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vode.f
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!>@author
!>Netlib 2019
!>@copyright Public Domain
!>@brief
!> Collection of routines for
!> non-stiff or stiff ordinary differential equation initial-value problem solver
!> backward differentiation formulae (variable coefficient formulae)
!> (double precision)
!>Downloaded from Netlib, 2019
!> with changes by Paul Connolly, University of Manchester
*DECK DVODE
SUBROUTINE DVODE (F, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
1 ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JAC, MF,
2 RPAR, IPAR)
EXTERNAL F, JAC
DOUBLE PRECISION Y, T, TOUT, RTOL, ATOL, RWORK, RPAR
INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK, LIW,
1 MF, IPAR
DIMENSION Y(*), RTOL(*), ATOL(*), RWORK(LRW), IWORK(LIW),
1 RPAR(*), IPAR(*)
C-----------------------------------------------------------------------
C DVODE: Variable-coefficient Ordinary Differential Equation solver,
C with fixed-leading-coefficient implementation.
C This version is in double precision.
C
C DVODE 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 DVODE is a package based on the EPISODE and EPISODEB packages, and
C on the ODEPACK user interface standard, with minor modifications.
C-----------------------------------------------------------------------
C Authors:
C Peter N. Brown and Alan C. Hindmarsh
C Center for Applied Scientific Computing, L-561
C Lawrence Livermore National Laboratory
C Livermore, CA 94551
C and
C George D. Byrne
C Illinois Institute of Technology
C Chicago, IL 60616
C-----------------------------------------------------------------------
C References:
C
C 1. P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, "VODE: A Variable
C Coefficient ODE Solver," SIAM J. Sci. Stat. Comput., 10 (1989),
C pp. 1038-1051. Also, LLNL Report UCRL-98412, June 1988.
C 2. G. D. Byrne and A. C. Hindmarsh, "A Polyalgorithm for the
C Numerical Solution of Ordinary Differential Equations,"
C ACM Trans. Math. Software, 1 (1975), pp. 71-96.
C 3. A. C. Hindmarsh and G. D. Byrne, "EPISODE: An Effective Package
C for the Integration of Systems of Ordinary Differential
C Equations," LLNL Report UCID-30112, Rev. 1, April 1977.
C 4. G. D. Byrne and A. C. Hindmarsh, "EPISODEB: An Experimental
C Package for the Integration of Systems of Ordinary Differential
C Equations with Banded Jacobians," LLNL Report UCID-30132, April
C 1976.
C 5. A. 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 6. K. R. Jackson and R. Sacks-Davis, "An Alternative Implementation
C of Variable Step-Size Multistep Formulas for Stiff ODEs," ACM
C Trans. Math. Software, 6 (1980), pp. 295-318.
C-----------------------------------------------------------------------
C Summary of usage.
C
C Communication between the user and the DVODE 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, RPAR, IPAR)
C DOUBLE PRECISION T, Y(NEQ), YDOT(NEQ), RPAR
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 four standard
C choices for MF (21, 22, 24, 25), and DVODE requires the Jacobian
C matrix in some form. In these cases (MF .gt. 0), DVODE will use a
C saved copy of the Jacobian matrix. If this is undesirable because of
C storage limitations, set MF to the corresponding negative value
C (-21, -22, -24, -25). (See full description of MF below.)
C The Jacobian matrix is regarded either as full (MF = 21 or 22),
C or banded (MF = 24 or 25). In the banded case, DVODE requires two
C half-bandwidth parameters ML and MU. These are, respectively, the
C widths of the lower and upper parts of the band, excluding the main
C diagonal. 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
C C. If the problem is stiff, you are encouraged to supply the Jacobian
C directly (MF = 21 or 24), but if this is not feasible, DVODE will
C compute it internally by difference quotients (MF = 22 or 25).
C If you are supplying the Jacobian, provide a subroutine of the form:
C SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD, RPAR, IPAR)
C DOUBLE PRECISION T, Y(NEQ), PD(NROWPD,NEQ), RPAR
C which supplies 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 the
C 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 rows of
C PD from the top down.
C In either case, only nonzero elements need be loaded.
C
C D. Write a main program which calls subroutine DVODE 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 DVODE. On the first call to DVODE, 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 input used.
C RWORK = Real work array of length at least:
C 20 + 16*NEQ for MF = 10,
C 22 + 9*NEQ + 2*NEQ**2 for MF = 21 or 22,
C 22 + 11*NEQ + (3*ML + 2*MU)*NEQ for MF = 24 or 25.
C LRW = Declared length of RWORK (in user's DIMENSION statement).
C IWORK = Integer work array of length at least:
C 30 for MF = 10,
C 30 + NEQ for MF = 21, 22, 24, or 25.
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 statement).
C JAC = Name of subroutine for Jacobian matrix (MF = 21 or 24).
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 21 for stiff (BDF) method, user-supplied full Jacobian.
C 22 for stiff method, internally generated full Jacobian.
C 24 for stiff method, user-supplied banded Jacobian.
C 25 for stiff method, internally generated banded Jacobian.
C RPAR,IPAR = user-defined real and integer arrays passed to F and JAC.
C Note that the main program must declare arrays Y, RWORK, IWORK,
C and possibly ATOL, RPAR, and IPAR.
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 DVODE 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 input.)
C -5 means repeated convergence failures. (Perhaps bad
C Jacobian 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
C F. To continue the integration after a successful return, simply
C reset TOUT and call DVODE 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 DVODE. 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 DVODE, 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.
C At the end of the run, statistical quantities of interest are
C printed. (See optional output in the full description below.)
C To generate Fortran source code, replace C in column 1 with a blank
C in the coding below.
C
C EXTERNAL FEX, JEX
C DOUBLE PRECISION ATOL, RPAR, RTOL, RWORK, T, TOUT, Y
C DIMENSION Y(3), ATOL(3), RWORK(67), IWORK(33)
C NEQ = 3
C Y(1) = 1.0D0
C Y(2) = 0.0D0
C Y(3) = 0.0D0
C T = 0.0D0
C TOUT = 0.4D0
C ITOL = 2
C RTOL = 1.D-4
C ATOL(1) = 1.D-8
C ATOL(2) = 1.D-14
C ATOL(3) = 1.D-6
C ITASK = 1
C ISTATE = 1
C IOPT = 0
C LRW = 67
C LIW = 33
C MF = 21
C DO 40 IOUT = 1,12
C CALL DVODE(FEX,NEQ,Y,T,TOUT,ITOL,RTOL,ATOL,ITASK,ISTATE,
C 1 IOPT,RWORK,LRW,IWORK,LIW,JEX,MF,RPAR,IPAR)
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),
C 1 IWORK(20),IWORK(21),IWORK(22)
C 60 FORMAT(/' No. steps =',I4,' No. f-s =',I4,
C 1 ' No. J-s =',I4,' No. LU-s =',I4/
C 2 ' No. nonlinear iterations =',I4/
C 3 ' No. nonlinear convergence failures =',I4/
C 4 ' No. error test failures =',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, RPAR, IPAR)
C DOUBLE PRECISION RPAR, T, Y, YDOT
C DIMENSION Y(NEQ), YDOT(NEQ)
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, RPAR, IPAR)
C DOUBLE PRECISION PD, RPAR, T, Y
C DIMENSION Y(NEQ), PD(NRPD,NEQ)
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 following output was obtained from the above program on a
C Cray-1 computer with the CFT compiler.
C
C At t = 4.0000e-01 y = 9.851680e-01 3.386314e-05 1.479817e-02
C At t = 4.0000e+00 y = 9.055255e-01 2.240539e-05 9.445214e-02
C At t = 4.0000e+01 y = 7.158108e-01 9.184883e-06 2.841800e-01
C At t = 4.0000e+02 y = 4.505032e-01 3.222940e-06 5.494936e-01
C At t = 4.0000e+03 y = 1.832053e-01 8.942690e-07 8.167938e-01
C At t = 4.0000e+04 y = 3.898560e-02 1.621875e-07 9.610142e-01
C At t = 4.0000e+05 y = 4.935882e-03 1.984013e-08 9.950641e-01
C At t = 4.0000e+06 y = 5.166183e-04 2.067528e-09 9.994834e-01
C At t = 4.0000e+07 y = 5.201214e-05 2.080593e-10 9.999480e-01
C At t = 4.0000e+08 y = 5.213149e-06 2.085271e-11 9.999948e-01
C At t = 4.0000e+09 y = 5.183495e-07 2.073399e-12 9.999995e-01
C At t = 4.0000e+10 y = 5.450996e-08 2.180399e-13 9.999999e-01
C
C No. steps = 595 No. f-s = 832 No. J-s = 13 No. LU-s = 112
C No. nonlinear iterations = 831
C No. nonlinear convergence failures = 0
C No. error test failures = 22
C-----------------------------------------------------------------------
C Full description of user interface to DVODE.
C
C The user interface to DVODE consists of the following parts.
C
C i. The call sequence to subroutine DVODE, 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
C * a description of optional input available through the
C call sequence,
C * a description of optional output (in the work arrays), and
C * instructions for interrupting and restarting a solution.
C
C ii. Descriptions of other routines in the DVODE 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 iii. Descriptions of COMMON blocks to be declared in overlay
C or similar environments.
C
C iv. Description of two routines in the DVODE 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 i. 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 input and optional output. (The term output here refers
C to the return from subroutine DVODE 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 in the 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, RPAR, IPAR)
C DOUBLE PRECISION T, Y(NEQ), YDOT(NEQ), RPAR
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 real and integer
C work arrays RPAR and IPAR, which are to be dimensioned
C in the main program.
C
C If quantities computed in the F routine are needed
C externally to DVODE, 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 DVINDY 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 not be increased during the problem, but
C can be decreased (with ISTATE = 3 in the input).
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. In the output, Y contains the computed solution
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.
C
C T = The independent variable. In the input, T is used only on
C the first call, as the initial point of the integration.
C In the 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 output, 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 iv 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 In the 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, ML, MU,
C and any of the optional input 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 to include
C the initial conditions in the output.)
C Thus the first call for which TOUT .ne. T requires
C ISTATE = 1 in the input.
C
C In the output, ISTATE has the following values and meanings.
C 1 means nothing was done, as TOUT was equal to T with
C ISTATE = 1 in the 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 optional input below.)
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
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 input, before
C calling the solver again.
C
C IOPT = An integer flag to specify whether or not any optional
C input is being used on this call. Input only.
C The optional input is listed separately below.
C IOPT = 0 means no optional input is being used.
C Default values will be used in all cases.
C IOPT = 1 means optional input is 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 + 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 = length of work space for matrix-related data:
C LWM = 0 if MITER = 0,
C LWM = 2*NEQ**2 + 2 if MITER = 1 or 2, and MF.gt.0,
C LWM = NEQ**2 + 2 if MITER = 1 or 2, and MF.lt.0,
C LWM = NEQ + 2 if MITER = 3,
C LWM = (3*ML+2*MU+2)*NEQ + 2 if MITER = 4 or 5, and MF.gt.0,
C LWM = (2*ML+MU+1)*NEQ + 2 if MITER = 4 or 5, and MF.lt.0.
C (See the MF description for METH and MITER.)
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 + 2*NEQ**2 for MF = 11 or 12,
C 22 + 16*NEQ + NEQ**2 for MF = -11 or -12,
C 22 + 17*NEQ for MF = 13,
C 22 + 18*NEQ + (3*ML+2*MU)*NEQ for MF = 14 or 15,
C 22 + 17*NEQ + (2*ML+MU)*NEQ for MF = -14 or -15,
C 20 + 9*NEQ for MF = 20,
C 22 + 9*NEQ + 2*NEQ**2 for MF = 21 or 22,
C 22 + 9*NEQ + NEQ**2 for MF = -21 or -22,
C 22 + 10*NEQ for MF = 23,
C 22 + 11*NEQ + (3*ML+2*MU)*NEQ for MF = 24 or 25.
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 input and optional output.
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 or 3 (MF = 10, 13, 20, 23), or
C 30 + NEQ otherwise (abs(MF) = 11,12,14,15,21,22,24,25).
C The first 30 words of IWORK are reserved for conditional and
C optional input and optional output.
C
C The following 2 words in IWORK are conditional input:
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 MITER 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 work arrays must not be altered between calls to DVODE
C for the same problem, except possibly for the conditional and
C optional input, 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 DVODE 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
C the scalar t and the vector y. It is to have the form
C SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD,
C RPAR, IPAR)
C DOUBLE PRECISION T, Y(NEQ), PD(NROWPD,NEQ), RPAR
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 the
C Jacobian matrix) in the 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 (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 In the band matrix case (MITER = 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 DVODE.
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 real and integer
C work arrays, RPAR and IPAR, whose dimensions are set by the
C user in the main program.
C
C MF = The method flag. Used only for input. The legal values of
C MF are 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25,
C -11, -12, -14, -15, -21, -22, -24, -25.
C MF is a signed two-digit integer, MF = JSV*(10*METH + MITER).
C JSV = SIGN(MF) indicates the Jacobian-saving strategy:
C JSV = 1 means a copy of the Jacobian is saved for reuse
C in the corrector iteration algorithm.
C JSV = -1 means a copy of the Jacobian is not saved
C (valid only for MITER = 1, 2, 4, or 5).
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 (BDF-s).
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 full (NEQ by NEQ) Jacobian.
C MITER = 2 means chord iteration with an internally
C generated (difference quotient) full Jacobian
C (using NEQ extra calls to F per df/dy value).
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 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+1 extra
C calls to F per df/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 RPAR User-specified array used to communicate real parameters
C to user-supplied subroutines. If RPAR is a vector, then
C it must be dimensioned in the user's main program. If it
C is unused or it is a scalar, then it need not be
C dimensioned.
C
C IPAR User-specified array used to communicate integer parameter
C to user-supplied subroutines. The comments on dimensioning
C RPAR apply to IPAR.
C-----------------------------------------------------------------------
C Optional Input.
C
C The following is a list of the optional input provided for in the
C call sequence. (See also Part ii.) 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 this input requires IOPT = 1, and in that
C case all of this input is examined. A value of zero for any
C of these optional input variables will cause the default value to be
C used. Thus to use a subset of the optional input, 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-----------------------------------------------------------------------
C Optional Output.
C
C As optional additional output from DVODE, the variables listed
C below are quantities related to the performance of DVODE
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 this output is defined
C on any successful return from DVODE, 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, output 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. In the output,
C TCUR will always be at least as far from the
C initial value of t as the current argument T,
C 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 NJE IWORK(13) The number of Jacobian evaluations 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.
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 NLU IWORK(19) The number of matrix LU decompositions so far.
C
C NNI IWORK(20) The number of nonlinear (Newton) iterations so far.
C
C NCFN IWORK(21) The number of convergence failures of the nonlinear
C solver so far.
C
C NETF IWORK(22) The number of error test failures of the integrator
C so far.
C
C The following two arrays are segments of the RWORK array which
C may also be of interest to the user as optional output.
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
C solution, evaluated at t = TCUR.
C
C ACOR LENRW-NEQ+1 Array of size NEQ used for the accumulated
C corrections on each step, scaled in the 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 DVODE.
C
C-----------------------------------------------------------------------
C Interrupting and Restarting
C
C If the integration of a given problem by DVODE is to be
C interrrupted and then later continued, such as when restarting
C an interrupted run or alternating between two or more ODE problems,
C the user should save, following the return from the last DVODE call
C prior to the interruption, the contents of the call sequence
C variables and internal COMMON blocks, and later restore these
C values before the next DVODE call for that problem. To save
C and restore the COMMON blocks, use subroutine DVSRCO, as
C described below in part ii.
C
C In addition, if non-default values for either LUN or MFLAG are
C desired, an extra call to XSETUN and/or XSETF should be made just
C before continuing the integration. See Part ii below for details.
C
C-----------------------------------------------------------------------
C Part ii. Other Routines Callable.
C
C The following are optional calls which the user may make to
C gain additional capabilities in conjunction with DVODE.
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 DVODE, 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 DVODE.
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 DVSRCO(RSAV,ISAV,JOB) Saves and restores the contents of
C the internal COMMON blocks used by
C DVODE. (See Part iii below.)
C RSAV must be a real array of length 49
C or more, and ISAV must be an integer
C array of length 40 or more.
C JOB=1 means save COMMON into RSAV/ISAV.
C JOB=2 means restore COMMON from RSAV/ISAV.
C DVSRCO is useful if one is
C interrupting a run and restarting
C later, or alternating between two or
C more problems solved with DVODE.
C
C CALL DVINDY(,,,,,) 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 DVODE.
C
C The detailed instructions for using DVINDY are as follows.
C The form of the call is:
C
C CALL DVINDY (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 DVODE).
C For valid results, T must lie between TCUR - HU and TCUR.
C (See optional output 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 output). The capability corresponding
C to K = 0, i.e. computing y(T), is already provided
C by DVODE directly. Since NQCUR .ge. 1, the first
C derivative dy/dt is always available with DVINDY.
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 iii. COMMON Blocks.
C If DVODE 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 DVODE,
C (2) the two internal COMMON blocks
C /DVOD01/ of length 81 (48 double precision words
C followed by 33 integer words),
C /DVOD02/ of length 9 (1 double precision word
C followed by 8 integer words),
C
C If DVODE 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 two COMMON blocks in his main program to insure
C that their contents are preserved.
C
C-----------------------------------------------------------------------
C Part iv. Optionally Replaceable Solver Routines.
C
C Below are descriptions of two routines in the DVODE 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 DVODE 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 comparison with
C errors in Y(i). The EWT array returned by DEWSET is passed to the
C DVNORM routine (See below.), and also used by DVODE 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 Output. 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 RVOD, H, HU
C COMMON /DVOD01/ RVOD(48), IVOD(33)
C COMMON /DVOD02/ HU, NCFN, NETF, NFE, NJE, NLU, NNI, NQU, NST
C NQ = IVOD(28)
C H = RVOD(21)
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 DVODE.
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 REVISION HISTORY (YYYYMMDD)
C 19890615 Date Written. Initial release.
C 19890922 Added interrupt/restart ability, minor changes throughout.
C 19910228 Minor revisions in line format, prologue, etc.
C 19920227 Modifications by D. Pang:
C (1) Applied subgennam to get generic intrinsic names.
C (2) Changed intrinsic names to generic in comments.
C (3) Added *DECK lines before each routine.
C 19920721 Names of routines and labeled Common blocks changed, so as
C to be unique in combined single/double precision code (ACH).
C 19920722 Minor revisions to prologue (ACH).
C 19920831 Conversion to double precision done (ACH).
C 19921106 Fixed minor bug: ETAQ,ETAQM1 in DVSTEP SAVE statement (ACH).
C 19921118 Changed LUNSAV/MFLGSV to IXSAV (ACH).
C 19941222 Removed MF overwrite; attached sign to H in estimated second
C deriv. in DVHIN; misc. comment changes throughout (ACH).
C 19970515 Minor corrections to comments in prologue, DVJAC (ACH).
C 19981111 Corrected Block B by adding final line, GO TO 200 (ACH).
C 20020430 Various upgrades (ACH): Use ODEPACK error handler package.
C Replaced D1MACH by DUMACH. Various changes to main
C prologue and other routine prologues.