG
GO TO (300, 530, 540, 400, 550), KGO
C
C KGO = 1:success; 2:error test failure; 3:convergence failure;
C 4:RES ordered return; 5:RES returned error.
C-----------------------------------------------------------------------
C Block F.
C The following block handles the case of a successful return from the
C core integrator (KFLAG = 0). Test for stop conditions.
C-----------------------------------------------------------------------
300 INIT = 1
GO TO (310, 400, 330, 340, 350), ITASK
C ITASK = 1. If TOUT has been reached, interpolate. -------------------
310 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
C ITASK = 3. Jump to exit if TOUT was reached. ------------------------
330 IF ((TN - TOUT)*H .GE. 0.0D0) GO TO 400
GO TO 250
C ITASK = 4. See if TOUT or TCRIT was reached. Adjust H if necessary.
340 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 345
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
345 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
JSTART = -2
GO TO 250
C ITASK = 5. see if TCRIT was reached and jump to exit. ---------------
350 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
C-----------------------------------------------------------------------
C Block G.
C The following block handles all successful returns from DLSOIBT.
C If ITASK .ne. 1, Y is loaded from YH and T is set accordingly.
C ISTATE is set to 2, and the optional outputs are loaded into the
C work arrays before returning.
C-----------------------------------------------------------------------
400 DO 410 I = 1,N
410 Y(I) = RWORK(I+LYH-1)
T = TN
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 420
IF (IHIT) T = TCRIT
420 ISTATE = 2
IF ( KFLAG .EQ. -3 ) ISTATE = 3
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
RETURN
C-----------------------------------------------------------------------
C Block H.
C The following block handles all unsuccessful returns other than
C those for illegal input. First the error message routine is called.
C If there was an error test or convergence test failure, IMXER is set.
C Then Y is loaded from YH and T is set to TN.
C The optional outputs are loaded into the work arrays before returning.
C-----------------------------------------------------------------------
C The maximum number of steps was taken before reaching TOUT. ----------
500 MSG = 'DLSOIBT- At current T (=R1), MXSTEP (=I1) steps '
CALL XERRWD (MSG, 50, 201, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' taken on this call before reaching TOUT '
CALL XERRWD (MSG, 50, 201, 0, 1, MXSTEP, 0, 1, TN, 0.0D0)
ISTATE = -1
GO TO 580
C EWT(i) .le. 0.0 for some i (not at start of problem). ----------------
510 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSOIBT- At T (=R1), EWT(I1) has become R2 .le. 0.'
CALL XERRWD (MSG, 50, 202, 0, 1, I, 0, 2, TN, EWTI)
ISTATE = -6
GO TO 590
C Too much accuracy requested for machine precision. -------------------
520 MSG = 'DLSOIBT- At T (=R1), too much accuracy requested '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' for precision of machine.. See TOLSF (=R2) '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 2, TN, TOLSF)
RWORK(14) = TOLSF
ISTATE = -2
GO TO 590
C KFLAG = -1. Error test failed repeatedly or with ABS(H) = HMIN. -----
530 MSG = 'DLSOIBT- At T (=R1) and step size H (=R2), the '
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = 'error test failed repeatedly or with ABS(H) = HMIN'
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 2, TN, H)
ISTATE = -4
GO TO 570
C KFLAG = -2. Convergence failed repeatedly or with ABS(H) = HMIN. ----
540 MSG = 'DLSOIBT- At T (=R1) and step size H (=R2), the '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' corrector convergence failed repeatedly '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' or with ABS(H) = HMIN '
CALL XERRWD (MSG, 30, 205, 0, 0, 0, 0, 2, TN, H)
ISTATE = -5
GO TO 570
C IRES = 3 returned by RES, despite retries by DSTODI.------------------
550 MSG = 'DLSOIBT- At T (=R1) residual routine returned '
CALL XERRWD (MSG, 50, 206, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' error IRES = 3 repeatedly. '
CALL XERRWD (MSG, 40, 206, 0, 0, 0, 0, 1, TN, 0.0D0)
ISTATE = -7
GO TO 590
C DAIGBT failed because a diagonal block of A matrix was singular. -----
560 IER = -IER
MSG='DLSOIBT- Attempt to initialize dy/dt failed: Matrix A has a'
CALL XERRWD (MSG, 60, 207, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' singular diagonal block, block no. = (I1) '
CALL XERRWD (MSG, 50, 207, 0, 1, IER, 0, 0, 0.0D0, 0.0D0)
ISTATE = -8
RETURN
C DAIGBT failed because RES set IRES to 2 or 3. ------------------------
565 MSG = 'DLSOIBT- Attempt to initialize dy/dt failed '
CALL XERRWD (MSG, 50, 208, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' because residual routine set its error flag '
CALL XERRWD (MSG, 50, 208, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' to IRES = (I1)'
CALL XERRWD (MSG, 20, 208, 0, 1, IER, 0, 0, 0.0D0, 0.0D0)
ISTATE = -8
RETURN
C Compute IMXER if relevant. -------------------------------------------
570 BIG = 0.0D0
IMXER = 1
DO 575 I = 1,N
SIZE = ABS(RWORK(I+LACOR-1)*RWORK(I+LEWT-1))
IF (BIG .GE. SIZE) GO TO 575
BIG = SIZE
IMXER = I
575 CONTINUE
IWORK(16) = IMXER
C Compute residual if relevant. ----------------------------------------
580 LYD0 = LYH + NYH
DO 585 I = 1,N
RWORK(I+LSAVF-1) = RWORK(I+LYD0-1)/H
585 Y(I) = RWORK(I+LYH-1)
IRES = 1
CALL RES (NEQ, TN, Y, RWORK(LSAVF), YDOTI, IRES)
NFE = NFE + 1
IF (IRES .LE. 1) GO TO 595
MSG = 'DLSOIBT- Residual routine set its flag IRES '
CALL XERRWD (MSG, 50, 210, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' to (I1) when called for final output. '
CALL XERRWD (MSG, 50, 210, 0, 1, IRES, 0, 0, 0.0D0, 0.0D0)
GO TO 595
C Set Y vector, T, and optional outputs. -------------------------------
590 DO 592 I = 1,N
592 Y(I) = RWORK(I+LYH-1)
595 T = TN
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
RETURN
C-----------------------------------------------------------------------
C Block I.
C The following block handles all error returns due to illegal input
C (ISTATE = -3), as detected before calling the core integrator.
C First the error message routine is called. If the illegal input
C is a negative ISTATE, the run is aborted (apparent infinite loop).
C-----------------------------------------------------------------------
601 MSG = 'DLSOIBT- ISTATE (=I1) illegal.'
CALL XERRWD (MSG, 30, 1, 0, 1, ISTATE, 0, 0, 0.0D0, 0.0D0)
IF (ISTATE .LT. 0) GO TO 800
GO TO 700
602 MSG = 'DLSOIBT- ITASK (=I1) illegal. '
CALL XERRWD (MSG, 30, 2, 0, 1, ITASK, 0, 0, 0.0D0, 0.0D0)
GO TO 700
603 MSG = 'DLSOIBT- ISTATE.gt.1 but DLSOIBT not initialized. '
CALL XERRWD (MSG, 50, 3, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
GO TO 700
604 MSG = 'DLSOIBT- NEQ (=I1) .lt. 1 '
CALL XERRWD (MSG, 30, 4, 0, 1, NEQ(1), 0, 0, 0.0D0, 0.0D0)
GO TO 700
605 MSG = 'DLSOIBT- ISTATE = 3 and NEQ increased (I1 to I2). '
CALL XERRWD (MSG, 50, 5, 0, 2, N, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
606 MSG = 'DLSOIBT- ITOL (=I1) illegal. '
CALL XERRWD (MSG, 30, 6, 0, 1, ITOL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
607 MSG = 'DLSOIBT- IOPT (=I1) illegal. '
CALL XERRWD (MSG, 30, 7, 0, 1, IOPT, 0, 0, 0.0D0, 0.0D0)
GO TO 700
608 MSG = 'DLSOIBT- MF (=I1) illegal. '
CALL XERRWD (MSG, 30, 8, 0, 1, MF, 0, 0, 0.0D0, 0.0D0)
GO TO 700
609 MSG = 'DLSOIBT- MB (=I1) or NB (=I2) illegal. '
CALL XERRWD (MSG, 40, 9, 0, 2, MB, NB, 0, 0.0D0, 0.0D0)
GO TO 700
610 MSG = 'DLSOIBT- NB (=I1) .lt. 4 illegal. '
CALL XERRWD (MSG, 40, 10, 0, 1, NB, 0, 0, 0.0D0, 0.0D0)
GO TO 700
611 MSG = 'DLSOIBT- MAXORD (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 11, 0, 1, MAXORD, 0, 0, 0.0D0, 0.0D0)
GO TO 700
612 MSG = 'DLSOIBT- MXSTEP (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 12, 0, 1, MXSTEP, 0, 0, 0.0D0, 0.0D0)
GO TO 700
613 MSG = 'DLSOIBT- MXHNIL (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 13, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
614 MSG = 'DLSOIBT- TOUT (=R1) behind T (=R2) '
CALL XERRWD (MSG, 40, 14, 0, 0, 0, 0, 2, TOUT, T)
MSG = ' Integration direction is given by H0 (=R1) '
CALL XERRWD (MSG, 50, 14, 0, 0, 0, 0, 1, H0, 0.0D0)
GO TO 700
615 MSG = 'DLSOIBT- HMAX (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 15, 0, 0, 0, 0, 1, HMAX, 0.0D0)
GO TO 700
616 MSG = 'DLSOIBT- HMIN (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 16, 0, 0, 0, 0, 1, HMIN, 0.0D0)
GO TO 700
617 MSG='DLSOIBT- RWORK length needed, LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 17, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
618 MSG='DLSOIBT- IWORK length needed, LENIW (=I1), exceeds LIW (=I2)'
CALL XERRWD (MSG, 60, 18, 0, 2, LENIW, LIW, 0, 0.0D0, 0.0D0)
GO TO 700
619 MSG = 'DLSOIBT- RTOL(=I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 19, 0, 1, I, 0, 1, RTOLI, 0.0D0)
GO TO 700
620 MSG = 'DLSOIBT- ATOL(=I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 20, 0, 1, I, 0, 1, ATOLI, 0.0D0)
GO TO 700
621 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSOIBT- EWT(I1) is R1 .le. 0.0 '
CALL XERRWD (MSG, 40, 21, 0, 1, I, 0, 1, EWTI, 0.0D0)
GO TO 700
622 MSG='DLSOIBT- TOUT(=R1) too close to T(=R2) to start integration.'
CALL XERRWD (MSG, 60, 22, 0, 0, 0, 0, 2, TOUT, T)
GO TO 700
623 MSG='DLSOIBT- ITASK = I1 and TOUT (=R1) behind TCUR - HU (= R2) '
CALL XERRWD (MSG, 60, 23, 0, 1, ITASK, 0, 2, TOUT, TP)
GO TO 700
624 MSG='DLSOIBT- ITASK = 4 or 5 and TCRIT (=R1) behind TCUR (=R2) '
CALL XERRWD (MSG, 60, 24, 0, 0, 0, 0, 2, TCRIT, TN)
GO TO 700
625 MSG='DLSOIBT- ITASK = 4 or 5 and TCRIT (=R1) behind TOUT (=R2) '
CALL XERRWD (MSG, 60, 25, 0, 0, 0, 0, 2, TCRIT, TOUT)
GO TO 700
626 MSG = 'DLSOIBT- At start of problem, too much accuracy '
CALL XERRWD (MSG, 50, 26, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' requested for precision of machine.. See TOLSF (=R1) '
CALL XERRWD (MSG, 60, 26, 0, 0, 0, 0, 1, TOLSF, 0.0D0)
RWORK(14) = TOLSF
GO TO 700
627 MSG = 'DLSOIBT- Trouble in DINTDY. ITASK = I1, TOUT = R1'
CALL XERRWD (MSG, 50, 27, 0, 1, ITASK, 0, 1, TOUT, 0.0D0)
C
700 ISTATE = -3
RETURN
C
800 MSG = 'DLSOIBT- Run aborted.. apparent infinite loop. '
CALL XERRWD (MSG, 50, 303, 2, 0, 0, 0, 0, 0.0D0, 0.0D0)
RETURN
C----------------------- End of Subroutine DLSOIBT ---------------------
END
*DECK DLSODIS
SUBROUTINE DLSODIS (RES, ADDA, JAC, NEQ, Y, YDOTI, T, TOUT, ITOL,
1 RTOL, ATOL, ITASK, ISTATE, IOPT, RWORK, LRW, IWORK, LIW, MF )
EXTERNAL RES, ADDA, JAC
INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK, LIW, MF
DOUBLE PRECISION Y, YDOTI, T, TOUT, RTOL, ATOL, RWORK
DIMENSION NEQ(*), Y(*), YDOTI(*), RTOL(*), ATOL(*), RWORK(LRW),
1 IWORK(LIW)
C-----------------------------------------------------------------------
C This is the 18 November 2003 version of
C DLSODIS: Livermore Solver for Ordinary Differential equations
C (Implicit form) with general Sparse Jacobian matrices.
C
C This version is in double precision.
C
C DLSODIS solves the initial value problem for linearly implicit
C systems of first order ODEs,
C A(t,y) * dy/dt = g(t,y) , where A(t,y) is a square matrix,
C or, in component form,
C ( a * ( dy / dt )) + ... + ( a * ( dy / dt )) =
C i,1 1 i,NEQ NEQ
C
C = g ( t, y , y ,..., y ) ( i = 1,...,NEQ )
C i 1 2 NEQ
C
C If A is singular, this is a differential-algebraic system.
C
C DLSODIS is a variant version of the DLSODI package, and is intended
C for stiff problems in which the matrix A and the Jacobian matrix
C d(g - A*s)/dy have arbitrary sparse structures.
C
C Authors: Alan C. Hindmarsh
C Center for Applied Scientific Computing, L-561
C Lawrence Livermore National Laboratory
C Livermore, CA 94551
C and
C Sheila Balsdon
C Zycor, Inc.
C Austin, TX 78741
C-----------------------------------------------------------------------
C References:
C 1. M. K. Seager and S. Balsdon, LSODIS, A Sparse Implicit
C ODE Solver, in Proceedings of the IMACS 10th World Congress,
C Montreal, August 8-13, 1982.
C
C 2. Alan C. Hindmarsh, LSODE and LSODI, Two New Initial Value
C Ordinary Differential Equation Solvers,
C ACM-SIGNUM Newsletter, vol. 15, no. 4 (1980), pp. 10-11.
C
C 3. S. C. Eisenstat, M. C. Gursky, M. H. Schultz, and A. H. Sherman,
C Yale Sparse Matrix Package: I. The Symmetric Codes,
C Int. J. Num. Meth. Eng., vol. 18 (1982), pp. 1145-1151.
C
C 4. S. C. Eisenstat, M. C. Gursky, M. H. Schultz, and A. H. Sherman,
C Yale Sparse Matrix Package: II. The Nonsymmetric Codes,
C Research Report No. 114, Dept. of Computer Sciences, Yale
C University, 1977.
C-----------------------------------------------------------------------
C Summary of Usage.
C
C Communication between the user and the DLSODIS package, for normal
C situations, is summarized here. This summary describes only a subset
C of the full set of options available. See the full description for
C details, including optional communication, nonstandard options,
C and instructions for special situations. See also the example
C problem (with program and output) following this summary.
C
C A. First, provide a subroutine of the form:
C SUBROUTINE RES (NEQ, T, Y, S, R, IRES)
C DOUBLE PRECISION T, Y(*), S(*), R(*)
C which computes the residual function
C r = g(t,y) - A(t,y) * s ,
C as a function of t and the vectors y and s. (s is an internally
C generated approximation to dy/dt.) The arrays Y and S are inputs
C to the RES routine and should not be altered. The residual
C vector is to be stored in the array R. The argument IRES should be
C ignored for casual use of DLSODIS. (For uses of IRES, see the
C paragraph on RES in the full description below.)
C
C B. DLSODIS must deal internally with the matrices A and dr/dy, where
C r is the residual function defined above. DLSODIS generates a linear
C combination of these two matrices in sparse form.
C The matrix structure is communicated by a method flag, MF:
C MF = 21 or 22 when the user provides the structures of
C matrix A and dr/dy,
C MF = 121 or 222 when the user does not provide structure
C information, and
C MF = 321 or 422 when the user provides the structure
C of matrix A.
C
C C. You must also provide a subroutine of the form:
C SUBROUTINE ADDA (NEQ, T, Y, J, IAN, JAN, P)
C DOUBLE PRECISION T, Y(*), P(*)
C INTEGER IAN(*), JAN(*)
C which adds the matrix A = A(t,y) to the contents of the array P.
C NEQ, T, Y, and J are input arguments and should not be altered.
C This routine should add the J-th column of matrix A to the array
C P (of length NEQ). I.e. add A(i,J) to P(i) for all relevant
C values of i. The arguments IAN and JAN should be ignored for normal
C situations. DLSODIS will call the ADDA routine with J = 1,2,...,NEQ.
C
C D. For the sake of efficiency, you are encouraged to supply the
C Jacobian matrix dr/dy in closed form, where r = g(t,y) - A(t,y)*s
C (s = a fixed vector) as above. If dr/dy is being supplied,
C use MF = 21, 121, or 321, and provide a subroutine of the form:
C SUBROUTINE JAC (NEQ, T, Y, S, J, IAN, JAN, PDJ)
C DOUBLE PRECISION T, Y(*), S(*), PDJ(*)
C INTEGER IAN(*), JAN(*)
C which computes dr/dy as a function of t, y, and s. Here NEQ, T, Y, S,
C and J are input arguments, and the JAC routine is to load the array
C PDJ (of length NEQ) with the J-th column of dr/dy. I.e. load PDJ(i)
C with dr(i)/dy(J) for all relevant values of i. The arguments IAN and
C JAN should be ignored for normal situations. DLSODIS will call the
C JAC routine with J = 1,2,...,NEQ.
C Only nonzero elements need be loaded. A crude approximation
C to dr/dy, possibly with fewer nonzero elememts, will suffice.
C Note that if A is independent of y (or this dependence
C is weak enough to be ignored) then JAC is to compute dg/dy.
C If it is not feasible to provide a JAC routine, use
C MF = 22, 222, or 422 and DLSODIS will compute an approximate
C Jacobian internally by difference quotients.
C
C E. Next decide whether or not to provide the initial value of the
C derivative vector dy/dt. If the initial value of A(t,y) is
C nonsingular (and not too ill-conditioned), you may let DLSODIS compute
C this vector (ISTATE = 0). (DLSODIS will solve the system A*s = g for
C s, with initial values of A and g.) If A(t,y) is initially
C singular, then the system is a differential-algebraic system, and
C you must make use of the particular form of the system to compute the
C initial values of y and dy/dt. In that case, use ISTATE = 1 and
C load the initial value of dy/dt into the array YDOTI.
C The input array YDOTI and the initial Y array must be consistent with
C the equations A*dy/dt = g. This implies that the initial residual
C r = g(t,y) - A(t,y)*YDOTI must be approximately zero.
C
C F. Write a main program which calls Subroutine DLSODIS once for
C each point at which answers are desired. This should also provide
C for possible use of logical unit 6 for output of error messages by
C DLSODIS. On the first call to DLSODIS, supply arguments as follows:
C RES = name of user subroutine for residual function r.
C ADDA = name of user subroutine for computing and adding A(t,y).
C JAC = name of user subroutine for Jacobian matrix dr/dy
C (MF = 121). If not used, pass a dummy name.
C Note: The names for the RES and ADDA routines and (if used) the
C JAC routine must be declared External in the calling program.
C NEQ = number of scalar equations in the system.
C Y = array of initial values, of length NEQ.
C YDOTI = array of length NEQ (containing initial dy/dt if ISTATE = 1).
C T = the initial value of the independent variable.
C TOUT = first point where output is desired (.ne. T).
C ITOL = 1 or 2 according as ATOL (below) is a scalar or array.
C RTOL = relative tolerance parameter (scalar).
C ATOL = absolute tolerance parameter (scalar or array).
C The estimated local error in y(i) will be controlled so as
C to be roughly less (in magnitude) than
C EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
C EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
C Thus the local error test passes if, in each component,
C either the absolute error is less than ATOL (or ATOL(i)),
C or the relative error is less than RTOL.
C Use RTOL = 0.0 for pure absolute error control, and
C use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative error
C control. Caution: Actual (global) errors may exceed these
C local tolerances, so choose them conservatively.
C ITASK = 1 for normal computation of output values of y at t = TOUT.
C ISTATE = integer flag (input and output). Set ISTATE = 1 if the
C initial dy/dt is supplied, and 0 otherwise.
C IOPT = 0 to indicate no optional inputs used.
C RWORK = real work array of length at least:
C 20 + (2 + 1./LENRAT)*NNZ + (11 + 9./LENRAT)*NEQ
C where:
C NNZ = the number of nonzero elements in the sparse
C iteration matrix P = A - con*dr/dy (con = scalar)
C (If NNZ is unknown, use an estimate of it.)
C LENRAT = the real to integer wordlength ratio (usually 1 in
C single precision and 2 in double precision).
C In any case, the required size of RWORK cannot generally
C be predicted in advance for any value of MF, and the
C value above is a rough estimate of a crude lower bound.
C Some experimentation with this size may be necessary.
C (When known, the correct required length is an optional
C output, available in IWORK(17).)
C LRW = declared length of RWORK (in user's dimension).
C IWORK = integer work array of length at least 30.
C LIW = declared length of IWORK (in user's dimension).
C MF = method flag. Standard values are:
C 121 for a user-supplied sparse Jacobian.
C 222 for an internally generated sparse Jacobian.
C For other choices of MF, see the paragraph on MF in
C the full description below.
C Note that the main program must declare arrays Y, YDOTI, RWORK, IWORK,
C and possibly ATOL.
C
C G. The output from the first call, or any call, is:
C Y = array of computed values of y(t) vector.
C T = corresponding value of independent variable (normally TOUT).
C ISTATE = 2 if DLSODIS was successful, negative otherwise.
C -1 means excess work done on this call (check all inputs).
C -2 means excess accuracy requested (tolerances too small).
C -3 means illegal input detected (see printed message).
C -4 means repeated error test failures (check all inputs).
C -5 means repeated convergence failures (perhaps bad Jacobian
C supplied or wrong choice of tolerances).
C -6 means error weight became zero during problem. (Solution
C component i vanished, and ATOL or ATOL(i) = 0.)
C -7 cannot occur in casual use.
C -8 means DLSODIS was unable to compute the initial dy/dt.
C in casual use, this means A(t,y) is initially singular.
C Supply YDOTI and use ISTATE = 1 on the first call.
C -9 means a fatal error return flag came from sparse solver
C CDRV by way of DPRJIS or DSOLSS. Should never happen.
C
C A return with ISTATE = -1, -4, or -5, may result from using
C an inappropriate sparsity structure, one that is quite
C different from the initial structure. Consider calling
C DLSODIS again with ISTATE = 3 to force the structure to be
C reevaluated. See the full description of ISTATE below.
C
C If DLSODIS returns ISTATE = -1, -4 or -5, then the output of
C DLSODIS also includes YDOTI = array containing residual vector
C r = g - A * dy/dt evaluated at the current t, y, and dy/dt.
C
C H. To continue the integration after a successful return, simply
C reset TOUT and call DLSODIS again. No other parameters need be reset.
C
C-----------------------------------------------------------------------
C Example Problem.
C
C The following is an example problem, with the coding needed
C for its solution by DLSODIS. The problem comes from the partial
C differential equation (the Burgers equation)
C du/dt = - u * du/dx + eta * d**2 u/dx**2, eta = .05,
C on -1 .le. x .le. 1. The boundary conditions are periodic:
C u(-1,t) = u(1,t) and du/dx(-1,t) = du/dx(1,t)
C The initial profile is a square wave,
C u = 1 in ABS(x) .lt. .5, u = .5 at ABS(x) = .5, u = 0 elsewhere.
C The PDE is discretized in x by a simplified Galerkin method,
C using piecewise linear basis functions, on a grid of 40 intervals.
C The result is a system A * dy/dt = g(y), of size NEQ = 40,
C where y(i) is the approximation to u at x = x(i), with
C x(i) = -1 + (i-1)*delx, delx = 2/NEQ = .05.
C The individual equations in the system are (in order):
C (1/6)dy(NEQ)/dt+(4/6)dy(1)/dt+(1/6)dy(2)/dt
C = r4d*(y(NEQ)**2-y(2)**2)+eodsq*(y(2)-2*y(1)+y(NEQ))
C for i = 2,3,...,nm1,
C (1/6)dy(i-1)/dt+(4/6)dy(i)/dt+(1/6)dy(i+1)/dt
C = r4d*(y(i-1)**2-y(i+1)**2)+eodsq*(y(i+1)-2*y(i)+y(i-1))
C and finally
C (1/6)dy(nm1)/dt+(4/6)dy(NEQ)/dt+(1/6)dy(1)/dt
C = r4d*(y(nm1)**2-y(1)**2)+eodsq*(y(1)-2*y(NEQ)+y(nm1))
C where r4d = 1/(4*delx), eodsq = eta/delx**2 and nm1 = NEQ-1.
C The following coding solves the problem with MF = 121, with output
C of solution statistics at t = .1, .2, .3, and .4, and of the
C solution vector at t = .4. Optional outputs (run statistics) are
C also printed.
C
C EXTERNAL RESID, ADDASP, JACSP
C DOUBLE PRECISION ATOL, RTOL, RW, T, TOUT, Y, YDOTI, R4D, EODSQ, DELX
C DIMENSION Y(40), YDOTI(40), RW(1409), IW(30)
C COMMON /TEST1/ R4D, EODSQ, NM1
C DATA ITOL/1/, RTOL/1.0D-3/, ATOL/1.0D-3/, ITASK/1/, IOPT/0/
C DATA NEQ/40/, LRW/1409/, LIW/30/, MF/121/
C
C DELX = 2.0/NEQ
C R4D = 0.25/DELX
C EODSQ = 0.05/DELX**2
C NM1 = NEQ - 1
C DO 10 I = 1,NEQ
C 10 Y(I) = 0.0
C Y(11) = 0.5
C DO 15 I = 12,30
C 15 Y(I) = 1.0
C Y(31) = 0.5
C T = 0.0
C TOUT = 0.1
C ISTATE = 0
C DO 30 IO = 1,4
C CALL DLSODIS (RESID, ADDASP, JACSP, NEQ, Y, YDOTI, T, TOUT,
C 1 ITOL, RTOL, ATOL, ITASK, ISTATE, IOPT, RW, LRW, IW, LIW, MF)
C WRITE(6,20) T,IW(11),RW(11)
C 20 FORMAT(' At t =',F5.2,' No. steps =',I4,
C 1 ' Last step =',D12.4)
C IF (ISTATE .NE. 2) GO TO 90
C TOUT = TOUT + 0.1
C 30 CONTINUE
C WRITE (6,40) (Y(I),I=1,NEQ)
C 40 FORMAT(/' Final solution values..'/8(5D12.4/))
C WRITE(6,50) IW(17),IW(18),IW(11),IW(12),IW(13)
C NNZLU = IW(25) + IW(26) + NEQ
C WRITE(6,60) IW(19),NNZLU
C 50 FORMAT(/' Required RW size =',I5,' IW size =',I4/
C 1 ' No. steps =',I4,' No. r-s =',I4,' No. J-s =',i4)
C 60 FORMAT(' No. of nonzeros in P matrix =',I4,
C 1 ' No. of nonzeros in LU =',I4)
C STOP
C 90 WRITE (6,95) ISTATE
C 95 FORMAT(///' Error halt.. ISTATE =',I3)
C STOP
C END
C
C SUBROUTINE GFUN (N, T, Y, G)
C DOUBLE PRECISION T, Y, G, R4D, EODSQ
C DIMENSION G(N), Y(N)
C COMMON /TEST1/ R4D, EODSQ, NM1
C G(1) = R4D*(Y(N)**2-Y(2)**2) + EODSQ*(Y(2)-2.0*Y(1)+Y(N))
C DO 10 I = 2,NM1
C G(I) = R4D*(Y(I-1)**2 - Y(I+1)**2)
C 1 + EODSQ*(Y(I+1) - 2.0*Y(I) + Y(I-1))
C 10 CONTINUE
C G(N) = R4D*(Y(NM1)**2-Y(1)**2) + EODSQ*(Y(1)-2.0*Y(N)+Y(NM1))
C RETURN
C END
C
C SUBROUTINE RESID (N, T, Y, S, R, IRES)
C DOUBLE PRECISION T, Y, S, R, R4D, EODSQ
C DIMENSION Y(N), S(N), R(N)
C COMMON /TEST1/ R4D, EODSQ, NM1
C CALL GFUN (N, T, Y, R)
C R(1) = R(1) - (S(N) + 4.0*S(1) + S(2))/6.0
C DO 10 I = 2,NM1
C 10 R(I) = R(I) - (S(I-1) + 4.0*S(I) + S(I+1))/6.0
C R(N) = R(N) - (S(NM1) + 4.0*S(N) + S(1))/6.0
C RETURN
C END
C
C SUBROUTINE ADDASP (N, T, Y, J, IP, JP, P)
C DOUBLE PRECISION T, Y, P
C DIMENSION Y(N), IP(*), JP(*), P(N)
C JM1 = J - 1
C JP1 = J + 1
C IF (J .EQ. N) JP1 = 1
C IF (J .EQ. 1) JM1 = N
C P(J) = P(J) + (2.0/3.0)
C P(JP1) = P(JP1) + (1.0/6.0)
C P(JM1) = P(JM1) + (1.0/6.0)
C RETURN
C END
C
C SUBROUTINE JACSP (N, T, Y, S, J, IP, JP, PDJ)
C DOUBLE PRECISION T, Y, S, PDJ, R4D, EODSQ
C DIMENSION Y(N), S(N), IP(*), JP(*), PDJ(N)
C COMMON /TEST1/ R4D, EODSQ, NM1
C JM1 = J - 1
C JP1 = J + 1
C IF (J .EQ. 1) JM1 = N
C IF (J .EQ. N) JP1 = 1
C PDJ(JM1) = -2.0*R4D*Y(J) + EODSQ
C PDJ(J) = -2.0*EODSQ
C PDJ(JP1) = 2.0*R4D*Y(J) + EODSQ
C RETURN
C END
C
C The output of this program (on a CDC-7600 in single precision)
C is as follows:
C
C At t = 0.10 No. steps = 15 Last step = 1.6863e-02
C At t = 0.20 No. steps = 19 Last step = 2.4101e-02
C At t = 0.30 No. steps = 22 Last step = 4.3143e-02
C At t = 0.40 No. steps = 24 Last step = 5.7819e-02
C
C Final solution values..
C 1.8371e-02 1.3578e-02 1.5864e-02 2.3805e-02 3.7245e-02
C 5.6630e-02 8.2538e-02 1.1538e-01 1.5522e-01 2.0172e-01
C 2.5414e-01 3.1150e-01 3.7259e-01 4.3608e-01 5.0060e-01
C 5.6482e-01 6.2751e-01 6.8758e-01 7.4415e-01 7.9646e-01
C 8.4363e-01 8.8462e-01 9.1853e-01 9.4500e-01 9.6433e-01
C 9.7730e-01 9.8464e-01 9.8645e-01 9.8138e-01 9.6584e-01
C 9.3336e-01 8.7497e-01 7.8213e-01 6.5315e-01 4.9997e-01
C 3.4672e-01 2.1758e-01 1.2461e-01 6.6208e-02 3.3784e-02
C
C Required RW size = 1409 IW size = 30
C No. steps = 24 No. r-s = 33 No. J-s = 8
C No. of nonzeros in P matrix = 120 No. of nonzeros in LU = 194
C
C-----------------------------------------------------------------------
C Full Description of User Interface to DLSODIS.
C
C The user interface to DLSODIS consists of the following parts.
C
C 1. The call sequence to Subroutine DLSODIS, which is a driver
C routine for the solver. This includes descriptions of both
C the call sequence arguments and of user-supplied routines.
C Following these descriptions is a description of
C optional inputs available through the call sequence, and then
C a description of optional outputs (in the work arrays).
C
C 2. Descriptions of other routines in the DLSODIS package that may be
C (optionally) called by the user. These provide the ability to
C alter error message handling, save and restore the internal
C Common, and obtain specified derivatives of the solution y(t).
C
C 3. Descriptions of Common blocks to be declared in overlay
C or similar environments, or to be saved when doing an interrupt
C of the problem and continued solution later.
C
C 4. Description of two routines in the DLSODIS package, either of
C which the user may replace with his/her own version, if desired.
C These relate to the measurement of errors.
C
C-----------------------------------------------------------------------
C Part 1. Call Sequence.
C
C The call sequence parameters used for input only are
C RES, ADDA, JAC, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK,
C IOPT, LRW, LIW, MF,
C and those used for both input and output are
C Y, T, ISTATE, YDOTI.
C The work arrays RWORK and IWORK are also used for conditional and
C optional inputs and optional outputs. (The term output here refers
C to the return from Subroutine DLSODIS to the user's calling program.)
C
C The legality of input parameters will be thoroughly checked on the
C initial call for the problem, but not checked thereafter unless a
C change in input parameters is flagged by ISTATE = 3 on input.
C
C The descriptions of the call arguments are as follows.
C
C RES = the name of the user-supplied subroutine which supplies
C the residual vector for the ODE system, defined by
C r = g(t,y) - A(t,y) * s
C as a function of the scalar t and the vectors
C s and y (s approximates dy/dt). This subroutine
C is to have the form
C SUBROUTINE RES (NEQ, T, Y, S, R, IRES)
C DOUBLE PRECISION T, Y(*), S(*), R(*)
C where NEQ, T, Y, S, and IRES are input, and R and
C IRES are output. Y, S, and R are arrays of length NEQ.
C On input, IRES indicates how DLSODIS will use the
C returned array R, as follows:
C IRES = 1 means that DLSODIS needs the full residual,
C r = g - A*s, exactly.
C IRES = -1 means that DLSODIS is using R only to compute
C the Jacobian dr/dy by difference quotients.
C The RES routine can ignore IRES, or it can omit some terms
C if IRES = -1. If A does not depend on y, then RES can
C just return R = g when IRES = -1. If g - A*s contains other
C additive terms that are independent of y, these can also be
C dropped, if done consistently, when IRES = -1.
C The subroutine should set the flag IRES if it
C encounters a halt condition or illegal input.
C Otherwise, it should not reset IRES. On output,
C IRES = 1 or -1 represents a normal return, and
C DLSODIS continues integrating the ODE. Leave IRES
C unchanged from its input value.
C IRES = 2 tells DLSODIS to immediately return control
C to the calling program, with ISTATE = 3. This lets
C the calling program change parameters of the problem
C if necessary.
C IRES = 3 represents an error condition (for example, an
C illegal value of y). DLSODIS tries to integrate the system
C without getting IRES = 3 from RES. If it cannot, DLSODIS
C returns with ISTATE = -7 or -1.
C On a return with ISTATE = 3, -1, or -7, the values
C of T and Y returned correspond to the last point reached
C successfully without getting the flag IRES = 2 or 3.
C The flag values IRES = 2 and 3 should not be used to
C handle switches or root-stop conditions. This is better
C done by calling DLSODIS in a one-step mode and checking the
C stopping function for a sign change at each step.
C If quantities computed in the RES routine are needed
C externally to DLSODIS, an extra call to RES should be made
C for this purpose, for consistent and accurate results.
C To get the current dy/dt for the S argument, use DINTDY.
C RES must be declared External in the calling
C program. See note below for more about RES.
C
C ADDA = the name of the user-supplied subroutine which adds the
C matrix A = A(t,y) to another matrix stored in sparse form.
C This subroutine is to have the form
C SUBROUTINE ADDA (NEQ, T, Y, J, IAN, JAN, P)
C DOUBLE PRECISION T, Y(*), P(*)
C INTEGER IAN(*), JAN(*)
C where NEQ, T, Y, J, IAN, JAN, and P are input. This routine
C should add the J-th column of matrix A to the array P, of
C length NEQ. Thus a(i,J) is to be added to P(i) for all
C relevant values of i. Here T and Y have the same meaning as
C in Subroutine RES, and J is a column index (1 to NEQ).
C IAN and JAN are undefined in calls to ADDA for structure
C determination (MOSS .ne. 0). Otherwise, IAN and JAN are
C structure descriptors, as defined under optional outputs
C below, and so can be used to determine the relevant row
C indices i, if desired.
C Calls to ADDA are made with J = 1,...,NEQ, in that
C order. ADDA must not alter its input arguments.
C ADDA must be declared External in the calling program.
C See note below for more information about ADDA.
C
C JAC = the name of the user-supplied subroutine which supplies
C the Jacobian matrix, dr/dy, where r = g - A*s. JAC is
C required if MITER = 1, or MOSS = 1 or 3. Otherwise a dummy
C name can be passed. This subroutine is to have the form
C SUBROUTINE JAC (NEQ, T, Y, S, J, IAN, JAN, PDJ)
C DOUBLE PRECISION T, Y(*), S(*), PDJ(*)
C INTEGER IAN(*), JAN(*)
C where NEQ, T, Y, S, J, IAN, and JAN are input. The
C array PDJ, of length NEQ, is to be loaded with column J
C of the Jacobian on output. Thus dr(i)/dy(J) is to be
C loaded into PDJ(i) for all relevant values of i.
C Here T, Y, and S have the same meaning as in Subroutine RES,
C and J is a column index (1 to NEQ). IAN and JAN
C are undefined in calls to JAC for structure determination
C (MOSS .ne. 0). Otherwise, IAN and JAN are structure
C descriptors, as defined under optional outputs below, and
C so can be used to determine the relevant row indices i, if
C desired.
C JAC need not provide dr/dy exactly. A crude
C approximation (possibly with greater sparsity) will do.
C In any case, PDJ is preset to zero by the solver,
C so that only the nonzero elements need be loaded by JAC.
C Calls to JAC are made with J = 1,...,NEQ, in that order, and
C each such set of calls is preceded by a call to RES with the
C same arguments NEQ, T, Y, S, and IRES. Thus to gain some
C efficiency intermediate quantities shared by both calculations
C may be saved in a user Common block by RES and not recomputed
C by JAC, if desired. JAC must not alter its input arguments.
C JAC must be declared External in the calling program.
C See note below for more about JAC.
C
C Note on RES, ADDA, and JAC:
C These subroutines may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in the subroutines) and/or Y has length
C exceeding NEQ(1). However, these subroutines should not
C alter NEQ(1), Y(1),...,Y(NEQ) or any other input variables.
C See the descriptions of NEQ and Y below.
C
C NEQ = the size of the system (number of first order ordinary
C differential equations or scalar algebraic equations).
C Used only for input.
C NEQ may be decreased, but not increased, during the problem.
C If NEQ is decreased (with ISTATE = 3 on input), the
C remaining components of Y should be left undisturbed, if
C these are to be accessed in RES, ADDA, or JAC.
C
C Normally, NEQ is a scalar, and it is generally referred to
C as a scalar in this user interface description. However,
C NEQ may be an array, with NEQ(1) set to the system size.
C (The DLSODIS package accesses only NEQ(1).) In either case,
C this parameter is passed as the NEQ argument in all calls
C to RES, ADDA, and JAC. Hence, if it is an array,
C locations NEQ(2),... may be used to store other integer data
C and pass it to RES, ADDA, or JAC. Each such subroutine
C must include NEQ in a Dimension statement in that case.
C
C Y = a real array for the vector of dependent variables, of
C length NEQ or more. Used for both input and output on the
C first call (ISTATE = 0 or 1), and only for output on other
C calls. On the first call, Y must contain the vector of
C initial values. On output, Y contains the computed solution
C vector, evaluated at T. If desired, the Y array may be used
C for other purposes between calls to the solver.
C
C This array is passed as the Y argument in all calls to RES,
C ADDA, and JAC. Hence its length may exceed NEQ,
C and locations Y(NEQ+1),... may be used to store other real
C data and pass it to RES, ADDA, or JAC. (The DLSODIS
C package accesses only Y(1),...,Y(NEQ). )
C
C YDOTI = a real array for the initial value of the vector
C dy/dt and for work space, of dimension at least NEQ.
C
C On input:
C If ISTATE = 0 then DLSODIS will compute the initial value
C of dy/dt, if A is nonsingular. Thus YDOTI will
C serve only as work space and may have any value.
C If ISTATE = 1 then YDOTI must contain the initial value
C of dy/dt.
C If ISTATE = 2 or 3 (continuation calls) then YDOTI
C may have any value.
C Note: If the initial value of A is singular, then
C DLSODIS cannot compute the initial value of dy/dt, so
C it must be provided in YDOTI, with ISTATE = 1.
C
C On output, when DLSODIS terminates abnormally with ISTATE =
C -1, -4, or -5, YDOTI will contain the residual
C r = g(t,y) - A(t,y)*(dy/dt). If r is large, t is near
C its initial value, and YDOTI is supplied with ISTATE = 1,
C there may have been an incorrect input value of
C YDOTI = dy/dt, or the problem (as given to DLSODIS)
C may not have a solution.
C
C If desired, the YDOTI array may be used for other
C purposes between calls to the solver.
C
C T = the independent variable. On input, T is used only on the
C first call, as the initial point of the integration.
C On output, after each call, T is the value at which a
C computed solution y is evaluated (usually the same as TOUT).
C On an error return, T is the farthest point reached.
C
C TOUT = the next value of t at which a computed solution is desired.
C Used only for input.
C
C When starting the problem (ISTATE = 0 or 1), TOUT may be
C equal to T for one call, then should .ne. T for the next
C call. For the initial T, an input value of TOUT .ne. T is
C used in order to determine the direction of the integration
C (i.e. the algebraic sign of the step sizes) and the rough
C scale of the problem. Integration in either direction
C (forward or backward in t) is permitted.
C
C If ITASK = 2 or 5 (one-step modes), TOUT is ignored after
C the first call (i.e. the first call with TOUT .ne. T).
C Otherwise, TOUT is required on every call.
C
C If ITASK = 1, 3, or 4, the values of TOUT need not be
C monotone, but a value of TOUT which backs up is limited
C to the current internal T interval, whose endpoints are
C TCUR - HU and TCUR (see optional outputs, below, for
C TCUR and HU).
C
C ITOL = an indicator for the type of error control. See
C description below under ATOL. Used only for input.
C
C RTOL = a relative error tolerance parameter, either a scalar or
C an array of length NEQ. See description below under ATOL.
C Input only.
C
C ATOL = an absolute error tolerance parameter, either a scalar or
C an array of length NEQ. Input only.
C
C The input parameters ITOL, RTOL, and ATOL determine
C the error control performed by the solver. The solver will
C control the vector E = (E(i)) of estimated local errors
C in y, according to an inequality of the form
C RMS-norm of ( E(i)/EWT(i) ) .le. 1,
C where EWT(i) = RTOL(i)*ABS(Y(i)) + ATOL(i),
C and the RMS-norm (root-mean-square norm) here is
C RMS-norm(v) = SQRT(sum v(i)**2 / NEQ). Here EWT = (EWT(i))
C is a vector of weights which must always be positive, and
C the values of RTOL and ATOL should all be non-negative.
C The following table gives the types (scalar/array) of
C RTOL and ATOL, and the corresponding form of EWT(i).
C
C ITOL RTOL ATOL EWT(i)
C 1 scalar scalar RTOL*ABS(Y(i)) + ATOL
C 2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
C 3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
C 4 array scalar RTOL(i)*ABS(Y(i)) + ATOL(i)
C
C When either of these parameters is a scalar, it need not
C be dimensioned in the user's calling program.
C
C If none of the above choices (with ITOL, RTOL, and ATOL
C fixed throughout the problem) is suitable, more general
C error controls can be obtained by substituting
C user-supplied routines for the setting of EWT and/or for
C the norm calculation. See Part 4 below.
C
C If global errors are to be estimated by making a repeated
C run on the same problem with smaller tolerances, then all
C components of RTOL and ATOL (i.e. of EWT) should be scaled
C down uniformly.
C
C ITASK = an index specifying the task to be performed.
C Input only. ITASK has the following values and meanings.
C 1 means normal computation of output values of y(t) at
C t = TOUT (by overshooting and interpolating).
C 2 means take one step only and return.
C 3 means stop at the first internal mesh point at or
C beyond t = TOUT and return.
C 4 means normal computation of output values of y(t) at
C t = TOUT but without overshooting t = TCRIT.
C TCRIT must be input as RWORK(1). TCRIT may be equal to
C or beyond TOUT, but not behind it in the direction of
C integration. This option is useful if the problem
C has a singularity at or beyond t = TCRIT.
C 5 means take one step, without passing TCRIT, and return.
C TCRIT must be input as RWORK(1).
C
C Note: If ITASK = 4 or 5 and the solver reaches TCRIT
C (within roundoff), it will return T = TCRIT (exactly) to
C indicate this (unless ITASK = 4 and TOUT comes before TCRIT,
C in which case answers at t = TOUT are returned first).
C
C ISTATE = an index used for input and output to specify the
C state of the calculation.
C
C On input, the values of ISTATE are as follows.
C 0 means this is the first call for the problem, and
C DLSODIS is to compute the initial value of dy/dt
C (while doing other initializations). See note below.
C 1 means this is the first call for the problem, and
C the initial value of dy/dt has been supplied in
C YDOTI (DLSODIS will do other initializations).
C See note below.
C 2 means this is not the first call, and the calculation
C is to continue normally, with no change in any input
C parameters except possibly TOUT and ITASK.
C (If ITOL, RTOL, and/or ATOL are changed between calls
C with ISTATE = 2, the new values will be used but not
C tested for legality.)
C 3 means this is not the first call, and the
C calculation is to continue normally, but with
C a change in input parameters other than
C TOUT and ITASK. Changes are allowed in
C NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, MF,
C the conditional inputs IA, JA, IC, and JC,
C and any of the optional inputs except H0.
C A call with ISTATE = 3 will cause the sparsity
C structure of the problem to be recomputed.
C (Structure information is reread from IA and JA if
C MOSS = 0, 3, or 4 and from IC and JC if MOSS = 0).
C Note: A preliminary call with TOUT = T is not counted
C as a first call here, as no initialization or checking of
C input is done. (Such a call is sometimes useful for the
C purpose of outputting the initial conditions.)
C Thus the first call for which TOUT .ne. T requires
C ISTATE = 0 or 1 on input.
C
C On output, ISTATE has the following values and meanings.
C 0 or 1 means nothing was done; TOUT = T and
C ISTATE = 0 or 1 on input.
C 2 means that the integration was performed successfully.
C 3 means that the user-supplied Subroutine RES signalled
C DLSODIS to halt the integration and return (IRES = 2).
C Integration as far as T was achieved with no occurrence
C of IRES = 2, but this flag was set on attempting the
C next step.
C -1 means an excessive amount of work (more than MXSTEP
C steps) was done on this call, before completing the
C requested task, but the integration was otherwise
C successful as far as T. (MXSTEP is an optional input
C and is normally 500.) To continue, the user may
C simply reset ISTATE to a value .gt. 1 and call again
C (the excess work step counter will be reset to 0).
C In addition, the user may increase MXSTEP to avoid
C this error return (see below on optional inputs).
C -2 means too much accuracy was requested for the precision
C of the machine being used. This was detected before
C completing the requested task, but the integration
C was successful as far as T. To continue, the tolerance
C parameters must be reset, and ISTATE must be set
C to 3. The optional output TOLSF may be used for this
C purpose. (Note: If this condition is detected before
C taking any steps, then an illegal input return
C (ISTATE = -3) occurs instead.)
C -3 means illegal input was detected, before taking any
C integration steps. See written message for details.
C Note: If the solver detects an infinite loop of calls
C to the solver with illegal input, it will cause
C the run to stop.
C -4 means there were repeated error test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C The problem may have a singularity, or the input
C may be inappropriate.
C -5 means there were repeated convergence test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C This may be caused by an inaccurate Jacobian matrix.
C -6 means EWT(i) became zero for some i during the
C integration. Pure relative error control (ATOL(i) = 0.0)
C was requested on a variable which has now vanished.
C the integration was successful as far as T.
C -7 means that the user-supplied Subroutine RES set
C its error flag (IRES = 3) despite repeated tries by
C DLSODIS to avoid that condition.
C -8 means that ISTATE was 0 on input but DLSODIS was unable
C to compute the initial value of dy/dt. See the
C printed message for details.
C -9 means a fatal error return flag came from the sparse
C solver CDRV by way of DPRJIS or DSOLSS (numerical
C factorization or backsolve). This should never happen.
C The integration was successful as far as T.
C
C Note: An error return with ISTATE = -1, -4, or -5
C may mean that the sparsity structure of the
C problem has changed significantly since it was last
C determined (or input). In that case, one can attempt to
C complete the integration by setting ISTATE = 3 on the next
C call, so that a new structure determination is done.
C
C Note: Since the normal output value of ISTATE is 2,
C it does not need to be reset for normal continuation.
C similarly, ISTATE (= 3) need not be reset if RES told
C DLSODIS to return because the calling program must change
C the parameters of the problem.
C Also, since a negative input value of ISTATE will be
C regarded as illegal, a negative output value requires the
C user to change it, and possibly other inputs, before
C calling the solver again.
C
C IOPT = an integer flag to specify whether or not any optional
C inputs are being used on this call. Input only.
C The optional inputs are listed separately below.
C IOPT = 0 means no optional inputs are being used.
C Default values will be used in all cases.
C IOPT = 1 means one or more optional inputs are being used.
C
C RWORK = a work array used for a mixture of real (double precision)
C and integer work space.
C The length of RWORK (in real words) must be at least
C 20 + NYH*(MAXORD + 1) + 3*NEQ + LWM where
C NYH = the initial value of NEQ,
C MAXORD = 12 (if METH = 1) or 5 (if METH = 2) (unless a
C smaller value is given as an optional input),
C LWM = 2*NNZ + 2*NEQ + (NNZ+9*NEQ)/LENRAT if MITER = 1,
C LWM = 2*NNZ + 2*NEQ + (NNZ+10*NEQ)/LENRAT if MITER = 2.
C in the above formulas,
C NNZ = number of nonzero elements in the iteration matrix
C P = A - con*J (con is a constant and J is the
C Jacobian matrix dr/dy).
C LENRAT = the real to integer wordlength ratio (usually 1 in
C single precision and 2 in double precision).
C (See the MF description for METH and MITER.)
C Thus if MAXORD has its default value and NEQ is constant,
C the minimum length of RWORK is:
C 20 + 16*NEQ + LWM for MF = 11, 111, 311, 12, 212, 412,
C 20 + 9*NEQ + LWM for MF = 21, 121, 321, 22, 222, 422.
C The above formula for LWM is only a crude lower bound.
C The required length of RWORK cannot be readily predicted
C in general, as it depends on the sparsity structure
C of the problem. Some experimentation may be necessary.
C
C The first 20 words of RWORK are reserved for conditional
C and optional inputs and optional outputs.
C
C The following word in RWORK is a conditional input:
C RWORK(1) = TCRIT = critical value of t which the solver
C is not to overshoot. Required if ITASK is
C 4 or 5, and ignored otherwise. (See ITASK.)
C
C LRW = the length of the array RWORK, as declared by the user.
C (This will be checked by the solver.)
C
C IWORK = an integer work array. The length of IWORK must be at least
C 32 + 2*NEQ + NZA + NZC for MOSS = 0,
C 30 for MOSS = 1 or 2,
C 31 + NEQ + NZA for MOSS = 3 or 4.
C (NZA is the number of nonzero elements in matrix A, and
C NZC is the number of nonzero elements in dr/dy.)
C
C In DLSODIS, IWORK is used for conditional and
C optional inputs and optional outputs.
C
C The following two blocks of words in IWORK are conditional
C inputs, required if MOSS = 0, 3, or 4, but not otherwise
C (see the description of MF for MOSS).
C IWORK(30+j) = IA(j) (j=1,...,NEQ+1)
C IWORK(31+NEQ+k) = JA(k) (k=1,...,NZA)
C The two arrays IA and JA describe the sparsity structure
C to be assumed for the matrix A. JA contains the row
C indices where nonzero elements occur, reading in columnwise
C order, and IA contains the starting locations in JA of the
C descriptions of columns 1,...,NEQ, in that order, with
C IA(1) = 1. Thus, for each column index j = 1,...,NEQ, the
C values of the row index i in column j where a nonzero
C element may occur are given by
C i = JA(k), where IA(j) .le. k .lt. IA(j+1).
C If NZA is the total number of nonzero locations assumed,
C then the length of the JA array is NZA, and IA(NEQ+1) must
C be NZA + 1. Duplicate entries are not allowed.
C The following additional blocks of words are required
C if MOSS = 0, but not otherwise. If LC = 31 + NEQ + NZA, then
C IWORK(LC+j) = IC(j) (j=1,...,NEQ+1), and
C IWORK(LC+NEQ+1+k) = JC(k) (k=1,...,NZC)
C The two arrays IC and JC describe the sparsity
C structure to be assumed for the Jacobian matrix dr/dy.
C They are used in the same manner as the above IA and JA
C arrays. If NZC is the number of nonzero locations
C assumed, then the length of the JC array is NZC, and
C IC(NEQ+1) must be NZC + 1. Duplicate entries are not
C allowed.
C
C LIW = the length of the array IWORK, as declared by the user.
C (This will be checked by the solver.)
C
C Note: The work arrays must not be altered between calls to DLSODIS
C for the same problem, except possibly for the conditional and
C optional inputs, and except for the last 3*NEQ words of RWORK.
C The latter space is used for internal scratch space, and so is
C available for use by the user outside DLSODIS between calls, if
C desired (but not for use by RES, ADDA, or JAC).
C
C MF = the method flag. Used only for input.
C MF has three decimal digits-- MOSS, METH, and MITER.
C For standard options:
C MF = 100*MOSS + 10*METH + MITER.
C MOSS indicates the method to be used to obtain the sparsity
C structure of the Jacobian matrix:
C MOSS = 0 means the user has supplied IA, JA, IC, and JC
C (see descriptions under IWORK above).
C MOSS = 1 means the user has supplied JAC (see below) and
C the structure will be obtained from NEQ initial
C calls to JAC and NEQ initial calls to ADDA.
C MOSS = 2 means the structure will be obtained from NEQ+1
C initial calls to RES and NEQ initial calls to ADDA
C MOSS = 3 like MOSS = 1, except user has supplied IA and JA.
C MOSS = 4 like MOSS = 2, except user has supplied IA and JA.
C METH indicates the basic linear multistep method:
C METH = 1 means the implicit Adams method.
C METH = 2 means the method based on Backward
C Differentiation Formulas (BDFs).
C The BDF method is strongly preferred for stiff problems,
C while the Adams method is preferred when the problem is
C not stiff. If the matrix A(t,y) is nonsingular,
C stiffness here can be taken to mean that of the explicit
C ODE system dy/dt = A-inverse * g. If A is singular,
C the concept of stiffness is not well defined.
C If you do not know whether the problem is stiff, we
C recommend using METH = 2. If it is stiff, the advantage
C of METH = 2 over METH = 1 will be great, while if it is
C not stiff, the advantage of METH = 1 will be slight.
C If maximum efficiency is important, some experimentation
C with METH may be necessary.
C MITER indicates the corrector iteration method:
C MITER = 1 means chord iteration with a user-supplied
C sparse Jacobian, given by Subroutine JAC.
C MITER = 2 means chord iteration with an internally
C generated (difference quotient) sparse
C Jacobian (using NGP extra calls to RES per
C dr/dy value, where NGP is an optional
C output described below.)
C If MITER = 1 or MOSS = 1 or 3 the user must supply a
C Subroutine JAC (the name is arbitrary) as described above
C under JAC. Otherwise, a dummy argument can be used.
C
C The standard choices for MF are:
C MF = 21 or 22 for a stiff problem with IA/JA and IC/JC
C supplied,
C MF = 121 for a stiff problem with JAC supplied, but not
C IA/JA or IC/JC,
C MF = 222 for a stiff problem with neither IA/JA, IC/JC/,
C nor JAC supplied,
C MF = 321 for a stiff problem with IA/JA and JAC supplied,
C but not IC/JC,
C MF = 422 for a stiff problem with IA/JA supplied, but not
C IC/JC or JAC.
C
C The sparseness structure can be changed during the problem
C by making a call to DLSODIS with ISTATE = 3.
C-----------------------------------------------------------------------
C Optional Inputs.
C
C The following is a list of the optional inputs provided for in the
C call sequence. (See also Part 2.) For each such input variable,
C this table lists its name as used in this documentation, its
C location in the call sequence, its meaning, and the default value.
C The use of any of these inputs requires IOPT = 1, and in that
C case all of these inputs are examined. A value of zero for any
C of these optional inputs will cause the default value to be used.
C Thus to use a subset of the optional inputs, simply preload
C locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively, and
C then set those of interest to nonzero values.
C
C Name Location Meaning and Default Value
C
C H0 RWORK(5) the step size to be attempted on the first step.
C The default value is determined by the solver.
C
C HMAX RWORK(6) the maximum absolute step size allowed.
C The default value is infinite.
C
C HMIN RWORK(7) the minimum absolute step size allowed.
C The default value is 0. (This lower bound is not
C enforced on the final step before reaching TCRIT
C when ITASK = 4 or 5.)
C
C MAXORD IWORK(5) the maximum order to be allowed. The default
C value is 12 if METH = 1, and 5 if METH = 2.
C If MAXORD exceeds the default value, it will
C be reduced to the default value.
C If MAXORD is changed during the problem, it may
C cause the current order to be reduced.
C
C MXSTEP IWORK(6) maximum number of (internally defined) steps
C allowed during one call to the solver.
C The default value is 500.
C
C MXHNIL IWORK(7) maximum number of messages printed (per problem)
C warning that T + H = T on a step (H = step size).
C This must be positive to result in a non-default
C value. The default value is 10.
C-----------------------------------------------------------------------
C Optional Outputs.
C
C As optional additional output from DLSODIS, the variables listed
C below are quantities related to the performance of DLSODIS
C which are available to the user. These are communicated by way of
C the work arrays, but also have internal mnemonic names as shown.
C Except where stated otherwise, all of these outputs are defined
C on any successful return from DLSODIS, and on any return with
C ISTATE = -1, -2, -4, -5, -6, or -7. On a return with -3 (illegal
C input) or -8, they will be unchanged from their existing values
C (if any), except possibly for TOLSF, LENRW, and LENIW.
C On any error return, outputs relevant to the error will be defined,
C as noted below.
C
C Name Location Meaning
C
C HU RWORK(11) the step size in t last used (successfully).
C
C HCUR RWORK(12) the step size to be attempted on the next step.
C
C TCUR RWORK(13) the current value of the independent variable
C which the solver has actually reached, i.e. the
C current internal mesh point in t. On output, TCUR
C will always be at least as far as the argument
C T, but may be farther (if interpolation was done).
C
C TOLSF RWORK(14) a tolerance scale factor, greater than 1.0,
C computed when a request for too much accuracy was
C detected (ISTATE = -3 if detected at the start of
C the problem, ISTATE = -2 otherwise). If ITOL is
C left unaltered but RTOL and ATOL are uniformly
C scaled up by a factor of TOLSF for the next call,
C then the solver is deemed likely to succeed.
C (The user may also ignore TOLSF and alter the
C tolerance parameters in any other way appropriate.)
C
C NST IWORK(11) the number of steps taken for the problem so far.
C
C NRE IWORK(12) the number of residual evaluations (RES calls)
C for the problem so far, excluding those for
C structure determination (MOSS = 2 or 4).
C
C NJE IWORK(13) the number of Jacobian evaluations (each involving
C an evaluation of A and dr/dy) for the problem so
C far, excluding those for structure determination
C (MOSS = 1 or 3). This equals the number of calls
C to ADDA and (if MITER = 1) JAC.
C
C NQU IWORK(14) the method order last used (successfully).
C
C NQCUR IWORK(15) the order to be attempted on the next step.
C
C IMXER IWORK(16) the index of the component of largest magnitude in
C the weighted local error vector ( E(i)/EWT(i) ),
C on an error return with ISTATE = -4 or -5.
C
C LENRW IWORK(17) the length of RWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C LENIW IWORK(18) the length of IWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C NNZ IWORK(19) the number of nonzero elements in the iteration
C matrix P = A - con*J (con is a constant and
C J is the Jacobian matrix dr/dy).
C
C NGP IWORK(20) the number of groups of column indices, used in
C difference quotient Jacobian aproximations if
C MITER = 2. This is also the number of extra RES
C evaluations needed for each Jacobian evaluation.
C
C NLU IWORK(21) the number of sparse LU decompositions for the
C problem so far. (Excludes the LU decomposition
C necessary when ISTATE = 0.)
C
C LYH IWORK(22) the base address in RWORK of the history array YH,
C described below in this list.
C
C IPIAN IWORK(23) the base address of the structure descriptor array
C IAN, described below in this list.
C
C IPJAN IWORK(24) the base address of the structure descriptor array
C JAN, described below in this list.
C
C NZL IWORK(25) the number of nonzero elements in the strict lower
C triangle of the LU factorization used in the chord
C iteration.
C
C NZU IWORK(26) the number of nonzero elements in the strict upper
C triangle of the LU factorization used in the chord
C iteration. The total number of nonzeros in the
C factorization is therefore NZL + NZU + NEQ.
C
C The following four arrays are segments of the RWORK array which
C may also be of interest to the user as optional outputs.
C For each array, the table below gives its internal name,
C its base address, and its description.
C For YH and ACOR, the base addresses are in RWORK (a real array).
C The integer arrays IAN and JAN are to be obtained by declaring an
C integer array IWK and identifying IWK(1) with RWORK(21), using either
C an equivalence statement or a subroutine call. Then the base
C addresses IPIAN (of IAN) and IPJAN (of JAN) in IWK are to be obtained
C as optional outputs IWORK(23) and IWORK(24), respectively.
C Thus IAN(1) is IWK(ipian), etc.
C
C Name Base Address Description
C
C IAN IPIAN (in IWK) structure descriptor array of size NEQ + 1.
C JAN IPJAN (in IWK) structure descriptor array of size NNZ.
C (see above) IAN and JAN together describe the sparsity
C structure of the iteration matrix
C P = A - con*J, as used by DLSODIS.
C JAN contains the row indices of the nonzero
C locations, reading in columnwise order, and
C IAN contains the starting locations in JAN of
C the descriptions of columns 1,...,NEQ, in
C that order, with IAN(1) = 1. Thus for each
C j = 1,...,NEQ, the row indices i of the
C nonzero locations in column j are
C i = JAN(k), IAN(j) .le. k .lt. IAN(j+1).
C Note that IAN(NEQ+1) = NNZ + 1.
C YH LYH the Nordsieck history array, of size NYH by
C (optional (NQCUR + 1), where NYH is the initial value
C output) of NEQ. For j = 0,1,...,NQCUR, column j+1
C of YH contains HCUR**j/factorial(j) times
C the j-th derivative of the interpolating
C polynomial currently representing the solution,
C evaluated at t = TCUR. The base address LYH
C is another optional output, listed above.
C
C ACOR LENRW-NEQ+1 array of size NEQ used for the accumulated
C corrections on each step, scaled on output to
C represent the estimated local error in y on the
C last step. This is the vector E in the
C description of the error control. It is defined
C only on a return from DLSODIS with ISTATE = 2.
C
C-----------------------------------------------------------------------
C Part 2. Other Routines Callable.
C
C The following are optional calls which the user may make to
C gain additional capabilities in conjunction with DLSODIS.
C (The routines XSETUN and XSETF are designed to conform to the
C SLATEC error handling package.)
C
C Form of Call Function
C CALL XSETUN(LUN) Set the logical unit number, LUN, for
C output of messages from DLSODIS, if
C The default is not desired.
C The default value of LUN is 6.
C
C CALL XSETF(MFLAG) Set a flag to control the printing of
C messages by DLSODIS.
C MFLAG = 0 means do not print. (Danger:
C This risks losing valuable information.)
C MFLAG = 1 means print (the default).
C
C Either of the above calls may be made at
C any time and will take effect immediately.
C
C CALL DSRCMS(RSAV,ISAV,JOB) saves and restores the contents of
C the internal Common blocks used by
C DLSODIS (see Part 3 below).
C RSAV must be a real array of length 224
C or more, and ISAV must be an integer
C array of length 71 or more.
C JOB=1 means save Common into RSAV/ISAV.
C JOB=2 means restore Common from RSAV/ISAV.
C DSRCMS is useful if one is
C interrupting a run and restarting
C later, or alternating between two or
C more problems solved with DLSODIS.
C
C CALL DINTDY(,,,,,) Provide derivatives of y, of various
C (see below) orders, at a specified point t, if
C desired. It may be called only after
C a successful return from DLSODIS.
C
C The detailed instructions for using DINTDY are as follows.
C The form of the call is:
C
C LYH = IWORK(22)
C CALL DINTDY (T, K, RWORK(LYH), NYH, DKY, IFLAG)
C
C The input parameters are:
C
C T = value of independent variable where answers are desired
C (normally the same as the T last returned by DLSODIS).
C For valid results, T must lie between TCUR - HU and TCUR.
C (See optional outputs for TCUR and HU.)
C K = integer order of the derivative desired. K must satisfy
C 0 .le. K .le. NQCUR, where NQCUR is the current order
C (see optional outputs). The capability corresponding
C to K = 0, i.e. computing y(t), is already provided
C by DLSODIS directly. Since NQCUR .ge. 1, the first
C derivative dy/dt is always available with DINTDY.
C LYH = the base address of the history array YH, obtained
C as an optional output as shown above.
C NYH = column length of YH, equal to the initial value of NEQ.
C
C The output parameters are:
C
C DKY = a real array of length NEQ containing the computed value
C of the K-th derivative of y(t).
C IFLAG = integer flag, returned as 0 if K and T were legal,
C -1 if K was illegal, and -2 if T was illegal.
C On an error return, a message is also written.
C-----------------------------------------------------------------------
C Part 3. Common Blocks.
C
C If DLSODIS is to be used in an overlay situation, the user
C must declare, in the primary overlay, the variables in:
C (1) the call sequence to DLSODIS, and
C (2) the two internal Common blocks
C /DLS001/ of length 255 (218 double precision words
C followed by 37 integer words),
C /DLSS01/ of length 40 (6 double precision words
C followed by 34 integer words).
C
C If DLSODIS is used on a system in which the contents of internal
C Common blocks are not preserved between calls, the user should
C declare the above Common blocks in the calling program to insure
C that their contents are preserved.
C
C If the solution of a given problem by DLSODIS is to be interrupted
C and then later continued, such as when restarting an interrupted run
C or alternating between two or more problems, the user should save,
C following the return from the last DLSODIS call prior to the
C interruption, the contents of the call sequence variables and the
C internal Common blocks, and later restore these values before the
C next DLSODIS call for that problem. To save and restore the Common
C blocks, use Subroutines DSRCMS (see Part 2 above).
C
C-----------------------------------------------------------------------
C Part 4. Optionally Replaceable Solver Routines.
C
C Below are descriptions of two routines in the DLSODIS package which
C relate to the measurement of errors. Either routine can be
C replaced by a user-supplied version, if desired. However, since such
C a replacement may have a major impact on performance, it should be
C done only when absolutely necessary, and only with great caution.
C (Note: The means by which the package version of a routine is
C superseded by the user's version may be system-dependent.)
C
C (a) DEWSET.
C The following subroutine is called just before each internal
C integration step, and sets the array of error weights, EWT, as
C described under ITOL/RTOL/ATOL above:
C SUBROUTINE DEWSET (NEQ, ITOL, RTOL, ATOL, YCUR, EWT)
C where NEQ, ITOL, RTOL, and ATOL are as in the DLSODIS call sequence,
C YCUR contains the current dependent variable vector, and
C EWT is the array of weights set by DEWSET.
C
C If the user supplies this subroutine, it must return in EWT(i)
C (i = 1,...,NEQ) a positive quantity suitable for comparing errors
C in y(i) to. The EWT array returned by DEWSET is passed to the DVNORM
C routine (see below), and also used by DLSODIS in the computation
C of the optional output IMXER, and the increments for difference
C quotient Jacobians.
C
C In the user-supplied version of DEWSET, it may be desirable to use
C the current values of derivatives of y. Derivatives up to order NQ
C are available from the history array YH, described above under
C optional outputs. In DEWSET, YH is identical to the YCUR array,
C extended to NQ + 1 columns with a column length of NYH and scale
C factors of H**j/factorial(j). On the first call for the problem,
C given by NST = 0, NQ is 1 and H is temporarily set to 1.0.
C NYH is the initial value of NEQ. The quantities NQ, H, and NST
C can be obtained by including in DEWSET the statements:
C DOUBLE PRECISION RLS
C COMMON /DLS001/ RLS(218),ILS(37)
C NQ = ILS(33)
C NST = ILS(34)
C H = RLS(212)
C Thus, for example, the current value of dy/dt can be obtained as
C YCUR(NYH+i)/H (i=1,...,NEQ) (and the division by H is
C unnecessary when NST = 0).
C
C (b) DVNORM.
C The following is a real function routine which computes the weighted
C root-mean-square norm of a vector v:
C D = DVNORM (N, V, W)
C where:
C N = the length of the vector,
C V = real array of length N containing the vector,
C W = real array of length N containing weights,
C D = SQRT( (1/N) * sum(V(i)*W(i))**2 ).
C DVNORM is called with N = NEQ and with W(i) = 1.0/EWT(i), where
C EWT is as set by Subroutine DEWSET.
C
C If the user supplies this function, it should return a non-negative
C value of DVNORM suitable for use in the error control in DLSODIS.
C None of the arguments should be altered by DVNORM.
C For example, a user-supplied DVNORM routine might:
C -substitute a max-norm of (V(i)*w(I)) for the RMS-norm, or
C -ignore some components of V in the norm, with the effect of
C suppressing the error control on those components of y.
C-----------------------------------------------------------------------
C
C***REVISION HISTORY (YYYYMMDD)
C 19820714 DATE WRITTEN
C 19830812 Major update, based on recent LSODI and LSODES revisions:
C Upgraded MDI in ODRV package: operates on M + M-transpose.
C Numerous revisions in use of work arrays;
C use wordlength ratio LENRAT; added IPISP & LRAT to Common;
C added optional outputs IPIAN/IPJAN;
C Added routine CNTNZU; added NZL and NZU to /LSS001/;
C changed ADJLR call logic; added optional outputs NZL & NZU;
C revised counter initializations; revised PREPI stmt. nos.;
C revised difference quotient increment;
C eliminated block /LSI001/, using IERPJ flag;
C revised STODI logic after PJAC return;
C revised tuning of H change and step attempts in STODI;
C corrections to main prologue and comments throughout.
C 19870320 Corrected jump on test of umax in CDRV routine.
C 20010125 Numerous revisions: corrected comments throughout;
C removed TRET from Common; rewrote EWSET with 4 loops;
C fixed t test in INTDY; added Cray directives in STODI;
C in STODI, fixed DELP init. and logic around PJAC call;
C combined routines to save/restore Common;
C passed LEVEL = 0 in error message calls (except run abort).
C 20010425 Major update: convert source lines to upper case;
C added *DECK lines; changed from 1 to * in dummy dimensions;
C changed names R1MACH/D1MACH to RUMACH/DUMACH;
C renamed routines for uniqueness across single/double prec.;
C converted intrinsic names to generic form;
C removed ILLIN and NTREP (data loaded) from Common;
C removed all 'own' variables from Common;
C changed error messages to quoted strings;
C replaced XERRWV/XERRWD with 1993 revised version;
C converted prologues, comments, error messages to mixed case;
C converted arithmetic IF statements to logical IF statements;
C numerous corrections to prologues and internal comments.
C 20010507 Converted single precision source to double precision.
C 20020502 Corrected declarations in descriptions of user routines.
C 20031021 Fixed address offset bugs in Subroutine DPREPI.
C 20031027 Changed 0. to 0.0D0 in Subroutine DPREPI.
C 20031105 Restored 'own' variables to Common blocks, to enable
C interrupt/restart feature.
C 20031112 Added SAVE statements for data-loaded constants.
C 20031117 Changed internal names NRE, LSAVR to NFE, LSAVF resp.
C
C-----------------------------------------------------------------------
C Other routines in the DLSODIS package.
C
C In addition to Subroutine DLSODIS, the DLSODIS package includes the
C following subroutines and function routines:
C DIPREPI acts as an interface between DLSODIS and DPREPI, and also
C does adjusting of work space pointers and work arrays.
C DPREPI is called by DIPREPI to compute sparsity and do sparse
C matrix preprocessing.
C DAINVGS computes the initial value of the vector
C dy/dt = A-inverse * g
C ADJLR adjusts the length of required sparse matrix work space.
C It is called by DPREPI.
C CNTNZU is called by DPREPI and counts the nonzero elements in the
C strict upper triangle of P + P-transpose.
C JGROUP is called by DPREPI to compute groups of Jacobian column
C indices for use when MITER = 2.
C DINTDY computes an interpolated value of the y vector at t = TOUT.
C DSTODI is the core integrator, which does one step of the
C integration and the associated error control.
C DCFODE sets all method coefficients and test constants.
C DPRJIS computes and preprocesses the Jacobian matrix J = dr/dy
C and the Newton iteration matrix P = A - h*l0*J.
C DSOLSS manages solution of linear system in chord iteration.
C DEWSET sets the error weight vector EWT before each step.
C DVNORM computes the weighted RMS-norm of a vector.
C DSRCMS is a user-callable routine to save and restore
C the contents of the internal Common blocks.
C ODRV constructs a reordering of the rows and columns of
C a matrix by the minimum degree algorithm. ODRV is a
C driver routine which calls Subroutines MD, MDI, MDM,
C MDP, MDU, and SRO. See Ref. 2 for details. (The ODRV
C module has been modified since Ref. 2, however.)
C CDRV performs reordering, symbolic factorization, numerical
C factorization, or linear system solution operations,
C depending on a path argument IPATH. CDRV is a
C driver routine which calls Subroutines NROC, NSFC,
C NNFC, NNSC, and NNTC. See Ref. 3 for details.
C DLSODIS uses CDRV to solve linear systems in which the
C coefficient matrix is P = A - con*J, where A is the
C matrix for the linear system A(t,y)*dy/dt = g(t,y),
C con is a scalar, and J is an approximation to
C the Jacobian dr/dy. Because CDRV deals with rowwise
C sparsity descriptions, CDRV works with P-transpose, not P.
C DLSODIS also uses CDRV to solve the linear system
C A(t,y)*dy/dt = g(t,y) for dy/dt when ISTATE = 0.
C (For this, CDRV works with A-transpose, not A.)
C DUMACH computes the unit roundoff in a machine-independent manner.
C XERRWD, XSETUN, XSETF, IXSAV, and IUMACH handle the printing of all
C error messages and warnings. XERRWD is machine-dependent.
C Note: DVNORM, DUMACH, IXSAV, and IUMACH are function routines.
C All the others are subroutines.
C
C-----------------------------------------------------------------------
EXTERNAL DPRJIS, DSOLSS
DOUBLE PRECISION DUMACH, DVNORM
INTEGER INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP,
1 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA,
2 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ,
3 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU
INTEGER I, I1, I2, IER, IGO, IFLAG, IMAX, IMUL, IMXER, IPFLAG,
1 IPGO, IREM, IRES, J, KGO, LENRAT, LENYHT, LENIW, LENRW,
2 LIA, LIC, LJA, LJC, LRTEM, LWTEM, LYD0, LYHD, LYHN, MF1,
3 MORD, MXHNL0, MXSTP0, NCOLM
DOUBLE PRECISION ROWNS,
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
DOUBLE PRECISION CON0, CONMIN, CCMXJ, PSMALL, RBIG, SETH
DOUBLE PRECISION ATOLI, AYI, BIG, EWTI, H0, HMAX, HMX, RH, RTOLI,
1 TCRIT, TDIST, TNEXT, TOL, TOLSF, TP, SIZE, SUM, W0
DIMENSION MORD(2)
LOGICAL IHIT
CHARACTER*60 MSG
SAVE LENRAT, MORD, MXSTP0, MXHNL0
C-----------------------------------------------------------------------
C The following two internal Common blocks contain
C (a) variables which are local to any subroutine but whose values must
C be preserved between calls to the routine ("own" variables), and
C (b) variables which are communicated between subroutines.
C The block DLS001 is declared in subroutines DLSODIS, DIPREPI, DPREPI,
C DINTDY, DSTODI, DPRJIS, and DSOLSS.
C The block DLSS01 is declared in subroutines DLSODIS, DAINVGS,
C DIPREPI, DPREPI, DPRJIS, and DSOLSS.
C Groups of variables are replaced by dummy arrays in the Common
C declarations in routines where those variables are not used.
C-----------------------------------------------------------------------
COMMON /DLS001/ ROWNS(209),
1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
2 INIT, MXSTEP, MXHNIL, NHNIL, NSLAST, NYH, IOWNS(6),
3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
COMMON /DLSS01/ CON0, CONMIN, CCMXJ, PSMALL, RBIG, SETH,
1 IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP,
2 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA,
3 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ,
4 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU
C
DATA MORD(1),MORD(2)/12,5/, MXSTP0/500/, MXHNL0/10/
C-----------------------------------------------------------------------
C In the Data statement below, set LENRAT equal to the ratio of
C the wordlength for a real number to that for an integer. Usually,
C LENRAT = 1 for single precision and 2 for double precision. If the
C true ratio is not an integer, use the next smaller integer (.ge. 1),
C-----------------------------------------------------------------------
DATA LENRAT/2/
C-----------------------------------------------------------------------
C Block A.
C This code block is executed on every call.
C It tests ISTATE and ITASK for legality and branches appropirately.
C If ISTATE .gt. 1 but the flag INIT shows that initialization has
C not yet been done, an error return occurs.
C If ISTATE = 0 or 1 and TOUT = T, return immediately.
C-----------------------------------------------------------------------
IF (ISTATE .LT. 0 .OR. ISTATE .GT. 3) GO TO 601
IF (ITASK .LT. 1 .OR. ITASK .GT. 5) GO TO 602
IF (ISTATE .LE. 1) GO TO 10
IF (INIT .EQ. 0) GO TO 603
IF (ISTATE .EQ. 2) GO TO 200
GO TO 20
10 INIT = 0
IF (TOUT .EQ. T) RETURN
C-----------------------------------------------------------------------
C Block B.
C The next code block is executed for the initial call (ISTATE = 0 or 1)
C or for a continuation call with parameter changes (ISTATE = 3).
C It contains checking of all inputs and various initializations.
C If ISTATE = 0 or 1, the final setting of work space pointers, the
C matrix preprocessing, and other initializations are done in Block C.
C
C First check legality of the non-optional inputs NEQ, ITOL, IOPT, and
C MF.
C-----------------------------------------------------------------------
20 IF (NEQ(1) .LE. 0) GO TO 604
IF (ISTATE .LE. 1) GO TO 25
IF (NEQ(1) .GT. N) GO TO 605
25 N = NEQ(1)
IF (ITOL .LT. 1 .OR. ITOL .GT. 4) GO TO 606
IF (IOPT .LT. 0 .OR. IOPT .GT. 1) GO TO 607
MOSS = MF/100
MF1 = MF - 100*MOSS
METH = MF1/10
MITER = MF1 - 10*METH
IF (MOSS .LT. 0 .OR. MOSS .GT. 4) GO TO 608
IF (MITER .EQ. 2 .AND. MOSS .EQ. 1) MOSS = MOSS + 1
IF (MITER .EQ. 2 .AND. MOSS .EQ. 3) MOSS = MOSS + 1
IF (MITER .EQ. 1 .AND. MOSS .EQ. 2) MOSS = MOSS - 1
IF (MITER .EQ. 1 .AND. MOSS .EQ. 4) MOSS = MOSS - 1
IF (METH .LT. 1 .OR. METH .GT. 2) GO TO 608
IF (MITER .LT. 1 .OR. MITER .GT. 2) GO TO 608
C Next process and check the optional inputs. --------------------------
IF (IOPT .EQ. 1) GO TO 40
MAXORD = MORD(METH)
MXSTEP = MXSTP0
MXHNIL = MXHNL0
IF (ISTATE .LE. 1) H0 = 0.0D0
HMXI = 0.0D0
HMIN = 0.0D0
GO TO 60
40 MAXORD = IWORK(5)
IF (MAXORD .LT. 0) GO TO 611
IF (MAXORD .EQ. 0) MAXORD = 100
MAXORD = MIN(MAXORD,MORD(METH))
MXSTEP = IWORK(6)
IF (MXSTEP .LT. 0) GO TO 612
IF (MXSTEP .EQ. 0) MXSTEP = MXSTP0
MXHNIL = IWORK(7)
IF (MXHNIL .LT. 0) GO TO 613
IF (MXHNIL .EQ. 0) MXHNIL = MXHNL0
IF (ISTATE .GT. 1) GO TO 50
H0 = RWORK(5)
IF ((TOUT - T)*H0 .LT. 0.0D0) GO TO 614
50 HMAX = RWORK(6)
IF (HMAX .LT. 0.0D0) GO TO 615
HMXI = 0.0D0
IF (HMAX .GT. 0.0D0) HMXI = 1.0D0/HMAX
HMIN = RWORK(7)
IF (HMIN .LT. 0.0D0) GO TO 616
C Check RTOL and ATOL for legality. ------------------------------------
60 RTOLI = RTOL(1)
ATOLI = ATOL(1)
DO 65 I = 1,N
IF (ITOL .GE. 3) RTOLI = RTOL(I)
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
IF (RTOLI .LT. 0.0D0) GO TO 619
IF (ATOLI .LT. 0.0D0) GO TO 620
65 CONTINUE
C-----------------------------------------------------------------------
C Compute required work array lengths, as far as possible, and test
C these against LRW and LIW. Then set tentative pointers for work
C arrays. Pointers to RWORK/IWORK segments are named by prefixing L to
C the name of the segment. E.g., the segment YH starts at RWORK(LYH).
C Segments of RWORK (in order) are denoted WM, YH, SAVR, EWT, ACOR.
C The required length of the matrix work space WM is not yet known,
C and so a crude minimum value is used for the initial tests of LRW
C and LIW, and YH is temporarily stored as far to the right in RWORK
C as possible, to leave the maximum amount of space for WM for matrix
C preprocessing. Thus if MOSS .ne. 2 or 4, some of the segments of
C RWORK are temporarily omitted, as they are not needed in the
C preprocessing. These omitted segments are: ACOR if ISTATE = 1,
C EWT and ACOR if ISTATE = 3 and MOSS = 1, and SAVR, EWT, and ACOR if
C ISTATE = 3 and MOSS = 0.
C-----------------------------------------------------------------------
LRAT = LENRAT
IF (ISTATE .LE. 1) NYH = N
IF (MITER .EQ. 1) LWMIN = 4*N + 10*N/LRAT
IF (MITER .EQ. 2) LWMIN = 4*N + 11*N/LRAT
LENYH = (MAXORD+1)*NYH
LREST = LENYH + 3*N
LENRW = 20 + LWMIN + LREST
IWORK(17) = LENRW
LENIW = 30
IF (MOSS .NE. 1 .AND. MOSS .NE. 2) LENIW = LENIW + N + 1
IWORK(18) = LENIW
IF (LENRW .GT. LRW) GO TO 617
IF (LENIW .GT. LIW) GO TO 618
LIA = 31
IF (MOSS .NE. 1 .AND. MOSS .NE. 2)
1 LENIW = LENIW + IWORK(LIA+N) - 1
IWORK(18) = LENIW
IF (LENIW .GT. LIW) GO TO 618
LJA = LIA + N + 1
LIA = MIN(LIA,LIW)
LJA = MIN(LJA,LIW)
LIC = LENIW + 1
IF (MOSS .EQ. 0) LENIW = LENIW + N + 1
IWORK(18) = LENIW
IF (LENIW .GT. LIW) GO TO 618
IF (MOSS .EQ. 0) LENIW = LENIW + IWORK(LIC+N) - 1
IWORK(18) = LENIW
IF (LENIW .GT. LIW) GO TO 618
LJC = LIC + N + 1
LIC = MIN(LIC,LIW)
LJC = MIN(LJC,LIW)
LWM = 21
IF (ISTATE .LE. 1) NQ = ISTATE
NCOLM = MIN(NQ+1,MAXORD+2)
LENYHM = NCOLM*NYH
LENYHT = LENYHM
IMUL = 2
IF (ISTATE .EQ. 3) IMUL = MOSS
IF (ISTATE .EQ. 3 .AND. MOSS .EQ. 3) IMUL = 1
IF (MOSS .EQ. 2 .OR. MOSS .EQ. 4) IMUL = 3
LRTEM = LENYHT + IMUL*N
LWTEM = LRW - 20 - LRTEM
LENWK = LWTEM
LYHN = LWM + LWTEM
LSAVF = LYHN + LENYHT
LEWT = LSAVF + N
LACOR = LEWT + N
ISTATC = ISTATE
IF (ISTATE .LE. 1) GO TO 100
C-----------------------------------------------------------------------
C ISTATE = 3. Move YH to its new location.
C Note that only the part of YH needed for the next step, namely
C MIN(NQ+1,MAXORD+2) columns, is actually moved.
C A temporary error weight array EWT is loaded if MOSS = 2 or 4.
C Sparse matrix processing is done in DIPREPI/DPREPI.
C If MAXORD was reduced below NQ, then the pointers are finally set
C so that SAVR is identical to (YH*,MAXORD+2)
C-----------------------------------------------------------------------
LYHD = LYH - LYHN
IMAX = LYHN - 1 + LENYHM
C Move YH. Move right if LYHD < 0; move left if LYHD > 0. -------------
IF (LYHD .LT. 0) THEN
DO 72 I = LYHN,IMAX
J = IMAX + LYHN - I
72 RWORK(J) = RWORK(J+LYHD)
ENDIF
IF (LYHD .GT. 0) THEN
DO 76 I = LYHN,IMAX
76 RWORK(I) = RWORK(I+LYHD)
ENDIF
80 LYH = LYHN
IWORK(22) = LYH
IF (MOSS .NE. 2 .AND. MOSS .NE. 4) GO TO 85
C Temporarily load EWT if MOSS = 2 or 4.
CALL DEWSET (N,ITOL,RTOL,ATOL,RWORK(LYH),RWORK(LEWT))
DO 82 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 621
82 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
85 CONTINUE
C DIPREPI and DPREPI do sparse matrix preprocessing. -------------------
LSAVF = MIN(LSAVF,LRW)
LEWT = MIN(LEWT,LRW)
LACOR = MIN(LACOR,LRW)
CALL DIPREPI (NEQ, Y, YDOTI, RWORK, IWORK(LIA), IWORK(LJA),
1 IWORK(LIC), IWORK(LJC), IPFLAG, RES, JAC, ADDA)
LENRW = LWM - 1 + LENWK + LREST
IWORK(17) = LENRW
IF (IPFLAG .NE. -1) IWORK(23) = IPIAN
IF (IPFLAG .NE. -1) IWORK(24) = IPJAN
IPGO = -IPFLAG + 1
GO TO (90, 628, 629, 630, 631, 632, 633, 634, 634), IPGO
90 IWORK(22) = LYH
LYD0 = LYH + N
IF (LENRW .GT. LRW) GO TO 617
C Set flag to signal changes to DSTODI.---------------------------------
JSTART = -1
IF (NQ .LE. MAXORD) GO TO 94
C MAXORD was reduced below NQ. Copy YH(*,MAXORD+2) into YDOTI. --------
DO 92 I = 1,N
92 YDOTI(I) = RWORK(I+LSAVF-1)
94 IF (N .EQ. NYH) GO TO 200
C NEQ was reduced. Zero part of YH to avoid undefined references. -----
I1 = LYH + L*NYH
I2 = LYH + (MAXORD + 1)*NYH - 1
IF (I1 .GT. I2) GO TO 200
DO 95 I = I1,I2
95 RWORK(I) = 0.0D0
GO TO 200
C-----------------------------------------------------------------------
C Block C.
C The next block is for the initial call only (ISTATE = 0 or 1).
C It contains all remaining initializations, the call to DAINVGS
C (if ISTATE = 0), the sparse matrix preprocessing, and the
C calculation if the initial step size.
C The error weights in EWT are inverted after being loaded.
C-----------------------------------------------------------------------
100 CONTINUE
LYH = LYHN
IWORK(22) = LYH
TN = T
NST = 0
NFE = 0
H = 1.0D0
NNZ = 0
NGP = 0
NZL = 0
NZU = 0
C Load the initial value vector in YH.----------------------------------
DO 105 I = 1,N
105 RWORK(I+LYH-1) = Y(I)
IF (ISTATE .NE. 1) GO TO 108
C Initial dy/dt was supplied. Load it into YH (LYD0 points to YH(*,2).)
LYD0 = LYH + NYH
DO 106 I = 1,N
106 RWORK(I+LYD0-1) = YDOTI(I)
108 CONTINUE
C Load and invert the EWT array. (H is temporarily set to 1.0.)--------
CALL DEWSET (N,ITOL,RTOL,ATOL,RWORK(LYH),RWORK(LEWT))
DO 110 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 621
110 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
C Call DIPREPI and DPREPI to do sparse matrix preprocessing.------------
LACOR = MIN(LACOR,LRW)
CALL DIPREPI (NEQ, Y, YDOTI, RWORK, IWORK(LIA), IWORK(LJA),
1 IWORK(LIC), IWORK(LJC), IPFLAG, RES, JAC, ADDA)
LENRW = LWM - 1 + LENWK + LREST
IWORK(17) = LENRW
IF (IPFLAG .NE. -1) IWORK(23) = IPIAN
IF (IPFLAG .NE. -1) IWORK(24) = IPJAN
IPGO = -IPFLAG + 1
GO TO (115, 628, 629, 630, 631, 632, 633, 634, 634), IPGO
115 IWORK(22) = LYH
IF (LENRW .GT. LRW) GO TO 617
C Compute initial dy/dt, if necessary, and load it into YH.-------------
LYD0 = LYH + N
IF (ISTATE .NE. 0) GO TO 120
CALL DAINVGS (NEQ, T, Y, RWORK(LWM), RWORK(LWM), RWORK(LACOR),
1 RWORK(LYD0), IER, RES, ADDA)
NFE = NFE + 1
IGO = IER + 1
GO TO (120, 565, 560, 560), IGO
C Check TCRIT for legality (ITASK = 4 or 5). ---------------------------
120 CONTINUE
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 125
TCRIT = RWORK(1)
IF ((TCRIT - TOUT)*(TOUT - T) .LT. 0.0D0) GO TO 625
IF (H0 .NE. 0.0D0 .AND. (T + H0 - TCRIT)*H0 .GT. 0.0D0)
1 H0 = TCRIT - T
C Initialize all remaining parameters. ---------------------------------
125 UROUND = DUMACH()
JSTART = 0
RWORK(LWM) = SQRT(UROUND)
NHNIL = 0
NJE = 0
NLU = 0
NSLAST = 0
HU = 0.0D0
NQU = 0
CCMAX = 0.3D0
MAXCOR = 3
MSBP = 20
MXNCF = 10
C-----------------------------------------------------------------------
C The coding below computes the step size, H0, to be attempted on the
C first step, unless the user has supplied a value for this.
C First check that TOUT - T differs significantly from zero.
C A scalar tolerance quantity TOL is computed, as MAX(RTOL(i))
C if this is positive, or MAX(ATOL(i)/ABS(Y(i))) otherwise, adjusted
C so as to be between 100*UROUND and 1.0E-3.
C Then the computed value H0 is given by..
C NEQ
C H0**2 = TOL / ( w0**-2 + (1/NEQ) * Sum ( YDOT(i)/ywt(i) )**2 )
C 1
C where w0 = MAX ( ABS(T), ABS(TOUT) ),
C YDOT(i) = i-th component of initial value of dy/dt,
C ywt(i) = EWT(i)/TOL (a weight for y(i)).
C The sign of H0 is inferred from the initial values of TOUT and T.
C-----------------------------------------------------------------------
IF (H0 .NE. 0.0D0) GO TO 180
TDIST = ABS(TOUT - T)
W0 = MAX(ABS(T),ABS(TOUT))
IF (TDIST .LT. 2.0D0*UROUND*W0) GO TO 622
TOL = RTOL(1)
IF (ITOL .LE. 2) GO TO 145
DO 140 I = 1,N
140 TOL = MAX(TOL,RTOL(I))
145 IF (TOL .GT. 0.0D0) GO TO 160
ATOLI = ATOL(1)
DO 150 I = 1,N
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
AYI = ABS(Y(I))
IF (AYI .NE. 0.0D0) TOL = MAX(TOL,ATOLI/AYI)
150 CONTINUE
160 TOL = MAX(TOL,100.0D0*UROUND)
TOL = MIN(TOL,0.001D0)
SUM = DVNORM (N, RWORK(LYD0), RWORK(LEWT))
SUM = 1.0D0/(TOL*W0*W0) + TOL*SUM**2
H0 = 1.0D0/SQRT(SUM)
H0 = MIN(H0,TDIST)
H0 = SIGN(H0,TOUT-T)
C Adjust H0 if necessary to meet HMAX bound. ---------------------------
180 RH = ABS(H0)*HMXI
IF (RH .GT. 1.0D0) H0 = H0/RH
C Load H with H0 and scale YH(*,2) by H0. ------------------------------
H = H0
DO 190 I = 1,N
190 RWORK(I+LYD0-1) = H0*RWORK(I+LYD0-1)
GO TO 270
C-----------------------------------------------------------------------
C Block D.
C The next code block is for continuation calls only (ISTATE = 2 or 3)
C and is to check stop conditions before taking a step.
C-----------------------------------------------------------------------
200 NSLAST = NST
GO TO (210, 250, 220, 230, 240), ITASK
210 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
220 TP = TN - HU*(1.0D0 + 100.0D0*UROUND)
IF ((TP - TOUT)*H .GT. 0.0D0) GO TO 623
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
GO TO 400
230 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
IF ((TCRIT - TOUT)*H .LT. 0.0D0) GO TO 625
IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 245
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
240 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. 0.0D0) GO TO 624
245 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
IF (ISTATE .EQ. 2) JSTART = -2
C-----------------------------------------------------------------------
C Block E.
C The next block is normally executed for all calls and contains
C the call to the one-step core integrator DSTODI.
C
C This is a looping point for the integration steps.
C
C First check for too many steps being taken, update EWT (if not at
C start of problem), check for too much accuracy being requested, and
C check for H below the roundoff level in T.
C-----------------------------------------------------------------------
250 CONTINUE
IF ((NST-NSLAST) .GE. MXSTEP) GO TO 500
CALL DEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 260 I = 1,N
IF (RWORK(I+LEWT-1) .LE. 0.0D0) GO TO 510
260 RWORK(I+LEWT-1) = 1.0D0/RWORK(I+LEWT-1)
270 TOLSF = UROUND*DVNORM (N, RWORK(LYH), RWORK(LEWT))
IF (TOLSF .LE. 1.0D0) GO TO 280
TOLSF = TOLSF*2.0D0
IF (NST .EQ. 0) GO TO 626
GO TO 520
280 IF ((TN + H) .NE. TN) GO TO 290
NHNIL = NHNIL + 1
IF (NHNIL .GT. MXHNIL) GO TO 290
MSG = 'DLSODIS- Warning..Internal T (=R1) and H (=R2) are'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' such that in the machine, T + H = T on the next step '
CALL XERRWD (MSG, 60, 101, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' (H = step size). Solver will continue anyway.'
CALL XERRWD (MSG, 50, 101, 0, 0, 0, 0, 2, TN, H)
IF (NHNIL .LT. MXHNIL) GO TO 290
MSG = 'DLSODIS- Above warning has been issued I1 times. '
CALL XERRWD (MSG, 50, 102, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' It will not be issued again for this problem.'
CALL XERRWD (MSG, 50, 102, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
290 CONTINUE
C-----------------------------------------------------------------------
C CALL DSTODI(NEQ,Y,YH,NYH,YH1,EWT,SAVF,SAVR,ACOR,WM,WM,RES,
C ADDA,JAC,DPRJIS,DSOLSS)
C Note: SAVF in DSTODI occupies the same space as YDOTI in DLSODIS.
C-----------------------------------------------------------------------
CALL DSTODI (NEQ, Y, RWORK(LYH), NYH, RWORK(LYH), RWORK(LEWT),
1 YDOTI, RWORK(LSAVF), RWORK(LACOR), RWORK(LWM),
2 RWORK(LWM), RES, ADDA, JAC, DPRJIS, DSOLSS )
KGO = 1 - KFLAG
GO TO (300, 530, 540, 400, 550, 555), KGO
C
C KGO = 1:success; 2:error test failure; 3:convergence failure;
C 4:RES ordered return; 5:RES returned error;
C 6:fatal error from CDRV via DPRJIS or DSOLSS.
C-----------------------------------------------------------------------
C Block F.
C The following block handles the case of a successful return from the
C core integrator (KFLAG = 0). Test for stop conditions.
C-----------------------------------------------------------------------
300 INIT = 1
GO TO (310, 400, 330, 340, 350), ITASK
C ITASK = 1. If TOUT has been reached, interpolate. -------------------
310 iF ((TN - TOUT)*H .LT. 0.0D0) GO TO 250
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
C ITASK = 3. Jump to exit if TOUT was reached. ------------------------
330 IF ((TN - TOUT)*H .GE. 0.0D0) GO TO 400
GO TO 250
C ITASK = 4. See if TOUT or TCRIT was reached. Adjust H if necessary.
340 IF ((TN - TOUT)*H .LT. 0.0D0) GO TO 345
CALL DINTDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
345 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(1.0D0 + 4.0D0*UROUND)
IF ((TNEXT - TCRIT)*H .LE. 0.0D0) GO TO 250
H = (TCRIT - TN)*(1.0D0 - 4.0D0*UROUND)
JSTART = -2
GO TO 250
C ITASK = 5. See if TCRIT was reached and jump to exit. ---------------
350 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. 100.0D0*UROUND*HMX
C-----------------------------------------------------------------------
C Block G.
C The following block handles all successful returns from DLSODIS.
C if ITASK .ne. 1, Y is loaded from YH and T is set accordingly.
C ISTATE is set to 2, and the optional outputs are loaded into the
C work arrays before returning.
C-----------------------------------------------------------------------
400 DO 410 I = 1,N
410 Y(I) = RWORK(I+LYH-1)
T = TN
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 420
IF (IHIT) T = TCRIT
420 ISTATE = 2
IF ( KFLAG .EQ. -3 ) ISTATE = 3
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(19) = NNZ
IWORK(20) = NGP
IWORK(21) = NLU
IWORK(25) = NZL
IWORK(26) = NZU
RETURN
C-----------------------------------------------------------------------
C Block H.
C The following block handles all unsuccessful returns other than
C those for illegal input. First the error message routine is called.
C If there was an error test or convergence test failure, IMXER is set.
C Then Y is loaded from YH and T is set to TN.
C The optional outputs are loaded into the work arrays before returning.
C-----------------------------------------------------------------------
C The maximum number of steps was taken before reaching TOUT. ----------
500 MSG = 'DLSODIS- At current T (=R1), MXSTEP (=I1) steps '
CALL XERRWD (MSG, 50, 201, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' taken on this call before reaching TOUT '
CALL XERRWD (MSG, 50, 201, 0, 1, MXSTEP, 0, 1, TN, 0.0D0)
ISTATE = -1
GO TO 580
C EWT(i) .le. 0.0 for some i (not at start of problem). ----------------
510 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODIS- At T (=R1), EWT(I1) has become R2 .le. 0.'
CALL XERRWD (MSG, 50, 202, 0, 1, I, 0, 2, TN, EWTI)
ISTATE = -6
GO TO 590
C Too much accuracy requested for machine precision. -------------------
520 MSG = 'DLSODIS- At T (=R1), too much accuracy requested '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' for precision of machine.. See TOLSF (=R2) '
CALL XERRWD (MSG, 50, 203, 0, 0, 0, 0, 2, TN, TOLSF)
RWORK(14) = TOLSF
ISTATE = -2
GO TO 590
C KFLAG = -1. Error test failed repeatedly or with ABS(H) = HMIN. -----
530 MSG = 'DLSODIS- At T (=R1) and step size H (=R2), the '
CALL XERRWD (MSG, 50, 204, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' error test failed repeatedly or with ABS(H) = HMIN '
CALL XERRWD (MSG, 60, 204, 0, 0, 0, 0, 2, TN, H)
ISTATE = -4
GO TO 570
C KFLAG = -2. Convergence failed repeatedly or with ABS(H) = HMIN. ----
540 MSG = 'DLSODIS- At T (=R1) and step size H (=R2), the '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' corrector convergence failed repeatedly '
CALL XERRWD (MSG, 50, 205, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' or with ABS(H) = HMIN '
CALL XERRWD (MSG, 30, 205, 0, 0, 0, 0, 2, TN, H)
ISTATE = -5
GO TO 570
C IRES = 3 returned by RES, despite retries by DSTODI. -----------------
550 MSG = 'DLSODIS- At T (=R1) residual routine returned '
CALL XERRWD (MSG, 50, 206, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' error IRES = 3 repeatedly.'
CALL XERRWD (MSG, 30, 206, 1, 0, 0, 0, 0, TN, 0.0D0)
ISTATE = -7
GO TO 590
C KFLAG = -5. Fatal error flag returned by DPRJIS or DSOLSS (CDRV). ---
555 MSG = 'DLSODIS- At T (=R1) and step size H (=R2), a fatal'
CALL XERRWD (MSG, 50, 207, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' error flag was returned by CDRV (by way of '
CALL XERRWD (MSG, 50, 207, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' Subroutine DPRJIS or DSOLSS) '
CALL XERRWD (MSG, 40, 207, 0, 0, 0, 0, 2, TN, H)
ISTATE = -9
GO TO 580
C DAINVGS failed because matrix A was singular. ------------------------
560 MSG='DLSODIS- Attempt to initialize dy/dt failed because matrix A'
CALL XERRWD (MSG, 60, 208, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' was singular. CDRV returned zero pivot error flag. '
CALL XERRWD (MSG, 60, 208, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = 'DAINVGS set its error flag to IER = (I1)'
CALL XERRWD (MSG, 40, 208, 0, 1, IER, 0, 0, 0.0D0, 0.0D0)
ISTATE = -8
RETURN
C DAINVGS failed because RES set IRES to 2 or 3. -----------------------
565 MSG = 'DLSODIS- Attempt to initialize dy/dt failed '
CALL XERRWD (MSG, 50, 209, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' because residual routine set its error flag '
CALL XERRWD (MSG, 50, 209, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' to IRES = (I1)'
CALL XERRWD (MSG, 20, 209, 0, 1, IER, 0, 0, 0.0D0, 0.0D0)
ISTATE = -8
RETURN
C Compute IMXER if relevant. -------------------------------------------
570 BIG = 0.0D0
IMXER = 1
DO 575 I = 1,N
SIZE = ABS(RWORK(I+LACOR-1)*RWORK(I+LEWT-1))
IF (BIG .GE. SIZE) GO TO 575
BIG = SIZE
IMXER = I
575 CONTINUE
IWORK(16) = IMXER
C Compute residual if relevant. ----------------------------------------
580 LYD0 = LYH + NYH
DO 585 I = 1, N
RWORK(I+LSAVF-1) = RWORK(I+LYD0-1) / H
585 Y(I) = RWORK(I+LYH-1)
IRES = 1
CALL RES (NEQ, TN, Y, RWORK(LSAVF), YDOTI, IRES)
NFE = NFE + 1
IF ( IRES .LE. 1 ) GO TO 595
MSG = 'DLSODIS- Residual routine set its flag IRES '
CALL XERRWD (MSG, 50, 210, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG = ' to (I1) when called for final output. '
CALL XERRWD (MSG, 50, 210, 0, 1, IRES, 0, 0, 0.0D0, 0.0D0)
GO TO 595
C set y vector, t, and optional outputs. -------------------------------
590 DO 592 I = 1,N
592 Y(I) = RWORK(I+LYH-1)
595 T = TN
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NJE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(19) = NNZ
IWORK(20) = NGP
IWORK(21) = NLU
IWORK(25) = NZL
IWORK(26) = NZU
RETURN
C-----------------------------------------------------------------------
C Block I.
C The following block handles all error returns due to illegal input
C (ISTATE = -3), as detected before calling the core integrator.
C First the error message routine is called. If the illegal input
C is a negative ISTATE, the run is aborted (apparent infinite loop).
C-----------------------------------------------------------------------
601 MSG = 'DLSODIS- ISTATE (=I1) illegal.'
CALL XERRWD (MSG, 30, 1, 0, 1, ISTATE, 0, 0, 0.0D0, 0.0D0)
IF (ISTATE .LT. 0) GO TO 800
GO TO 700
602 MSG = 'DLSODIS- ITASK (=I1) illegal. '
CALL XERRWD (MSG, 30, 2, 0, 1, ITASK, 0, 0, 0.0D0, 0.0D0)
GO TO 700
603 MSG = 'DLSODIS-ISTATE .gt. 1 but DLSODIS not initialized.'
CALL XERRWD (MSG, 50, 3, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
GO TO 700
604 MSG = 'DLSODIS- NEQ (=I1) .lt. 1 '
CALL XERRWD (MSG, 30, 4, 0, 1, NEQ(1), 0, 0, 0.0D0, 0.0D0)
GO TO 700
605 MSG = 'DLSODIS- ISTATE = 3 and NEQ increased (I1 to I2). '
CALL XERRWD (MSG, 50, 5, 0, 2, N, NEQ(1), 0, 0.0D0, 0.0D0)
GO TO 700
606 MSG = 'DLSODIS- ITOL (=I1) illegal. '
CALL XERRWD (MSG, 30, 6, 0, 1, ITOL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
607 MSG = 'DLSODIS- IOPT (=I1) illegal. '
CALL XERRWD (MSG, 30, 7, 0, 1, IOPT, 0, 0, 0.0D0, 0.0D0)
GO TO 700
608 MSG = 'DLSODIS- MF (=I1) illegal. '
CALL XERRWD (MSG, 30, 8, 0, 1, MF, 0, 0, 0.0D0, 0.0D0)
GO TO 700
611 MSG = 'DLSODIS- MAXORD (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 11, 0, 1, MAXORD, 0, 0, 0.0D0, 0.0D0)
GO TO 700
612 MSG = 'DLSODIS- MXSTEP (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 12, 0, 1, MXSTEP, 0, 0, 0.0D0, 0.0D0)
GO TO 700
613 MSG = 'DLSODIS- MXHNIL (=I1) .lt. 0 '
CALL XERRWD (MSG, 30, 13, 0, 1, MXHNIL, 0, 0, 0.0D0, 0.0D0)
GO TO 700
614 MSG = 'DLSODIS- TOUT (=R1) behind T (=R2) '
CALL XERRWD (MSG, 40, 14, 0, 0, 0, 0, 2, TOUT, T)
MSG = ' Integration direction is given by H0 (=R1) '
CALL XERRWD (MSG, 50, 14, 0, 0, 0, 0, 1, H0, 0.0D0)
GO TO 700
615 MSG = 'DLSODIS- HMAX (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 15, 0, 0, 0, 0, 1, HMAX, 0.0D0)
GO TO 700
616 MSG = 'DLSODIS- HMIN (=R1) .lt. 0.0 '
CALL XERRWD (MSG, 30, 16, 0, 0, 0, 0, 1, HMIN, 0.0D0)
GO TO 700
617 MSG = 'DLSODIS- RWORK length is insufficient to proceed. '
CALL XERRWD (MSG, 50, 17, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' Length needed is .ge. LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 17, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
618 MSG = 'DLSODIS- IWORK length is insufficient to proceed. '
CALL XERRWD (MSG, 50, 18, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' Length needed is .ge. LENIW (=I1), exceeds LIW (=I2)'
CALL XERRWD (MSG, 60, 18, 0, 2, LENIW, LIW, 0, 0.0D0, 0.0D0)
GO TO 700
619 MSG = 'DLSODIS- RTOL(=I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 19, 0, 1, I, 0, 1, RTOLI, 0.0D0)
GO TO 700
620 MSG = 'DLSODIS- ATOL(=I1) is R1 .lt. 0.0 '
CALL XERRWD (MSG, 40, 20, 0, 1, I, 0, 1, ATOLI, 0.0D0)
GO TO 700
621 EWTI = RWORK(LEWT+I-1)
MSG = 'DLSODIS- EWT(I1) is R1 .le. 0.0 '
CALL XERRWD (MSG, 40, 21, 0, 1, I, 0, 1, EWTI, 0.0D0)
GO TO 700
622 MSG='DLSODIS- TOUT(=R1) too close to T(=R2) to start integration.'
CALL XERRWD (MSG, 60, 22, 0, 0, 0, 0, 2, TOUT, T)
GO TO 700
623 MSG='DLSODIS- ITASK = I1 and TOUT (=R1) behind TCUR - HU (= R2) '
CALL XERRWD (MSG, 60, 23, 0, 1, ITASK, 0, 2, TOUT, TP)
GO TO 700
624 MSG='DLSODIS- ITASK = 4 or 5 and TCRIT (=R1) behind TCUR (=R2) '
CALL XERRWD (MSG, 60, 24, 0, 0, 0, 0, 2, TCRIT, TN)
GO TO 700
625 MSG='DLSODIS- ITASK = 4 or 5 and TCRIT (=R1) behind TOUT (=R2) '
CALL XERRWD (MSG, 60, 25, 0, 0, 0, 0, 2, TCRIT, TOUT)
GO TO 700
626 MSG = 'DLSODIS- At start of problem, too much accuracy '
CALL XERRWD (MSG, 50, 26, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' requested for precision of machine.. See TOLSF (=R1) '
CALL XERRWD (MSG, 60, 26, 0, 0, 0, 0, 1, TOLSF, 0.0D0)
RWORK(14) = TOLSF
GO TO 700
627 MSG = 'DLSODIS- Trouble in DINTDY. ITASK = I1, TOUT = R1'
CALL XERRWD (MSG, 50, 27, 0, 1, ITASK, 0, 1, TOUT, 0.0D0)
GO TO 700
628 MSG='DLSODIS- RWORK length insufficient (for Subroutine DPREPI). '
CALL XERRWD (MSG, 60, 28, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' Length needed is .ge. LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 28, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
629 MSG='DLSODIS- RWORK length insufficient (for Subroutine JGROUP). '
CALL XERRWD (MSG, 60, 29, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' Length needed is .ge. LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 29, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
630 MSG='DLSODIS- RWORK length insufficient (for Subroutine ODRV). '
CALL XERRWD (MSG, 60, 30, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' Length needed is .ge. LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 30, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
631 MSG='DLSODIS- Error from ODRV in Yale Sparse Matrix Package. '
CALL XERRWD (MSG, 60, 31, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
IMUL = (IYS - 1)/N
IREM = IYS - IMUL*N
MSG=' At T (=R1), ODRV returned error flag = I1*NEQ + I2. '
CALL XERRWD (MSG, 60, 31, 0, 2, IMUL, IREM, 1, TN, 0.0D0)
GO TO 700
632 MSG='DLSODIS- RWORK length insufficient (for Subroutine CDRV). '
CALL XERRWD (MSG, 60, 32, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
MSG=' Length needed is .ge. LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWD (MSG, 60, 32, 0, 2, LENRW, LRW, 0, 0.0D0, 0.0D0)
GO TO 700
633 MSG='DLSODIS- Error from CDRV in Yale Sparse Matrix Package. '
CALL XERRWD (MSG, 60, 33, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
IMUL = (IYS - 1)/N
IREM = IYS - IMUL*N
MSG=' At T (=R1), CDRV returned error flag = I1*NEQ + I2. '
CALL XERRWD (MSG, 60, 33, 0, 2, IMUL, IREM, 1, TN, 0.0D0)
IF (IMUL .EQ. 2) THEN
MSG=' Duplicate entry in sparsity structure descriptors. '
CALL XERRWD (MSG, 60, 33, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
IF (IMUL .EQ. 3 .OR. IMUL .EQ. 6) THEN
MSG=' Insufficient storage for NSFC (called by CDRV). '
CALL XERRWD (MSG, 60, 33, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
ENDIF
GO TO 700
634 MSG='DLSODIS- At T (=R1) residual routine (called by DPREPI) '
CALL XERRWD (MSG, 60, 34, 0, 0, 0, 0, 0, 0.0D0, 0.0D0)
IER = -IPFLAG - 5
MSG = ' returned error IRES (=I1)'
CALL XERRWD (MSG, 30, 34, 0, 1, IER, 0, 1, TN, 0.0D0)
C
700 ISTATE = -3
RETURN
C
800 MSG = 'DLSODIS- Run aborted.. apparent infinite loop. '
CALL XERRWD (MSG, 50, 303, 2, 0, 0, 0, 0, 0.0D0, 0.0D0)
RETURN
C----------------------- End of Subroutine DLSODIS ---------------------
END