*DECK DDRIV2
SUBROUTINE DDRIV2 (N, T, Y, F, TOUT, MSTATE, NROOT, EPS, EWT,
8 MINT, WORK, LENW, IWORK, LENIW, G, IERFLG)
C***BEGIN PROLOGUE DDRIV2
C***PURPOSE The function of DDRIV2 is to solve N ordinary differential
C equations of the form dY(I)/dT = F(Y(I),T), given the
C initial conditions Y(I) = YI. The program has options to
C allow the solution of both stiff and non-stiff differential
C equations. DDRIV2 uses double precision arithmetic.
C***LIBRARY SLATEC (SDRIVE)
C***CATEGORY I1A2, I1A1B
C***TYPE DOUBLE PRECISION (SDRIV2-S, DDRIV2-D, CDRIV2-C)
C***KEYWORDS DOUBLE PRECISION, GEAR'S METHOD, INITIAL VALUE PROBLEMS,
C ODE, ORDINARY DIFFERENTIAL EQUATIONS, SDRIVE, STIFF
C***AUTHOR Kahaner, D. K., (NIST)
C National Institute of Standards and Technology
C Gaithersburg, MD 20899
C Sutherland, C. D., (LANL)
C Mail Stop D466
C Los Alamos National Laboratory
C Los Alamos, NM 87545
C***DESCRIPTION
C
C I. PARAMETERS .....................................................
C
C (REMEMBER--To run DDRIV2 correctly in double precision, ALL
C non-integer arguments in the call sequence, including
C arrays, MUST be declared double precision.)
C
C The user should use parameter names in the call sequence of DDRIV2
C for those quantities whose value may be altered by DDRIV2. The
C parameters in the call sequence are:
C
C N = (Input) The number of differential equations.
C
C T = The independent variable. On input for the first call, T
C is the initial point. On output, T is the point at which
C the solution is given.
C
C Y = The vector of dependent variables. Y is used as input on
C the first call, to set the initial values. On output, Y
C is the computed solution vector. This array Y is passed
C in the call sequence of the user-provided routines F and
C G. Thus parameters required by F and G can be stored in
C this array in components N+1 and above. (Note: Changes
C by the user to the first N components of this array will
C take effect only after a restart, i.e., after setting
C MSTATE to +1(-1).)
C
C F = A subroutine supplied by the user. The name must be
C declared EXTERNAL in the user's calling program. This
C subroutine is of the form:
C SUBROUTINE F (N, T, Y, YDOT)
C DOUBLE PRECISION Y(*), YDOT(*)
C .
C .
C YDOT(1) = ...
C .
C .
C YDOT(N) = ...
C END (Sample)
C This computes YDOT = F(Y,T), the right hand side of the
C differential equations. Here Y is a vector of length at
C least N. The actual length of Y is determined by the
C user's declaration in the program which calls DDRIV2.
C Thus the dimensioning of Y in F, while required by FORTRAN
C convention, does not actually allocate any storage. When
C this subroutine is called, the first N components of Y are
C intermediate approximations to the solution components.
C The user should not alter these values. Here YDOT is a
C vector of length N. The user should only compute YDOT(I)
C for I from 1 to N. Normally a return from F passes
C control back to DDRIV2. However, if the user would like
C to abort the calculation, i.e., return control to the
C program which calls DDRIV2, he should set N to zero.
C DDRIV2 will signal this by returning a value of MSTATE
C equal to +6(-6). Altering the value of N in F has no
C effect on the value of N in the call sequence of DDRIV2.
C
C TOUT = (Input) The point at which the solution is desired.
C
C MSTATE = An integer describing the status of integration. The user
C must initialize MSTATE to +1 or -1. If MSTATE is
C positive, the routine will integrate past TOUT and
C interpolate the solution. This is the most efficient
C mode. If MSTATE is negative, the routine will adjust its
C internal step to reach TOUT exactly (useful if a
C singularity exists beyond TOUT.) The meaning of the
C magnitude of MSTATE:
C 1 (Input) Means the first call to the routine. This
C value must be set by the user. On all subsequent
C calls the value of MSTATE should be tested by the
C user. Unless DDRIV2 is to be reinitialized, only the
C sign of MSTATE may be changed by the user. (As a
C convenience to the user who may wish to put out the
C initial conditions, DDRIV2 can be called with
C MSTATE=+1(-1), and TOUT=T. In this case the program
C will return with MSTATE unchanged, i.e.,
C MSTATE=+1(-1).)
C 2 (Output) Means a successful integration. If a normal
C continuation is desired (i.e., a further integration
C in the same direction), simply advance TOUT and call
C again. All other parameters are automatically set.
C 3 (Output)(Unsuccessful) Means the integrator has taken
C 1000 steps without reaching TOUT. The user can
C continue the integration by simply calling DDRIV2
C again. Other than an error in problem setup, the
C most likely cause for this condition is trying to
C integrate a stiff set of equations with the non-stiff
C integrator option. (See description of MINT below.)
C 4 (Output)(Unsuccessful) Means too much accuracy has
C been requested. EPS has been increased to a value
C the program estimates is appropriate. The user can
C continue the integration by simply calling DDRIV2
C again.
C 5 (Output) A root was found at a point less than TOUT.
C The user can continue the integration toward TOUT by
C simply calling DDRIV2 again.
C 6 (Output)(Unsuccessful) N has been set to zero in
C SUBROUTINE F.
C 7 (Output)(Unsuccessful) N has been set to zero in
C FUNCTION G. See description of G below.
C 8 (Output)(Successful) For MSTATE negative, T is beyond
C TOUT. The solution was obtained by interpolation.
C The user can continue the integration by simply
C advancing TOUT and calling DDRIV2 again.
C 9 (Output)(Unsuccessful) The solution could not be
C obtained. The value of IERFLG (see description
C below) for a "Recoverable" situation indicates the
C type of difficulty encountered: either an illegal
C value for a parameter or an inability to continue the
C solution. For this condition the user should take
C corrective action and reset MSTATE to +1(-1) before
C calling DDRIV2 again. Otherwise the program will
C terminate the run.
C
C NROOT = (Input) The number of equations whose roots are desired.
C If NROOT is zero, the root search is not active. This
C option is useful for obtaining output at points which are
C not known in advance, but depend upon the solution, e.g.,
C when some solution component takes on a specified value.
C The root search is carried out using the user-written
C function G (see description of G below.) DDRIV2 attempts
C to find the value of T at which one of the equations
C changes sign. DDRIV2 can find at most one root per
C equation per internal integration step, and will then
C return the solution either at TOUT or at a root, whichever
C occurs first in the direction of integration. The initial
C point is never reported as a root. The index of the
C equation whose root is being reported is stored in the
C sixth element of IWORK.
C NOTE: NROOT is never altered by this program.
C
C EPS = On input, the requested relative accuracy in all solution
C components. EPS = 0 is allowed. On output, the adjusted
C relative accuracy if the input value was too small. The
C value of EPS should be set as large as is reasonable,
C because the amount of work done by DDRIV2 increases as
C EPS decreases.
C
C EWT = (Input) Problem zero, i.e., the smallest physically
C meaningful value for the solution. This is used inter-
C nally to compute an array YWT(I) = MAX(ABS(Y(I)), EWT).
C One step error estimates divided by YWT(I) are kept less
C than EPS. Setting EWT to zero provides pure relative
C error control. However, setting EWT smaller than
C necessary can adversely affect the running time.
C
C MINT = (Input) The integration method flag.
C MINT = 1 Means the Adams methods, and is used for
C non-stiff problems.
C MINT = 2 Means the stiff methods of Gear (i.e., the
C backward differentiation formulas), and is
C used for stiff problems.
C MINT = 3 Means the program dynamically selects the
C Adams methods when the problem is non-stiff
C and the Gear methods when the problem is
C stiff.
C MINT may not be changed without restarting, i.e., setting
C the magnitude of MSTATE to 1.
C
C WORK
C LENW = (Input)
C WORK is an array of LENW double precision words used
C internally for temporary storage. The user must allocate
C space for this array in the calling program by a statement
C such as
C DOUBLE PRECISION WORK(...)
C The length of WORK should be at least
C 16*N + 2*NROOT + 250 if MINT is 1, or
C N*N + 10*N + 2*NROOT + 250 if MINT is 2, or
C N*N + 17*N + 2*NROOT + 250 if MINT is 3,
C and LENW should be set to the value used. The contents of
C WORK should not be disturbed between calls to DDRIV2.
C
C IWORK
C LENIW = (Input)
C IWORK is an integer array of length LENIW used internally
C for temporary storage. The user must allocate space for
C this array in the calling program by a statement such as
C INTEGER IWORK(...)
C The length of IWORK should be at least
C 50 if MINT is 1, or
C N+50 if MINT is 2 or 3,
C and LENIW should be set to the value used. The contents
C of IWORK should not be disturbed between calls to DDRIV2.
C
C G = A double precision FORTRAN function supplied by the user
C if NROOT is not 0. In this case, the name must be
C declared EXTERNAL in the user's calling program. G is
C repeatedly called with different values of IROOT to
C obtain the value of each of the NROOT equations for which
C a root is desired. G is of the form:
C DOUBLE PRECISION FUNCTION G (N, T, Y, IROOT)
C DOUBLE PRECISION Y(*)
C GO TO (10, ...), IROOT
C 10 G = ...
C .
C .
C END (Sample)
C Here, Y is a vector of length at least N, whose first N
C components are the solution components at the point T.
C The user should not alter these values. The actual length
C of Y is determined by the user's declaration in the
C program which calls DDRIV2. Thus the dimensioning of Y in
C G, while required by FORTRAN convention, does not actually
C allocate any storage. Normally a return from G passes
C control back to DDRIV2. However, if the user would like
C to abort the calculation, i.e., return control to the
C program which calls DDRIV2, he should set N to zero.
C DDRIV2 will signal this by returning a value of MSTATE
C equal to +7(-7). In this case, the index of the equation
C being evaluated is stored in the sixth element of IWORK.
C Altering the value of N in G has no effect on the value of
C N in the call sequence of DDRIV2.
C
C IERFLG = An error flag. The error number associated with a
C diagnostic message (see Section II-A below) is the same as
C the corresponding value of IERFLG. The meaning of IERFLG:
C 0 The routine completed successfully. (No message is
C issued.)
C 3 (Warning) The number of steps required to reach TOUT
C exceeds MXSTEP.
C 4 (Warning) The value of EPS is too small.
C 11 (Warning) For MSTATE negative, T is beyond TOUT.
C The solution was obtained by interpolation.
C 15 (Warning) The integration step size is below the
C roundoff level of T. (The program issues this
C message as a warning but does not return control to
C the user.)
C 22 (Recoverable) N is not positive.
C 23 (Recoverable) MINT is less than 1 or greater than 3 .
C 26 (Recoverable) The magnitude of MSTATE is either 0 or
C greater than 9 .
C 27 (Recoverable) EPS is less than zero.
C 32 (Recoverable) Insufficient storage has been allocated
C for the WORK array.
C 33 (Recoverable) Insufficient storage has been allocated
C for the IWORK array.
C 41 (Recoverable) The integration step size has gone
C to zero.
C 42 (Recoverable) The integration step size has been
C reduced about 50 times without advancing the
C solution. The problem setup may not be correct.
C 999 (Fatal) The magnitude of MSTATE is 9 .
C
C II. OTHER COMMUNICATION TO THE USER ...............................
C
C A. The solver communicates to the user through the parameters
C above. In addition it writes diagnostic messages through the
C standard error handling program XERMSG. A complete description
C of XERMSG is given in "Guide to the SLATEC Common Mathematical
C Library" by Kirby W. Fong et al.. At installations which do not
C have this error handling package the short but serviceable
C routine, XERMSG, available with this package, can be used. That
C program uses the file named OUTPUT to transmit messages.
C
C B. The first three elements of WORK and the first five elements of
C IWORK will contain the following statistical data:
C AVGH The average step size used.
C HUSED The step size last used (successfully).
C AVGORD The average order used.
C IMXERR The index of the element of the solution vector that
C contributed most to the last error test.
C NQUSED The order last used (successfully).
C NSTEP The number of steps taken since last initialization.
C NFE The number of evaluations of the right hand side.
C NJE The number of evaluations of the Jacobian matrix.
C
C III. REMARKS ......................................................
C
C A. On any return from DDRIV2 all information necessary to continue
C the calculation is contained in the call sequence parameters,
C including the work arrays. Thus it is possible to suspend one
C problem, integrate another, and then return to the first.
C
C B. If this package is to be used in an overlay situation, the user
C must declare in the primary overlay the variables in the call
C sequence to DDRIV2.
C
C C. When the routine G is not required, difficulties associated with
C an unsatisfied external can be avoided by using the name of the
C routine which calculates the right hand side of the differential
C equations in place of G in the call sequence of DDRIV2.
C
C IV. USAGE .........................................................
C
C PROGRAM SAMPLE
C EXTERNAL F
C PARAMETER(MINT = 1, NROOT = 0, N = ...,
C 8 LENW = 16*N + 2*NROOT + 250, LENIW = 50)
C C N is the number of equations
C DOUBLE PRECISION EPS, EWT, T, TOUT, WORK(LENW), Y(N)
C INTEGER IWORK(LENIW)
C OPEN(FILE='TAPE6', UNIT=6, STATUS='NEW')
C C Initial point
C T = 0.
C C Set initial conditions
C DO 10 I = 1,N
C 10 Y(I) = ...
C TOUT = T
C EWT = ...
C MSTATE = 1
C EPS = ...
C 20 CALL DDRIV2 (N, T, Y, F, TOUT, MSTATE, NROOT, EPS, EWT,
C 8 MINT, WORK, LENW, IWORK, LENIW, F, IERFLG)
C C Next to last argument is not
C C F if rootfinding is used.
C IF (MSTATE .GT. 2) STOP
C WRITE(6, 100) TOUT, (Y(I), I=1,N)
C TOUT = TOUT + 1.
C IF (TOUT .LE. 10.) GO TO 20
C 100 FORMAT(...)
C END (Sample)
C
C***REFERENCES C. W. Gear, Numerical Initial Value Problems in
C Ordinary Differential Equations, Prentice-Hall, 1971.
C***ROUTINES CALLED DDRIV3, XERMSG
C***REVISION HISTORY (YYMMDD)
C 790601 DATE WRITTEN
C 900329 Initial submission to SLATEC.
C***END PROLOGUE DDRIV2
EXTERNAL F, G
DOUBLE PRECISION EPS, EWT, EWTCOM(1), G, HMAX, T, TOUT,
8 WORK(*), Y(*)
INTEGER IWORK(*)
INTEGER IERFLG, IERROR, IMPL, LENIW, LENW, MINT, MITER, ML,
8 MSTATE, MU, MXORD, MXSTEP, N, NDE, NROOT, NSTATE, NTASK
CHARACTER INTGR1*8
PARAMETER(IMPL = 0, MXSTEP = 1000)
C***FIRST EXECUTABLE STATEMENT DDRIV2
IF (ABS(MSTATE) .EQ. 9) THEN
IERFLG = 999
CALL XERMSG('SLATEC', 'DDRIV2',
8 'Illegal input. The magnitude of MSTATE IS 9 .',
8 IERFLG, 2)
RETURN
ELSE IF (ABS(MSTATE) .EQ. 0 .OR. ABS(MSTATE) .GT. 9) THEN
WRITE(INTGR1, '(I8)') MSTATE
IERFLG = 26
CALL XERMSG('SLATEC', 'DDRIV2',
8 'Illegal input. The magnitude of MSTATE, '//INTGR1//
8 ' is not in the range 1 to 8 .', IERFLG, 1)
MSTATE = SIGN(9, MSTATE)
RETURN
END IF
IF (MINT .LT. 1 .OR. MINT .GT. 3) THEN
WRITE(INTGR1, '(I8)') MINT
IERFLG = 23
CALL XERMSG('SLATEC', 'DDRIV2',
8 'Illegal input. Improper value for the integration method '//
8 'flag, '//INTGR1//' .', IERFLG, 1)
MSTATE = SIGN(9, MSTATE)
RETURN
END IF
IF (MSTATE .GE. 0) THEN
NSTATE = MSTATE
NTASK = 1
ELSE
NSTATE = - MSTATE
NTASK = 3
END IF
EWTCOM(1) = EWT
IF (EWT .NE. 0.D0) THEN
IERROR = 3
ELSE
IERROR = 2
END IF
IF (MINT .EQ. 1) THEN
MITER = 0
MXORD = 12
ELSE IF (MINT .EQ. 2) THEN
MITER = 2
MXORD = 5
ELSE IF (MINT .EQ. 3) THEN
MITER = 2
MXORD = 12
END IF
HMAX = 2.D0*ABS(TOUT - T)
CALL DDRIV3 (N, T, Y, F, NSTATE, TOUT, NTASK, NROOT, EPS, EWTCOM,
8 IERROR, MINT, MITER, IMPL, ML, MU, MXORD, HMAX, WORK,
8 LENW, IWORK, LENIW, F, F, NDE, MXSTEP, G, F, IERFLG)
IF (NSTATE .LE. 7) THEN
MSTATE = SIGN(NSTATE, MSTATE)
ELSE IF (NSTATE .EQ. 11) THEN
MSTATE = SIGN(8, MSTATE)
ELSE IF (NSTATE .GT. 11) THEN
MSTATE = SIGN(9, MSTATE)
END IF
RETURN
END