*DECK DDEBDF
SUBROUTINE DDEBDF (DF, NEQ, T, Y, TOUT, INFO, RTOL, ATOL, IDID,
+ RWORK, LRW, IWORK, LIW, RPAR, IPAR, DJAC)
C***BEGIN PROLOGUE DDEBDF
C***PURPOSE Solve an initial value problem in ordinary differential
C equations using backward differentiation formulas. It is
C intended primarily for stiff problems.
C***LIBRARY SLATEC (DEPAC)
C***CATEGORY I1A2
C***TYPE DOUBLE PRECISION (DEBDF-S, DDEBDF-D)
C***KEYWORDS BACKWARD DIFFERENTIATION FORMULAS, DEPAC,
C INITIAL VALUE PROBLEMS, ODE,
C ORDINARY DIFFERENTIAL EQUATIONS, STIFF
C***AUTHOR Shampine, L. F., (SNLA)
C Watts, H. A., (SNLA)
C***DESCRIPTION
C
C This is the backward differentiation code in the package of
C differential equation solvers DEPAC, consisting of the codes
C DDERKF, DDEABM, and DDEBDF. Design of the package was by
C L. F. Shampine and H. A. Watts. It is documented in
C SAND-79-2374 , DEPAC - Design of a User Oriented Package of ODE
C Solvers.
C DDEBDF is a driver for a modification of the code LSODE written by
C A. C. Hindmarsh
C Lawrence Livermore Laboratory
C Livermore, California 94550
C
C **********************************************************************
C ** DEPAC PACKAGE OVERVIEW **
C **********************************************************************
C
C You have a choice of three differential equation solvers from
C DEPAC. The following brief descriptions are meant to aid you
C in choosing the most appropriate code for your problem.
C
C DDERKF is a fifth order Runge-Kutta code. It is the simplest of
C the three choices, both algorithmically and in the use of the
C code. DDERKF is primarily designed to solve non-stiff and mild-
C ly stiff differential equations when derivative evaluations are
C not expensive. It should generally not be used to get high
C accuracy results nor answers at a great many specific points.
C Because DDERKF has very low overhead costs, it will usually
C result in the least expensive integration when solving
C problems requiring a modest amount of accuracy and having
C equations that are not costly to evaluate. DDERKF attempts to
C discover when it is not suitable for the task posed.
C
C DDEABM is a variable order (one through twelve) Adams code. Its
C complexity lies somewhere between that of DDERKF and DDEBDF.
C DDEABM is primarily designed to solve non-stiff and mildly
C stiff differential equations when derivative evaluations are
C expensive, high accuracy results are needed or answers at
C many specific points are required. DDEABM attempts to discover
C when it is not suitable for the task posed.
C
C DDEBDF is a variable order (one through five) backward
C differentiation formula code. It is the most complicated of
C the three choices. DDEBDF is primarily designed to solve stiff
C differential equations at crude to moderate tolerances.
C If the problem is very stiff at all, DDERKF and DDEABM will be
C quite inefficient compared to DDEBDF. However, DDEBDF will be
C inefficient compared to DDERKF and DDEABM on non-stiff problems
C because it uses much more storage, has a much larger overhead,
C and the low order formulas will not give high accuracies
C efficiently.
C
C The concept of stiffness cannot be described in a few words.
C If you do not know the problem to be stiff, try either DDERKF
C or DDEABM. Both of these codes will inform you of stiffness
C when the cost of solving such problems becomes important.
C
C **********************************************************************
C ** ABSTRACT **
C **********************************************************************
C
C Subroutine DDEBDF uses the backward differentiation formulas of
C orders one through five to integrate a system of NEQ first order
C ordinary differential equations of the form
C DU/DX = DF(X,U)
C when the vector Y(*) of initial values for U(*) at X=T is given.
C The subroutine integrates from T to TOUT. It is easy to continue the
C integration to get results at additional TOUT. This is the interval
C mode of operation. It is also easy for the routine to return with
C the solution at each intermediate step on the way to TOUT. This is
C the intermediate-output mode of operation.
C
C **********************************************************************
C * Description of The Arguments To DDEBDF (An Overview) *
C **********************************************************************
C
C The Parameters are:
C
C DF -- This is the name of a subroutine which you provide to
C define the differential equations.
C
C NEQ -- This is the number of (first order) differential
C equations to be integrated.
C
C T -- This is a DOUBLE PRECISION value of the independent
C variable.
C
C Y(*) -- This DOUBLE PRECISION array contains the solution
C components at T.
C
C TOUT -- This is a DOUBLE PRECISION point at which a solution is
C desired.
C
C INFO(*) -- The basic task of the code is to integrate the
C differential equations from T to TOUT and return an
C answer at TOUT. INFO(*) is an INTEGER array which is used
C to communicate exactly how you want this task to be
C carried out.
C
C RTOL, ATOL -- These DOUBLE PRECISION quantities
C represent relative and absolute error tolerances which you
C provide to indicate how accurately you wish the solution
C to be computed. You may choose them to be both scalars
C or else both vectors.
C
C IDID -- This scalar quantity is an indicator reporting what
C the code did. You must monitor this INTEGER variable to
C decide what action to take next.
C
C RWORK(*), LRW -- RWORK(*) is a DOUBLE PRECISION work array of
C length LRW which provides the code with needed storage
C space.
C
C IWORK(*), LIW -- IWORK(*) is an INTEGER work array of length LIW
C which provides the code with needed storage space and an
C across call flag.
C
C RPAR, IPAR -- These are DOUBLE PRECISION and INTEGER parameter
C arrays which you can use for communication between your
C calling program and the DF subroutine (and the DJAC
C subroutine).
C
C DJAC -- This is the name of a subroutine which you may choose to
C provide for defining the Jacobian matrix of partial
C derivatives DF/DU.
C
C Quantities which are used as input items are
C NEQ, T, Y(*), TOUT, INFO(*),
C RTOL, ATOL, RWORK(1), LRW,
C IWORK(1), IWORK(2), and LIW.
C
C Quantities which may be altered by the code are
C T, Y(*), INFO(1), RTOL, ATOL,
C IDID, RWORK(*) and IWORK(*).
C
C **********************************************************************
C * INPUT -- What To Do On The First Call To DDEBDF *
C **********************************************************************
C
C The first call of the code is defined to be the start of each new
C problem. Read through the descriptions of all the following items,
C provide sufficient storage space for designated arrays, set
C appropriate variables for the initialization of the problem, and
C give information about how you want the problem to be solved.
C
C
C DF -- Provide a subroutine of the form
C DF(X,U,UPRIME,RPAR,IPAR)
C to define the system of first order differential equations
C which is to be solved. For the given values of X and the
C vector U(*)=(U(1),U(2),...,U(NEQ)) , the subroutine must
C evaluate the NEQ components of the system of differential
C equations DU/DX=DF(X,U) and store the derivatives in the
C array UPRIME(*), that is, UPRIME(I) = * DU(I)/DX * for
C equations I=1,...,NEQ.
C
C Subroutine DF must not alter X or U(*). You must declare
C the name DF in an external statement in your program that
C calls DDEBDF. You must dimension U and UPRIME in DF.
C
C RPAR and IPAR are DOUBLE PRECISION and INTEGER parameter
C arrays which you can use for communication between your
C calling program and subroutine DF. They are not used or
C altered by DDEBDF. If you do not need RPAR or IPAR,
C ignore these parameters by treating them as dummy
C arguments. If you do choose to use them, dimension them in
C your calling program and in DF as arrays of appropriate
C length.
C
C NEQ -- Set it to the number of differential equations.
C (NEQ .GE. 1)
C
C T -- Set it to the initial point of the integration.
C You must use a program variable for T because the code
C changes its value.
C
C Y(*) -- Set this vector to the initial values of the NEQ solution
C components at the initial point. You must dimension Y at
C least NEQ in your calling program.
C
C TOUT -- Set it to the first point at which a solution is desired.
C You can take TOUT = T, in which case the code
C will evaluate the derivative of the solution at T and
C return. Integration either forward in T (TOUT .GT. T)
C or backward in T (TOUT .LT. T) is permitted.
C
C The code advances the solution from T to TOUT using
C step sizes which are automatically selected so as to
C achieve the desired accuracy. If you wish, the code will
C return with the solution and its derivative following
C each intermediate step (intermediate-output mode) so that
C you can monitor them, but you still must provide TOUT in
C accord with the basic aim of the code.
C
C The first step taken by the code is a critical one
C because it must reflect how fast the solution changes near
C the initial point. The code automatically selects an
C initial step size which is practically always suitable for
C the problem. By using the fact that the code will not
C step past TOUT in the first step, you could, if necessary,
C restrict the length of the initial step size.
C
C For some problems it may not be permissible to integrate
C past a point TSTOP because a discontinuity occurs there
C or the solution or its derivative is not defined beyond
C TSTOP. When you have declared a TSTOP point (see INFO(4)
C and RWORK(1)), you have told the code not to integrate
C past TSTOP. In this case any TOUT beyond TSTOP is invalid
C input.
C
C INFO(*) -- Use the INFO array to give the code more details about
C how you want your problem solved. This array should be
C dimensioned of length 15 to accommodate other members of
C DEPAC or possible future extensions, though DDEBDF uses
C only the first six entries. You must respond to all of
C the following items which are arranged as questions. The
C simplest use of the code corresponds to answering all
C questions as YES ,i.e. setting all entries of INFO to 0.
C
C INFO(1) -- This parameter enables the code to initialize
C itself. You must set it to indicate the start of every
C new problem.
C
C **** Is this the first call for this problem ...
C YES -- Set INFO(1) = 0
C NO -- Not applicable here.
C See below for continuation calls. ****
C
C INFO(2) -- How much accuracy you want of your solution
C is specified by the error tolerances RTOL and ATOL.
C The simplest use is to take them both to be scalars.
C To obtain more flexibility, they can both be vectors.
C The code must be told your choice.
C
C **** Are both error tolerances RTOL, ATOL scalars ...
C YES -- Set INFO(2) = 0
C and input scalars for both RTOL and ATOL
C NO -- Set INFO(2) = 1
C and input arrays for both RTOL and ATOL ****
C
C INFO(3) -- The code integrates from T in the direction
C of TOUT by steps. If you wish, it will return the
C computed solution and derivative at the next
C intermediate step (the intermediate-output mode) or
C TOUT, whichever comes first. This is a good way to
C proceed if you want to see the behavior of the solution.
C If you must have solutions at a great many specific
C TOUT points, this code will compute them efficiently.
C
C **** Do you want the solution only at
C TOUT (and NOT at the next intermediate step) ...
C YES -- Set INFO(3) = 0
C NO -- Set INFO(3) = 1 ****
C
C INFO(4) -- To handle solutions at a great many specific
C values TOUT efficiently, this code may integrate past
C TOUT and interpolate to obtain the result at TOUT.
C Sometimes it is not possible to integrate beyond some
C point TSTOP because the equation changes there or it is
C not defined past TSTOP. Then you must tell the code
C not to go past.
C
C **** Can the integration be carried out without any
C restrictions on the independent variable T ...
C YES -- Set INFO(4)=0
C NO -- Set INFO(4)=1
C and define the stopping point TSTOP by
C setting RWORK(1)=TSTOP ****
C
C INFO(5) -- To solve stiff problems it is necessary to use the
C Jacobian matrix of partial derivatives of the system
C of differential equations. If you do not provide a
C subroutine to evaluate it analytically (see the
C description of the item DJAC in the call list), it will
C be approximated by numerical differencing in this code.
C Although it is less trouble for you to have the code
C compute partial derivatives by numerical differencing,
C the solution will be more reliable if you provide the
C derivatives via DJAC. Sometimes numerical differencing
C is cheaper than evaluating derivatives in DJAC and
C sometimes it is not - this depends on your problem.
C
C If your problem is linear, i.e. has the form
C DU/DX = DF(X,U) = J(X)*U + G(X) for some matrix J(X)
C and vector G(X), the Jacobian matrix DF/DU = J(X).
C Since you must provide a subroutine to evaluate DF(X,U)
C analytically, it is little extra trouble to provide
C subroutine DJAC for evaluating J(X) analytically.
C Furthermore, in such cases, numerical differencing is
C much more expensive than analytic evaluation.
C
C **** Do you want the code to evaluate the partial
C derivatives automatically by numerical differences ...
C YES -- Set INFO(5)=0
C NO -- Set INFO(5)=1
C and provide subroutine DJAC for evaluating the
C Jacobian matrix ****
C
C INFO(6) -- DDEBDF will perform much better if the Jacobian
C matrix is banded and the code is told this. In this
C case, the storage needed will be greatly reduced,
C numerical differencing will be performed more cheaply,
C and a number of important algorithms will execute much
C faster. The differential equation is said to have
C half-bandwidths ML (lower) and MU (upper) if equation I
C involves only unknowns Y(J) with
C I-ML .LE. J .LE. I+MU
C for all I=1,2,...,NEQ. Thus, ML and MU are the widths
C of the lower and upper parts of the band, respectively,
C with the main diagonal being excluded. If you do not
C indicate that the equation has a banded Jacobian,
C the code works with a full matrix of NEQ**2 elements
C (stored in the conventional way). Computations with
C banded matrices cost less time and storage than with
C full matrices if 2*ML+MU .LT. NEQ. If you tell the
C code that the Jacobian matrix has a banded structure and
C you want to provide subroutine DJAC to compute the
C partial derivatives, then you must be careful to store
C the elements of the Jacobian matrix in the special form
C indicated in the description of DJAC.
C
C **** Do you want to solve the problem using a full
C (dense) Jacobian matrix (and not a special banded
C structure) ...
C YES -- Set INFO(6)=0
C NO -- Set INFO(6)=1
C and provide the lower (ML) and upper (MU)
C bandwidths by setting
C IWORK(1)=ML
C IWORK(2)=MU ****
C
C RTOL, ATOL -- You must assign relative (RTOL) and absolute (ATOL)
C error tolerances to tell the code how accurately you want
C the solution to be computed. They must be defined as
C program variables because the code may change them. You
C have two choices --
C Both RTOL and ATOL are scalars. (INFO(2)=0)
C Both RTOL and ATOL are vectors. (INFO(2)=1)
C In either case all components must be non-negative.
C
C The tolerances are used by the code in a local error test
C at each step which requires roughly that
C ABS(LOCAL ERROR) .LE. RTOL*ABS(Y)+ATOL
C for each vector component.
C (More specifically, a root-mean-square norm is used to
C measure the size of vectors, and the error test uses the
C magnitude of the solution at the beginning of the step.)
C
C The true (global) error is the difference between the true
C solution of the initial value problem and the computed
C approximation. Practically all present day codes,
C including this one, control the local error at each step
C and do not even attempt to control the global error
C directly. Roughly speaking, they produce a solution Y(T)
C which satisfies the differential equations with a
C residual R(T), DY(T)/DT = DF(T,Y(T)) + R(T) ,
C and, almost always, R(T) is bounded by the error
C tolerances. Usually, but not always, the true accuracy of
C the computed Y is comparable to the error tolerances. This
C code will usually, but not always, deliver a more accurate
C solution if you reduce the tolerances and integrate again.
C By comparing two such solutions you can get a fairly
C reliable idea of the true error in the solution at the
C bigger tolerances.
C
C Setting ATOL=0. results in a pure relative error test on
C that component. Setting RTOL=0. results in a pure abso-
C lute error test on that component. A mixed test with non-
C zero RTOL and ATOL corresponds roughly to a relative error
C test when the solution component is much bigger than ATOL
C and to an absolute error test when the solution component
C is smaller than the threshold ATOL.
C
C Proper selection of the absolute error control parameters
C ATOL requires you to have some idea of the scale of the
C solution components. To acquire this information may mean
C that you will have to solve the problem more than once. In
C the absence of scale information, you should ask for some
C relative accuracy in all the components (by setting RTOL
C values non-zero) and perhaps impose extremely small
C absolute error tolerances to protect against the danger of
C a solution component becoming zero.
C
C The code will not attempt to compute a solution at an
C accuracy unreasonable for the machine being used. It will
C advise you if you ask for too much accuracy and inform
C you as to the maximum accuracy it believes possible.
C
C RWORK(*) -- Dimension this DOUBLE PRECISION work array of length
C LRW in your calling program.
C
C RWORK(1) -- If you have set INFO(4)=0, you can ignore this
C optional input parameter. Otherwise you must define a
C stopping point TSTOP by setting RWORK(1) = TSTOP.
C (For some problems it may not be permissible to integrate
C past a point TSTOP because a discontinuity occurs there
C or the solution or its derivative is not defined beyond
C TSTOP.)
C
C LRW -- Set it to the declared length of the RWORK array.
C You must have
C LRW .GE. 250+10*NEQ+NEQ**2
C for the full (dense) Jacobian case (when INFO(6)=0), or
C LRW .GE. 250+10*NEQ+(2*ML+MU+1)*NEQ
C for the banded Jacobian case (when INFO(6)=1).
C
C IWORK(*) -- Dimension this INTEGER work array of length LIW in
C your calling program.
C
C IWORK(1), IWORK(2) -- If you have set INFO(6)=0, you can ignore
C these optional input parameters. Otherwise you must define
C the half-bandwidths ML (lower) and MU (upper) of the
C Jacobian matrix by setting IWORK(1) = ML and
C IWORK(2) = MU. (The code will work with a full matrix
C of NEQ**2 elements unless it is told that the problem has
C a banded Jacobian, in which case the code will work with
C a matrix containing at most (2*ML+MU+1)*NEQ elements.)
C
C LIW -- Set it to the declared length of the IWORK array.
C You must have LIW .GE. 56+NEQ.
C
C RPAR, IPAR -- These are parameter arrays, of DOUBLE PRECISION and
C INTEGER type, respectively. You can use them for
C communication between your program that calls DDEBDF and
C the DF subroutine (and the DJAC subroutine). They are not
C used or altered by DDEBDF. If you do not need RPAR or
C IPAR, ignore these parameters by treating them as dummy
C arguments. If you do choose to use them, dimension them in
C your calling program and in DF (and in DJAC) as arrays of
C appropriate length.
C
C DJAC -- If you have set INFO(5)=0, you can ignore this parameter
C by treating it as a dummy argument. (For some compilers
C you may have to write a dummy subroutine named DJAC in
C order to avoid problems associated with missing external
C routine names.) Otherwise, you must provide a subroutine
C of the form
C DJAC(X,U,PD,NROWPD,RPAR,IPAR)
C to define the Jacobian matrix of partial derivatives DF/DU
C of the system of differential equations DU/DX = DF(X,U).
C For the given values of X and the vector
C U(*)=(U(1),U(2),...,U(NEQ)), the subroutine must evaluate
C the non-zero partial derivatives DF(I)/DU(J) for each
C differential equation I=1,...,NEQ and each solution
C component J=1,...,NEQ , and store these values in the
C matrix PD. The elements of PD are set to zero before each
C call to DJAC so only non-zero elements need to be defined.
C
C Subroutine DJAC must not alter X, U(*), or NROWPD. You
C must declare the name DJAC in an external statement in
C your program that calls DDEBDF. NROWPD is the row
C dimension of the PD matrix and is assigned by the code.
C Therefore you must dimension PD in DJAC according to
C DIMENSION PD(NROWPD,1)
C You must also dimension U in DJAC.
C
C The way you must store the elements into the PD matrix
C depends on the structure of the Jacobian which you
C indicated by INFO(6).
C *** INFO(6)=0 -- Full (Dense) Jacobian ***
C When you evaluate the (non-zero) partial derivative
C of equation I with respect to variable J, you must
C store it in PD according to
C PD(I,J) = * DF(I)/DU(J) *
C *** INFO(6)=1 -- Banded Jacobian with ML Lower and MU
C Upper Diagonal Bands (refer to INFO(6) description of
C ML and MU) ***
C When you evaluate the (non-zero) partial derivative
C of equation I with respect to variable J, you must
C store it in PD according to
C IROW = I - J + ML + MU + 1
C PD(IROW,J) = * DF(I)/DU(J) *
C
C RPAR and IPAR are DOUBLE PRECISION and INTEGER parameter
C arrays which you can use for communication between your
C calling program and your Jacobian subroutine DJAC. They
C are not altered by DDEBDF. If you do not need RPAR or
C IPAR, ignore these parameters by treating them as dummy
C arguments. If you do choose to use them, dimension them
C in your calling program and in DJAC as arrays of
C appropriate length.
C
C **********************************************************************
C * OUTPUT -- After any return from DDEBDF *
C **********************************************************************
C
C The principal aim of the code is to return a computed solution at
C TOUT, although it is also possible to obtain intermediate results
C along the way. To find out whether the code achieved its goal
C or if the integration process was interrupted before the task was
C completed, you must check the IDID parameter.
C
C
C T -- The solution was successfully advanced to the
C output value of T.
C
C Y(*) -- Contains the computed solution approximation at T.
C You may also be interested in the approximate derivative
C of the solution at T. It is contained in
C RWORK(21),...,RWORK(20+NEQ).
C
C IDID -- Reports what the code did
C
C *** Task Completed ***
C Reported by positive values of IDID
C
C IDID = 1 -- A step was successfully taken in the
C intermediate-output mode. The code has not
C yet reached TOUT.
C
C IDID = 2 -- The integration to TOUT was successfully
C completed (T=TOUT) by stepping exactly to TOUT.
C
C IDID = 3 -- The integration to TOUT was successfully
C completed (T=TOUT) by stepping past TOUT.
C Y(*) is obtained by interpolation.
C
C *** Task Interrupted ***
C Reported by negative values of IDID
C
C IDID = -1 -- A large amount of work has been expended.
C (500 steps attempted)
C
C IDID = -2 -- The error tolerances are too stringent.
C
C IDID = -3 -- The local error test cannot be satisfied
C because you specified a zero component in ATOL
C and the corresponding computed solution
C component is zero. Thus, a pure relative error
C test is impossible for this component.
C
C IDID = -4,-5 -- Not applicable for this code but used
C by other members of DEPAC.
C
C IDID = -6 -- DDEBDF had repeated convergence test failures
C on the last attempted step.
C
C IDID = -7 -- DDEBDF had repeated error test failures on
C the last attempted step.
C
C IDID = -8,..,-32 -- Not applicable for this code but
C used by other members of DEPAC or possible
C future extensions.
C
C *** Task Terminated ***
C Reported by the value of IDID=-33
C
C IDID = -33 -- The code has encountered trouble from which
C it cannot recover. A message is printed
C explaining the trouble and control is returned
C to the calling program. For example, this
C occurs when invalid input is detected.
C
C RTOL, ATOL -- These quantities remain unchanged except when
C IDID = -2. In this case, the error tolerances have been
C increased by the code to values which are estimated to be
C appropriate for continuing the integration. However, the
C reported solution at T was obtained using the input values
C of RTOL and ATOL.
C
C RWORK, IWORK -- Contain information which is usually of no
C interest to the user but necessary for subsequent calls.
C However, you may find use for
C
C RWORK(11)--which contains the step size H to be
C attempted on the next step.
C
C RWORK(12)--If the tolerances have been increased by the
C code (IDID = -2) , they were multiplied by the
C value in RWORK(12).
C
C RWORK(13)--which contains the current value of the
C independent variable, i.e. the farthest point
C integration has reached. This will be
C different from T only when interpolation has
C been performed (IDID=3).
C
C RWORK(20+I)--which contains the approximate derivative
C of the solution component Y(I). In DDEBDF, it
C is never obtained by calling subroutine DF to
C evaluate the differential equation using T and
C Y(*), except at the initial point of
C integration.
C
C **********************************************************************
C ** INPUT -- What To Do To Continue The Integration **
C ** (calls after the first) **
C **********************************************************************
C
C This code is organized so that subsequent calls to continue the
C integration involve little (if any) additional effort on your
C part. You must monitor the IDID parameter in order to determine
C what to do next.
C
C Recalling that the principal task of the code is to integrate
C from T to TOUT (the interval mode), usually all you will need
C to do is specify a new TOUT upon reaching the current TOUT.
C
C Do not alter any quantity not specifically permitted below,
C in particular do not alter NEQ, T, Y(*), RWORK(*), IWORK(*) or
C the differential equation in subroutine DF. Any such alteration
C constitutes a new problem and must be treated as such, i.e.
C you must start afresh.
C
C You cannot change from vector to scalar error control or vice
C versa (INFO(2)) but you can change the size of the entries of
C RTOL, ATOL. Increasing a tolerance makes the equation easier
C to integrate. Decreasing a tolerance will make the equation
C harder to integrate and should generally be avoided.
C
C You can switch from the intermediate-output mode to the
C interval mode (INFO(3)) or vice versa at any time.
C
C If it has been necessary to prevent the integration from going
C past a point TSTOP (INFO(4), RWORK(1)), keep in mind that the
C code will not integrate to any TOUT beyond the currently
C specified TSTOP. Once TSTOP has been reached you must change
C the value of TSTOP or set INFO(4)=0. You may change INFO(4)
C or TSTOP at any time but you must supply the value of TSTOP in
C RWORK(1) whenever you set INFO(4)=1.
C
C Do not change INFO(5), INFO(6), IWORK(1), or IWORK(2)
C unless you are going to restart the code.
C
C The parameter INFO(1) is used by the code to indicate the
C beginning of a new problem and to indicate whether integration
C is to be continued. You must input the value INFO(1) = 0
C when starting a new problem. You must input the value
C INFO(1) = 1 if you wish to continue after an interrupted task.
C Do not set INFO(1) = 0 on a continuation call unless you
C want the code to restart at the current T.
C
C *** Following a Completed Task ***
C If
C IDID = 1, call the code again to continue the integration
C another step in the direction of TOUT.
C
C IDID = 2 or 3, define a new TOUT and call the code again.
C TOUT must be different from T. You cannot change
C the direction of integration without restarting.
C
C *** Following an Interrupted Task ***
C To show the code that you realize the task was
C interrupted and that you want to continue, you
C must take appropriate action and reset INFO(1) = 1
C If
C IDID = -1, the code has attempted 500 steps.
C If you want to continue, set INFO(1) = 1 and
C call the code again. An additional 500 steps
C will be allowed.
C
C IDID = -2, the error tolerances RTOL, ATOL have been
C increased to values the code estimates appropriate
C for continuing. You may want to change them
C yourself. If you are sure you want to continue
C with relaxed error tolerances, set INFO(1)=1 and
C call the code again.
C
C IDID = -3, a solution component is zero and you set the
C corresponding component of ATOL to zero. If you
C are sure you want to continue, you must first
C alter the error criterion to use positive values
C for those components of ATOL corresponding to zero
C solution components, then set INFO(1)=1 and call
C the code again.
C
C IDID = -4,-5 --- cannot occur with this code but used
C by other members of DEPAC.
C
C IDID = -6, repeated convergence test failures occurred
C on the last attempted step in DDEBDF. An inaccu-
C rate Jacobian may be the problem. If you are
C absolutely certain you want to continue, restart
C the integration at the current T by setting
C INFO(1)=0 and call the code again.
C
C IDID = -7, repeated error test failures occurred on the
C last attempted step in DDEBDF. A singularity in
C the solution may be present. You should re-
C examine the problem being solved. If you are
C absolutely certain you want to continue, restart
C the integration at the current T by setting
C INFO(1)=0 and call the code again.
C
C IDID = -8,..,-32 --- cannot occur with this code but
C used by other members of DDEPAC or possible future
C extensions.
C
C *** Following a Terminated Task ***
C If
C IDID = -33, you cannot continue the solution of this
C problem. An attempt to do so will result in your
C run being terminated.
C
C **********************************************************************
C
C ***** Warning *****
C
C If DDEBDF is to be used in an overlay situation, you must save and
C restore certain items used internally by DDEBDF (values in the
C common block DDEBD1). This can be accomplished as follows.
C
C To save the necessary values upon return from DDEBDF, simply call
C DSVCO(RWORK(22+NEQ),IWORK(21+NEQ)).
C
C To restore the necessary values before the next call to DDEBDF,
C simply call DRSCO(RWORK(22+NEQ),IWORK(21+NEQ)).
C
C***REFERENCES L. F. Shampine and H. A. Watts, DEPAC - design of a user
C oriented package of ODE solvers, Report SAND79-2374,
C Sandia Laboratories, 1979.
C***ROUTINES CALLED DLSOD, XERMSG
C***COMMON BLOCKS DDEBD1
C***REVISION HISTORY (YYMMDD)
C 820301 DATE WRITTEN
C 890831 Modified array declarations. (WRB)
C 891024 Changed references from DVNORM to DHVNRM. (WRB)
C 891024 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 900326 Removed duplicate information from DESCRIPTION section.
C (WRB)
C 900510 Convert XERRWV calls to XERMSG calls, make Prologue comments
C consistent with DEBDF. (RWC)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE DDEBDF
INTEGER IACOR, IBAND, IBEGIN, ICOMI, ICOMR, IDELSN, IDID, IER,
1 IEWT, IINOUT, IINTEG, IJAC, ILRW, INFO, INIT,
2 IOWNS, IPAR, IQUIT, ISAVF, ITOL, ITSTAR, ITSTOP, IWM,
3 IWORK, IYH, IYPOUT, JSTART, KFLAG, KSTEPS, L, LIW, LRW,
4 MAXORD, METH, MITER, ML, MU, N, NEQ, NFE, NJE, NQ, NQU,
5 NST
DOUBLE PRECISION ATOL, EL0, H, HMIN, HMXI, HU, ROWNS, RPAR,
1 RTOL, RWORK, T, TN, TOLD, TOUT, UROUND, Y
LOGICAL INTOUT
CHARACTER*8 XERN1, XERN2
CHARACTER*16 XERN3
C
DIMENSION Y(*),INFO(15),RTOL(*),ATOL(*),RWORK(*),IWORK(*),
1 RPAR(*),IPAR(*)
C
COMMON /DDEBD1/ TOLD,ROWNS(210),EL0,H,HMIN,HMXI,HU,TN,UROUND,
1 IQUIT,INIT,IYH,IEWT,IACOR,ISAVF,IWM,KSTEPS,IBEGIN,
2 ITOL,IINTEG,ITSTOP,IJAC,IBAND,IOWNS(6),IER,JSTART,
3 KFLAG,L,METH,MITER,MAXORD,N,NQ,NST,NFE,NJE,NQU
C
EXTERNAL DF, DJAC
C
C CHECK FOR AN APPARENT INFINITE LOOP
C
C***FIRST EXECUTABLE STATEMENT DDEBDF
IF (INFO(1) .EQ. 0) IWORK(LIW) = 0
C
IF (IWORK(LIW).GE. 5) THEN
IF (T .EQ. RWORK(21+NEQ)) THEN
WRITE (XERN3, '(1PE15.6)') T
CALL XERMSG ('SLATEC', 'DDEBDF',
* 'AN APPARENT INFINITE LOOP HAS BEEN DETECTED.$$' //
* 'YOU HAVE MADE REPEATED CALLS AT T = ' // XERN3 //
* ' AND THE INTEGRATION HAS NOT ADVANCED. CHECK THE ' //
* 'WAY YOU HAVE SET PARAMETERS FOR THE CALL TO THE ' //
* 'CODE, PARTICULARLY INFO(1).', 13, 2)
RETURN
ENDIF
ENDIF
C
IDID = 0
C
C CHECK VALIDITY OF INFO PARAMETERS
C
IF (INFO(1) .NE. 0 .AND. INFO(1) .NE. 1) THEN
WRITE (XERN1, '(I8)') INFO(1)
CALL XERMSG ('SLATEC', 'DDEBDF', 'INFO(1) MUST BE SET TO 0 ' //
* 'FOR THE START OF A NEW PROBLEM, AND MUST BE SET TO 1 ' //
* 'FOLLOWING AN INTERRUPTED TASK. YOU ARE ATTEMPTING TO ' //
* 'CONTINUE THE INTEGRATION ILLEGALLY BY CALLING THE ' //
* 'CODE WITH INFO(1) = ' // XERN1, 3, 1)
IDID = -33
ENDIF
C
IF (INFO(2) .NE. 0 .AND. INFO(2) .NE. 1) THEN
WRITE (XERN1, '(I8)') INFO(2)
CALL XERMSG ('SLATEC', 'DDEBDF', 'INFO(2) MUST BE 0 OR 1 ' //
* 'INDICATING SCALAR AND VECTOR ERROR TOLERANCES, ' //
* 'RESPECTIVELY. YOU HAVE CALLED THE CODE WITH INFO(2) = ' //
* XERN1, 4, 1)
IDID = -33
ENDIF
C
IF (INFO(3) .NE. 0 .AND. INFO(3) .NE. 1) THEN
WRITE (XERN1, '(I8)') INFO(3)
CALL XERMSG ('SLATEC', 'DDEBDF', 'INFO(3) MUST BE 0 OR 1 ' //
* 'INDICATING THE INTERVAL OR INTERMEDIATE-OUTPUT MODE OF ' //
* 'INTEGRATION, RESPECTIVELY. YOU HAVE CALLED THE CODE ' //
* 'WITH INFO(3) = ' // XERN1, 5, 1)
IDID = -33
ENDIF
C
IF (INFO(4) .NE. 0 .AND. INFO(4) .NE. 1) THEN
WRITE (XERN1, '(I8)') INFO(4)
CALL XERMSG ('SLATEC', 'DDEBDF', 'INFO(4) MUST BE 0 OR 1 ' //
* 'INDICATING WHETHER OR NOT THE INTEGRATION INTERVAL IS ' //
* 'TO BE RESTRICTED BY A POINT TSTOP. YOU HAVE CALLED ' //
* 'THE CODE WITH INFO(4) = ' // XERN1, 14, 1)
IDID = -33
ENDIF
C
IF (INFO(5) .NE. 0 .AND. INFO(5) .NE. 1) THEN
WRITE (XERN1, '(I8)') INFO(5)
CALL XERMSG ('SLATEC', 'DDEBDF', 'INFO(5) MUST BE 0 OR 1 ' //
* 'INDICATING WHETHER THE CODE IS TOLD TO FORM THE ' //
* 'JACOBIAN MATRIX BY NUMERICAL DIFFERENCING OR YOU ' //
* 'PROVIDE A SUBROUTINE TO EVALUATE IT ANALYTICALLY. ' //
* 'YOU HAVE CALLED THE CODE WITH INFO(5) = ' // XERN1, 15, 1)
IDID = -33
ENDIF
C
IF (INFO(6) .NE. 0 .AND. INFO(6) .NE. 1) THEN
WRITE (XERN1, '(I8)') INFO(6)
CALL XERMSG ('SLATEC', 'DDEBDF', 'INFO(6) MUST BE 0 OR 1 ' //
* 'INDICATING WHETHER THE CODE IS TOLD TO TREAT THE ' //
* 'JACOBIAN AS A FULL (DENSE) MATRIX OR AS HAVING A ' //
* 'SPECIAL BANDED STRUCTURE. YOU HAVE CALLED THE CODE ' //
* 'WITH INFO(6) = ' // XERN1, 16, 1)
IDID = -33
ENDIF
C
ILRW = NEQ
IF (INFO(6) .NE. 0) THEN
C
C CHECK BANDWIDTH PARAMETERS
C
ML = IWORK(1)
MU = IWORK(2)
ILRW = 2*ML + MU + 1
C
IF (ML.LT.0 .OR. ML.GE.NEQ .OR. MU.LT.0 .OR. MU.GE.NEQ) THEN
WRITE (XERN1, '(I8)') ML
WRITE (XERN2, '(I8)') MU
CALL XERMSG ('SLATEC', 'DDEBDF', 'YOU HAVE SET INFO(6) ' //
* '= 1, TELLING THE CODE THAT THE JACOBIAN MATRIX HAS ' //
* 'A SPECIAL BANDED STRUCTURE. HOWEVER, THE LOWER ' //
* '(UPPER) BANDWIDTHS ML (MU) VIOLATE THE CONSTRAINTS ' //
* 'ML,MU .GE. 0 AND ML,MU .LT. NEQ. YOU HAVE CALLED ' //
* 'THE CODE WITH ML = ' // XERN1 // ' AND MU = ' // XERN2,
* 17, 1)
IDID = -33
ENDIF
ENDIF
C
C CHECK LRW AND LIW FOR SUFFICIENT STORAGE ALLOCATION
C
IF (LRW .LT. 250 + (10 + ILRW)*NEQ) THEN
WRITE (XERN1, '(I8)') LRW
IF (INFO(6) .EQ. 0) THEN
CALL XERMSG ('SLATEC', 'DDEBDF', 'LENGTH OF ARRAY RWORK ' //
* 'MUST BE AT LEAST 250 + 10*NEQ + NEQ*NEQ.$$' //
* 'YOU HAVE CALLED THE CODE WITH LRW = ' // XERN1, 1, 1)
ELSE
CALL XERMSG ('SLATEC', 'DDEBDF', 'LENGTH OF ARRAY RWORK ' //
* 'MUST BE AT LEAST 250 + 10*NEQ + (2*ML+MU+1)*NEQ.$$' //
* 'YOU HAVE CALLED THE CODE WITH LRW = ' // XERN1, 18, 1)
ENDIF
IDID = -33
ENDIF
C
IF (LIW .LT. 56 + NEQ) THEN
WRITE (XERN1, '(I8)') LIW
CALL XERMSG ('SLATEC', 'DDEBDF', 'LENGTH OF ARRAY IWORK ' //
* 'BE AT LEAST 56 + NEQ. YOU HAVE CALLED THE CODE WITH ' //
* 'LIW = ' // XERN1, 2, 1)
IDID = -33
ENDIF
C
C COMPUTE THE INDICES FOR THE ARRAYS TO BE STORED IN THE WORK
C ARRAY AND RESTORE COMMON BLOCK DATA
C
ICOMI = 21 + NEQ
IINOUT = ICOMI + 33
C
IYPOUT = 21
ITSTAR = 21 + NEQ
ICOMR = 22 + NEQ
C
IF (INFO(1) .NE. 0) INTOUT = IWORK(IINOUT) .NE. (-1)
C CALL DRSCO(RWORK(ICOMR),IWORK(ICOMI))
C
IYH = ICOMR + 218
IEWT = IYH + 6*NEQ
ISAVF = IEWT + NEQ
IACOR = ISAVF + NEQ
IWM = IACOR + NEQ
IDELSN = IWM + 2 + ILRW*NEQ
C
IBEGIN = INFO(1)
ITOL = INFO(2)
IINTEG = INFO(3)
ITSTOP = INFO(4)
IJAC = INFO(5)
IBAND = INFO(6)
RWORK(ITSTAR) = T
C
CALL DLSOD(DF,NEQ,T,Y,TOUT,RTOL,ATOL,IDID,RWORK(IYPOUT),
1 RWORK(IYH),RWORK(IYH),RWORK(IEWT),RWORK(ISAVF),
2 RWORK(IACOR),RWORK(IWM),IWORK(1),DJAC,INTOUT,
3 RWORK(1),RWORK(12),RWORK(IDELSN),RPAR,IPAR)
C
IWORK(IINOUT) = -1
IF (INTOUT) IWORK(IINOUT) = 1
C
IF (IDID .NE. (-2)) IWORK(LIW) = IWORK(LIW) + 1
IF (T .NE. RWORK(ITSTAR)) IWORK(LIW) = 0
C CALL DSVCO(RWORK(ICOMR),IWORK(ICOMI))
RWORK(11) = H
RWORK(13) = TN
INFO(1) = IBEGIN
C
RETURN
END