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opkdmain.f
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*DECK DLSODE
SUBROUTINE DLSODE (F, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
1 ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JAC, MF)
EXTERNAL F, JAC
INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK, LIW, MF
DOUBLE PRECISION Y, T, TOUT, RTOL, ATOL, RWORK
DIMENSION NEQ(*), Y(*), RTOL(*), ATOL(*), RWORK(LRW), IWORK(LIW)
!***BEGIN PROLOGUE DLSODE
!***PURPOSE Livermore Solver for Ordinary Differential Equations.
! DLSODE solves the initial-value problem for stiff or
! nonstiff systems of first-order ODE's,
! dy/dt = f(t,y), or, in component form,
! dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(N)), i=1,...,N.
!***CATEGORY I1A
!***TYPE DOUBLE PRECISION (SLSODE-S, DLSODE-D)
!***KEYWORDS ORDINARY DIFFERENTIAL EQUATIONS, INITIAL VALUE PROBLEM,
! STIFF, NONSTIFF
!***AUTHOR Hindmarsh, Alan C., (LLNL)
! Center for Applied Scientific Computing, L-561
! Lawrence Livermore National Laboratory
! Livermore, CA 94551.
!***DESCRIPTION
!
! NOTE: The "Usage" and "Arguments" sections treat only a subset of
! available options, in condensed fashion. The options
! covered and the information supplied will support most
! standard uses of DLSODE.
!
! For more sophisticated uses, full details on all options are
! given in the concluding section, headed "Long Description."
! A synopsis of the DLSODE Long Description is provided at the
! beginning of that section; general topics covered are:
! - Elements of the call sequence; optional input and output
! - Optional supplemental routines in the DLSODE package
! - internal COMMON block
!
! *Usage:
! Communication between the user and the DLSODE package, for normal
! situations, is summarized here. This summary describes a subset
! of the available options. See "Long Description" for complete
! details, including optional communication, nonstandard options,
! and instructions for special situations.
!
! A sample program is given in the "Examples" section.
!
! Refer to the argument descriptions for the definitions of the
! quantities that appear in the following sample declarations.
!
! For MF = 10,
! PARAMETER (LRW = 20 + 16*NEQ, LIW = 20)
! For MF = 21 or 22,
! PARAMETER (LRW = 22 + 9*NEQ + NEQ**2, LIW = 20 + NEQ)
! For MF = 24 or 25,
! PARAMETER (LRW = 22 + 10*NEQ + (2*ML+MU)*NEQ,
! * LIW = 20 + NEQ)
!
! EXTERNAL F, JAC
! INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK(LIW),
! * LIW, MF
! DOUBLE PRECISION Y(NEQ), T, TOUT, RTOL, ATOL(ntol), RWORK(LRW)
!
! CALL DLSODE (F, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
! * ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JAC, MF)
!
! *Arguments:
! F :EXT Name of subroutine for right-hand-side vector f.
! This name must be declared EXTERNAL in calling
! program. The form of F must be:
!
! SUBROUTINE F (NEQ, T, Y, YDOT)
! INTEGER NEQ
! DOUBLE PRECISION T, Y(*), YDOT(*)
!
! The inputs are NEQ, T, Y. F is to set
!
! YDOT(i) = f(i,T,Y(1),Y(2),...,Y(NEQ)),
! i = 1, ..., NEQ .
!
! NEQ :IN Number of first-order ODE's.
!
! Y :INOUT Array of values of the y(t) vector, of length NEQ.
! Input: For the first call, Y should contain the
! values of y(t) at t = T. (Y is an input
! variable only if ISTATE = 1.)
! Output: On return, Y will contain the values at the
! new t-value.
!
! T :INOUT Value of the independent variable. On return it
! will be the current value of t (normally TOUT).
!
! TOUT :IN Next point where output is desired (.NE. T).
!
! ITOL :IN 1 or 2 according as ATOL (below) is a scalar or
! an array.
!
! RTOL :IN Relative tolerance parameter (scalar).
!
! ATOL :IN Absolute tolerance parameter (scalar or array).
! If ITOL = 1, ATOL need not be dimensioned.
! If ITOL = 2, ATOL must be dimensioned at least NEQ.
!
! The estimated local error in Y(i) will be controlled
! so as to be roughly less (in magnitude) than
!
! EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
! EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
!
! Thus the local error test passes if, in each
! component, either the absolute error is less than
! ATOL (or ATOL(i)), or the relative error is less
! than RTOL.
!
! Use RTOL = 0.0 for pure absolute error control, and
! use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative
! error control. Caution: Actual (global) errors may
! exceed these local tolerances, so choose them
! conservatively.
!
! ITASK :IN Flag indicating the task DLSODE is to perform.
! Use ITASK = 1 for normal computation of output
! values of y at t = TOUT.
!
! ISTATE:INOUT Index used for input and output to specify the state
! of the calculation.
! Input:
! 1 This is the first call for a problem.
! 2 This is a subsequent call.
! Output:
! 1 Nothing was done, because TOUT was equal to T.
! 2 DLSODE was successful (otherwise, negative).
! Note that ISTATE need not be modified after a
! successful return.
! -1 Excess work done on this call (perhaps wrong
! MF).
! -2 Excess accuracy requested (tolerances too
! small).
! -3 Illegal input detected (see printed message).
! -4 Repeated error test failures (check all
! inputs).
! -5 Repeated convergence failures (perhaps bad
! Jacobian supplied or wrong choice of MF or
! tolerances).
! -6 Error weight became zero during problem
! (solution component i vanished, and ATOL or
! ATOL(i) = 0.).
!
! IOPT :IN Flag indicating whether optional inputs are used:
! 0 No.
! 1 Yes. (See "Optional inputs" under "Long
! Description," Part 1.)
!
! RWORK :WORK Real work array of length at least:
! 20 + 16*NEQ for MF = 10,
! 22 + 9*NEQ + NEQ**2 for MF = 21 or 22,
! 22 + 10*NEQ + (2*ML + MU)*NEQ for MF = 24 or 25.
!
! LRW :IN Declared length of RWORK (in user's DIMENSION
! statement).
!
! IWORK :WORK Integer work array of length at least:
! 20 for MF = 10,
! 20 + NEQ for MF = 21, 22, 24, or 25.
!
! If MF = 24 or 25, input in IWORK(1),IWORK(2) the
! lower and upper Jacobian half-bandwidths ML,MU.
!
! On return, IWORK contains information that may be
! of interest to the user:
!
! Name Location Meaning
! ----- --------- -----------------------------------------
! NST IWORK(11) Number of steps taken for the problem so
! far.
! NFE IWORK(12) Number of f evaluations for the problem
! so far.
! NJE IWORK(13) Number of Jacobian evaluations (and of
! matrix LU decompositions) for the problem
! so far.
! NQU IWORK(14) Method order last used (successfully).
! LENRW IWORK(17) Length of RWORK actually required. This
! is defined on normal returns and on an
! illegal input return for insufficient
! storage.
! LENIW IWORK(18) Length of IWORK actually required. This
! is defined on normal returns and on an
! illegal input return for insufficient
! storage.
!
! LIW :IN Declared length of IWORK (in user's DIMENSION
! statement).
!
! JAC :EXT Name of subroutine for Jacobian matrix (MF =
! 21 or 24). If used, this name must be declared
! EXTERNAL in calling program. If not used, pass a
! dummy name. The form of JAC must be:
!
! SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD)
! INTEGER NEQ, ML, MU, NROWPD
! DOUBLE PRECISION T, Y(*), PD(NROWPD,*)
!
! See item c, under "Description" below for more
! information about JAC.
!
! MF :IN Method flag. Standard values are:
! 10 Nonstiff (Adams) method, no Jacobian used.
! 21 Stiff (BDF) method, user-supplied full Jacobian.
! 22 Stiff method, internally generated full
! Jacobian.
! 24 Stiff method, user-supplied banded Jacobian.
! 25 Stiff method, internally generated banded
! Jacobian.
!
! *Description:
! DLSODE solves the initial value problem for stiff or nonstiff
! systems of first-order ODE's,
!
! dy/dt = f(t,y) ,
!
! or, in component form,
!
! dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(NEQ))
! (i = 1, ..., NEQ) .
!
! DLSODE is a package based on the GEAR and GEARB packages, and on
! the October 23, 1978, version of the tentative ODEPACK user
! interface standard, with minor modifications.
!
! The steps in solving such a problem are as follows.
!
! a. First write a subroutine of the form
!
! SUBROUTINE F (NEQ, T, Y, YDOT)
! INTEGER NEQ
! DOUBLE PRECISION T, Y(*), YDOT(*)
!
! which supplies the vector function f by loading YDOT(i) with
! f(i).
!
! b. Next determine (or guess) whether or not the problem is stiff.
! Stiffness occurs when the Jacobian matrix df/dy has an
! eigenvalue whose real part is negative and large in magnitude
! compared to the reciprocal of the t span of interest. If the
! problem is nonstiff, use method flag MF = 10. If it is stiff,
! there are four standard choices for MF, and DLSODE requires the
! Jacobian matrix in some form. This matrix is regarded either
! as full (MF = 21 or 22), or banded (MF = 24 or 25). In the
! banded case, DLSODE requires two half-bandwidth parameters ML
! and MU. These are, respectively, the widths of the lower and
! upper parts of the band, excluding the main diagonal. Thus the
! band consists of the locations (i,j) with
!
! i - ML <= j <= i + MU ,
!
! and the full bandwidth is ML + MU + 1 .
!
! c. If the problem is stiff, you are encouraged to supply the
! Jacobian directly (MF = 21 or 24), but if this is not feasible,
! DLSODE will compute it internally by difference quotients (MF =
! 22 or 25). If you are supplying the Jacobian, write a
! subroutine of the form
!
! SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD)
! INTEGER NEQ, ML, MU, NRWOPD
! DOUBLE PRECISION T, Y(*), PD(NROWPD,*)
!
! which provides df/dy by loading PD as follows:
! - For a full Jacobian (MF = 21), load PD(i,j) with df(i)/dy(j),
! the partial derivative of f(i) with respect to y(j). (Ignore
! the ML and MU arguments in this case.)
! - For a banded Jacobian (MF = 24), load PD(i-j+MU+1,j) with
! df(i)/dy(j); i.e., load the diagonal lines of df/dy into the
! rows of PD from the top down.
! - In either case, only nonzero elements need be loaded.
!
! d. Write a main program that calls subroutine DLSODE once for each
! point at which answers are desired. This should also provide
! for possible use of logical unit 6 for output of error messages
! by DLSODE.
!
! Before the first call to DLSODE, set ISTATE = 1, set Y and T to
! the initial values, and set TOUT to the first output point. To
! continue the integration after a successful return, simply
! reset TOUT and call DLSODE again. No other parameters need be
! reset.
!
! *Examples:
! The following is a simple example problem, with the coding needed
! for its solution by DLSODE. The problem is from chemical kinetics,
! and consists of the following three rate equations:
!
! dy1/dt = -.04*y1 + 1.E4*y2*y3
! dy2/dt = .04*y1 - 1.E4*y2*y3 - 3.E7*y2**2
! dy3/dt = 3.E7*y2**2
!
! on the interval from t = 0.0 to t = 4.E10, with initial conditions
! y1 = 1.0, y2 = y3 = 0. The problem is stiff.
!
! The following coding solves this problem with DLSODE, using
! MF = 21 and printing results at t = .4, 4., ..., 4.E10. It uses
! ITOL = 2 and ATOL much smaller for y2 than for y1 or y3 because y2
! has much smaller values. At the end of the run, statistical
! quantities of interest are printed.
!
! EXTERNAL FEX, JEX
! INTEGER IOPT, IOUT, ISTATE, ITASK, ITOL, IWORK(23), LIW, LRW,
! * MF, NEQ
! DOUBLE PRECISION ATOL(3), RTOL, RWORK(58), T, TOUT, Y(3)
! NEQ = 3
! Y(1) = 1.D0
! Y(2) = 0.D0
! Y(3) = 0.D0
! T = 0.D0
! TOUT = .4D0
! ITOL = 2
! RTOL = 1.D-4
! ATOL(1) = 1.D-6
! ATOL(2) = 1.D-10
! ATOL(3) = 1.D-6
! ITASK = 1
! ISTATE = 1
! IOPT = 0
! LRW = 58
! LIW = 23
! MF = 21
! DO 40 IOUT = 1,12
! CALL DLSODE (FEX, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
! * ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JEX, MF)
! WRITE(6,20) T, Y(1), Y(2), Y(3)
! 20 FORMAT(' At t =',D12.4,' y =',3D14.6)
! IF (ISTATE .LT. 0) GO TO 80
! 40 TOUT = TOUT*10.D0
! WRITE(6,60) IWORK(11), IWORK(12), IWORK(13)
! 60 FORMAT(/' No. steps =',i4,', No. f-s =',i4,', No. J-s =',i4)
! STOP
! 80 WRITE(6,90) ISTATE
! 90 FORMAT(///' Error halt.. ISTATE =',I3)
! STOP
! END
!
! SUBROUTINE FEX (NEQ, T, Y, YDOT)
! INTEGER NEQ
! DOUBLE PRECISION T, Y(3), YDOT(3)
! YDOT(1) = -.04D0*Y(1) + 1.D4*Y(2)*Y(3)
! YDOT(3) = 3.D7*Y(2)*Y(2)
! YDOT(2) = -YDOT(1) - YDOT(3)
! RETURN
! END
!
! SUBROUTINE JEX (NEQ, T, Y, ML, MU, PD, NRPD)
! INTEGER NEQ, ML, MU, NRPD
! DOUBLE PRECISION T, Y(3), PD(NRPD,3)
! PD(1,1) = -.04D0
! PD(1,2) = 1.D4*Y(3)
! PD(1,3) = 1.D4*Y(2)
! PD(2,1) = .04D0
! PD(2,3) = -PD(1,3)
! PD(3,2) = 6.D7*Y(2)
! PD(2,2) = -PD(1,2) - PD(3,2)
! RETURN
! END
!
! The output from this program (on a Cray-1 in single precision)
! is as follows.
!
! At t = 4.0000e-01 y = 9.851726e-01 3.386406e-05 1.479357e-02
! At t = 4.0000e+00 y = 9.055142e-01 2.240418e-05 9.446344e-02
! At t = 4.0000e+01 y = 7.158050e-01 9.184616e-06 2.841858e-01
! At t = 4.0000e+02 y = 4.504846e-01 3.222434e-06 5.495122e-01
! At t = 4.0000e+03 y = 1.831701e-01 8.940379e-07 8.168290e-01
! At t = 4.0000e+04 y = 3.897016e-02 1.621193e-07 9.610297e-01
! At t = 4.0000e+05 y = 4.935213e-03 1.983756e-08 9.950648e-01
! At t = 4.0000e+06 y = 5.159269e-04 2.064759e-09 9.994841e-01
! At t = 4.0000e+07 y = 5.306413e-05 2.122677e-10 9.999469e-01
! At t = 4.0000e+08 y = 5.494530e-06 2.197825e-11 9.999945e-01
! At t = 4.0000e+09 y = 5.129458e-07 2.051784e-12 9.999995e-01
! At t = 4.0000e+10 y = -7.170603e-08 -2.868241e-13 1.000000e+00
!
! No. steps = 330, No. f-s = 405, No. J-s = 69
!
! *Accuracy:
! The accuracy of the solution depends on the choice of tolerances
! RTOL and ATOL. Actual (global) errors may exceed these local
! tolerances, so choose them conservatively.
!
! *Cautions:
! The work arrays should not be altered between calls to DLSODE for
! the same problem, except possibly for the conditional and optional
! inputs.
!
! *Portability:
! Since NEQ is dimensioned inside DLSODE, some compilers may object
! to a call to DLSODE with NEQ a scalar variable. In this event,
! use DIMENSION NEQ(1). Similar remarks apply to RTOL and ATOL.
!
! Note to Cray users:
! For maximum efficiency, use the CFT77 compiler. Appropriate
! compiler optimization directives have been inserted for CFT77.
!
! *Reference:
! Alan C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE
! Solvers," in Scientific Computing, R. S. Stepleman, et al., Eds.
! (North-Holland, Amsterdam, 1983), pp. 55-64.
!
! *Long Description:
! The following complete description of the user interface to
! DLSODE consists of four parts:
!
! 1. The call sequence to subroutine DLSODE, which is a driver
! routine for the solver. This includes descriptions of both
! the call sequence arguments and user-supplied routines.
! Following these descriptions is a description of optional
! inputs available through the call sequence, and then a
! description of optional outputs in the work arrays.
!
! 2. Descriptions of other routines in the DLSODE package that may
! be (optionally) called by the user. These provide the ability
! to alter error message handling, save and restore the internal
! COMMON, and obtain specified derivatives of the solution y(t).
!
! 3. Descriptions of COMMON block to be declared in overlay or
! similar environments, or to be saved when doing an interrupt
! of the problem and continued solution later.
!
! 4. Description of two routines in the DLSODE package, either of
! which the user may replace with his own version, if desired.
! These relate to the measurement of errors.
!
!
! Part 1. Call Sequence
! ----------------------
!
! Arguments
! ---------
! The call sequence parameters used for input only are
!
! F, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK, IOPT, LRW, LIW, JAC, MF,
!
! and those used for both input and output are
!
! Y, T, ISTATE.
!
! The work arrays RWORK and IWORK are also used for conditional and
! optional inputs and optional outputs. (The term output here
! refers to the return from subroutine DLSODE to the user's calling
! program.)
!
! The legality of input parameters will be thoroughly checked on the
! initial call for the problem, but not checked thereafter unless a
! change in input parameters is flagged by ISTATE = 3 on input.
!
! The descriptions of the call arguments are as follows.
!
! F The name of the user-supplied subroutine defining the ODE
! system. The system must be put in the first-order form
! dy/dt = f(t,y), where f is a vector-valued function of
! the scalar t and the vector y. Subroutine F is to compute
! the function f. It is to have the form
!
! SUBROUTINE F (NEQ, T, Y, YDOT)
! DOUBLE PRECISION T, Y(*), YDOT(*)
!
! where NEQ, T, and Y are input, and the array YDOT =
! f(T,Y) is output. Y and YDOT are arrays of length NEQ.
! Subroutine F should not alter Y(1),...,Y(NEQ). F must be
! declared EXTERNAL in the calling program.
!
! Subroutine F may access user-defined quantities in
! NEQ(2),... and/or in Y(NEQ(1)+1),..., if NEQ is an array
! (dimensioned in F) and/or Y has length exceeding NEQ(1).
! See the descriptions of NEQ and Y below.
!
! If quantities computed in the F routine are needed
! externally to DLSODE, an extra call to F should be made
! for this purpose, for consistent and accurate results.
! If only the derivative dy/dt is needed, use DINTDY
! instead.
!
! NEQ The size of the ODE system (number of first-order
! ordinary differential equations). Used only for input.
! NEQ may be decreased, but not increased, during the
! problem. If NEQ is decreased (with ISTATE = 3 on input),
! the remaining components of Y should be left undisturbed,
! if these are to be accessed in F and/or JAC.
!
! Normally, NEQ is a scalar, and it is generally referred
! to as a scalar in this user interface description.
! However, NEQ may be an array, with NEQ(1) set to the
! system size. (The DLSODE package accesses only NEQ(1).)
! In either case, this parameter is passed as the NEQ
! argument in all calls to F and JAC. Hence, if it is an
! array, locations NEQ(2),... may be used to store other
! integer data and pass it to F and/or JAC. Subroutines
! F and/or JAC must include NEQ in a DIMENSION statement
! in that case.
!
! Y A real array for the vector of dependent variables, of
! length NEQ or more. Used for both input and output on
! the first call (ISTATE = 1), and only for output on
! other calls. On the first call, Y must contain the
! vector of initial values. On output, Y contains the
! computed solution vector, evaluated at T. If desired,
! the Y array may be used for other purposes between
! calls to the solver.
!
! This array is passed as the Y argument in all calls to F
! and JAC. Hence its length may exceed NEQ, and locations
! Y(NEQ+1),... may be used to store other real data and
! pass it to F and/or JAC. (The DLSODE package accesses
! only Y(1),...,Y(NEQ).)
!
! T The independent variable. On input, T is used only on
! the first call, as the initial point of the integration.
! On output, after each call, T is the value at which a
! computed solution Y is evaluated (usually the same as
! TOUT). On an error return, T is the farthest point
! reached.
!
! TOUT The next value of T at which a computed solution is
! desired. Used only for input.
!
! When starting the problem (ISTATE = 1), TOUT may be equal
! to T for one call, then should not equal T for the next
! call. For the initial T, an input value of TOUT .NE. T
! is used in order to determine the direction of the
! integration (i.e., the algebraic sign of the step sizes)
! and the rough scale of the problem. Integration in
! either direction (forward or backward in T) is permitted.
!
! If ITASK = 2 or 5 (one-step modes), TOUT is ignored
! after the first call (i.e., the first call with
! TOUT .NE. T). Otherwise, TOUT is required on every call.
!
! If ITASK = 1, 3, or 4, the values of TOUT need not be
! monotone, but a value of TOUT which backs up is limited
! to the current internal T interval, whose endpoints are
! TCUR - HU and TCUR. (See "Optional Outputs" below for
! TCUR and HU.)
!
!
! ITOL An indicator for the type of error control. See
! description below under ATOL. Used only for input.
!
! RTOL A relative error tolerance parameter, either a scalar or
! an array of length NEQ. See description below under
! ATOL. Input only.
!
! ATOL An absolute error tolerance parameter, either a scalar or
! an array of length NEQ. Input only.
!
! The input parameters ITOL, RTOL, and ATOL determine the
! error control performed by the solver. The solver will
! control the vector e = (e(i)) of estimated local errors
! in Y, according to an inequality of the form
!
! rms-norm of ( e(i)/EWT(i) ) <= 1,
!
! where
!
! EWT(i) = RTOL(i)*ABS(Y(i)) + ATOL(i),
!
! and the rms-norm (root-mean-square norm) here is
!
! rms-norm(v) = SQRT(sum v(i)**2 / NEQ).
!
! Here EWT = (EWT(i)) is a vector of weights which must
! always be positive, and the values of RTOL and ATOL
! should all be nonnegative. The following table gives the
! types (scalar/array) of RTOL and ATOL, and the
! corresponding form of EWT(i).
!
! ITOL RTOL ATOL EWT(i)
! ---- ------ ------ -----------------------------
! 1 scalar scalar RTOL*ABS(Y(i)) + ATOL
! 2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
! 3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
! 4 array array RTOL(i)*ABS(Y(i)) + ATOL(i)
!
! When either of these parameters is a scalar, it need not
! be dimensioned in the user's calling program.
!
! If none of the above choices (with ITOL, RTOL, and ATOL
! fixed throughout the problem) is suitable, more general
! error controls can be obtained by substituting
! user-supplied routines for the setting of EWT and/or for
! the norm calculation. See Part 4 below.
!
! If global errors are to be estimated by making a repeated
! run on the same problem with smaller tolerances, then all
! components of RTOL and ATOL (i.e., of EWT) should be
! scaled down uniformly.
!
! ITASK An index specifying the task to be performed. Input
! only. ITASK has the following values and meanings:
! 1 Normal computation of output values of y(t) at
! t = TOUT (by overshooting and interpolating).
! 2 Take one step only and return.
! 3 Stop at the first internal mesh point at or beyond
! t = TOUT and return.
! 4 Normal computation of output values of y(t) at
! t = TOUT but without overshooting t = TCRIT. TCRIT
! must be input as RWORK(1). TCRIT may be equal to or
! beyond TOUT, but not behind it in the direction of
! integration. This option is useful if the problem
! has a singularity at or beyond t = TCRIT.
! 5 Take one step, without passing TCRIT, and return.
! TCRIT must be input as RWORK(1).
!
! Note: If ITASK = 4 or 5 and the solver reaches TCRIT
! (within roundoff), it will return T = TCRIT (exactly) to
! indicate this (unless ITASK = 4 and TOUT comes before
! TCRIT, in which case answers at T = TOUT are returned
! first).
!
! ISTATE An index used for input and output to specify the state
! of the calculation.
!
! On input, the values of ISTATE are as follows:
! 1 This is the first call for the problem
! (initializations will be done). See "Note" below.
! 2 This is not the first call, and the calculation is to
! continue normally, with no change in any input
! parameters except possibly TOUT and ITASK. (If ITOL,
! RTOL, and/or ATOL are changed between calls with
! ISTATE = 2, the new values will be used but not
! tested for legality.)
! 3 This is not the first call, and the calculation is to
! continue normally, but with a change in input
! parameters other than TOUT and ITASK. Changes are
! allowed in NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, MF,
! ML, MU, and any of the optional inputs except H0.
! (See IWORK description for ML and MU.)
!
! Note: A preliminary call with TOUT = T is not counted as
! a first call here, as no initialization or checking of
! input is done. (Such a call is sometimes useful for the
! purpose of outputting the initial conditions.) Thus the
! first call for which TOUT .NE. T requires ISTATE = 1 on
! input.
!
! On output, ISTATE has the following values and meanings:
! 1 Nothing was done, as TOUT was equal to T with
! ISTATE = 1 on input.
! 2 The integration was performed successfully.
! -1 An excessive amount of work (more than MXSTEP steps)
! was done on this call, before completing the
! requested task, but the integration was otherwise
! successful as far as T. (MXSTEP is an optional input
! and is normally 500.) To continue, the user may
! simply reset ISTATE to a value >1 and call again (the
! excess work step counter will be reset to 0). In
! addition, the user may increase MXSTEP to avoid this
! error return; see "Optional Inputs" below.
! -2 Too much accuracy was requested for the precision of
! the machine being used. This was detected before
! completing the requested task, but the integration
! was successful as far as T. To continue, the
! tolerance parameters must be reset, and ISTATE must
! be set to 3. The optional output TOLSF may be used
! for this purpose. (Note: If this condition is
! detected before taking any steps, then an illegal
! input return (ISTATE = -3) occurs instead.)
! -3 Illegal input was detected, before taking any
! integration steps. See written message for details.
! (Note: If the solver detects an infinite loop of
! calls to the solver with illegal input, it will cause
! the run to stop.)
! -4 There were repeated error-test failures on one
! attempted step, before completing the requested task,
! but the integration was successful as far as T. The
! problem may have a singularity, or the input may be
! inappropriate.
! -5 There were repeated convergence-test failures on one
! attempted step, before completing the requested task,
! but the integration was successful as far as T. This
! may be caused by an inaccurate Jacobian matrix, if
! one is being used.
! -6 EWT(i) became zero for some i during the integration.
! Pure relative error control (ATOL(i)=0.0) was
! requested on a variable which has now vanished. The
! integration was successful as far as T.
!
! Note: Since the normal output value of ISTATE is 2, it
! does not need to be reset for normal continuation. Also,
! since a negative input value of ISTATE will be regarded
! as illegal, a negative output value requires the user to
! change it, and possibly other inputs, before calling the
! solver again.
!
! IOPT An integer flag to specify whether any optional inputs
! are being used on this call. Input only. The optional
! inputs are listed under a separate heading below.
! 0 No optional inputs are being used. Default values
! will be used in all cases.
! 1 One or more optional inputs are being used.
!
! RWORK A real working array (double precision). The length of
! RWORK must be at least
!
! 20 + NYH*(MAXORD + 1) + 3*NEQ + LWM
!
! where
! NYH = the initial value of NEQ,
! MAXORD = 12 (if METH = 1) or 5 (if METH = 2) (unless a
! smaller value is given as an optional input),
! LWM = 0 if MITER = 0,
! LWM = NEQ**2 + 2 if MITER = 1 or 2,
! LWM = NEQ + 2 if MITER = 3, and
! LWM = (2*ML + MU + 1)*NEQ + 2
! if MITER = 4 or 5.
! (See the MF description below for METH and MITER.)
!
! Thus if MAXORD has its default value and NEQ is constant,
! this length is:
! 20 + 16*NEQ for MF = 10,
! 22 + 16*NEQ + NEQ**2 for MF = 11 or 12,
! 22 + 17*NEQ for MF = 13,
! 22 + 17*NEQ + (2*ML + MU)*NEQ for MF = 14 or 15,
! 20 + 9*NEQ for MF = 20,
! 22 + 9*NEQ + NEQ**2 for MF = 21 or 22,
! 22 + 10*NEQ for MF = 23,
! 22 + 10*NEQ + (2*ML + MU)*NEQ for MF = 24 or 25.
!
! The first 20 words of RWORK are reserved for conditional
! and optional inputs and optional outputs.
!
! The following word in RWORK is a conditional input:
! RWORK(1) = TCRIT, the critical value of t which the
! solver is not to overshoot. Required if ITASK
! is 4 or 5, and ignored otherwise. See ITASK.
!
! LRW The length of the array RWORK, as declared by the user.
! (This will be checked by the solver.)
!
! IWORK An integer work array. Its length must be at least
! 20 if MITER = 0 or 3 (MF = 10, 13, 20, 23), or
! 20 + NEQ otherwise (MF = 11, 12, 14, 15, 21, 22, 24, 25).
! (See the MF description below for MITER.) The first few
! words of IWORK are used for conditional and optional
! inputs and optional outputs.
!
! The following two words in IWORK are conditional inputs:
! IWORK(1) = ML These are the lower and upper half-
! IWORK(2) = MU bandwidths, respectively, of the banded
! Jacobian, excluding the main diagonal.
! The band is defined by the matrix locations
! (i,j) with i - ML <= j <= i + MU. ML and MU
! must satisfy 0 <= ML,MU <= NEQ - 1. These are
! required if MITER is 4 or 5, and ignored
! otherwise. ML and MU may in fact be the band
! parameters for a matrix to which df/dy is only
! approximately equal.
!
! LIW The length of the array IWORK, as declared by the user.
! (This will be checked by the solver.)
!
! Note: The work arrays must not be altered between calls to DLSODE
! for the same problem, except possibly for the conditional and
! optional inputs, and except for the last 3*NEQ words of RWORK.
! The latter space is used for internal scratch space, and so is
! available for use by the user outside DLSODE between calls, if
! desired (but not for use by F or JAC).
!
! JAC The name of the user-supplied routine (MITER = 1 or 4) to
! compute the Jacobian matrix, df/dy, as a function of the
! scalar t and the vector y. (See the MF description below
! for MITER.) It is to have the form
!
! SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD)
! DOUBLE PRECISION T, Y(*), PD(NROWPD,*)
!
! where NEQ, T, Y, ML, MU, and NROWPD are input and the
! array PD is to be loaded with partial derivatives
! (elements of the Jacobian matrix) on output. PD must be
! given a first dimension of NROWPD. T and Y have the same
! meaning as in subroutine F.
!
! In the full matrix case (MITER = 1), ML and MU are
! ignored, and the Jacobian is to be loaded into PD in
! columnwise manner, with df(i)/dy(j) loaded into PD(i,j).
!
! In the band matrix case (MITER = 4), the elements within
! the band are to be loaded into PD in columnwise manner,
! with diagonal lines of df/dy loaded into the rows of PD.
! Thus df(i)/dy(j) is to be loaded into PD(i-j+MU+1,j). ML
! and MU are the half-bandwidth parameters (see IWORK).
! The locations in PD in the two triangular areas which
! correspond to nonexistent matrix elements can be ignored
! or loaded arbitrarily, as they are overwritten by DLSODE.
!
! JAC need not provide df/dy exactly. A crude approximation
! (possibly with a smaller bandwidth) will do.
!
! In either case, PD is preset to zero by the solver, so
! that only the nonzero elements need be loaded by JAC.
! Each call to JAC is preceded by a call to F with the same
! arguments NEQ, T, and Y. Thus to gain some efficiency,
! intermediate quantities shared by both calculations may
! be saved in a user COMMON block by F and not recomputed
! by JAC, if desired. Also, JAC may alter the Y array, if
! desired. JAC must be declared EXTERNAL in the calling
! program.
!
! Subroutine JAC may access user-defined quantities in
! NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
! (dimensioned in JAC) and/or Y has length exceeding
! NEQ(1). See the descriptions of NEQ and Y above.
!
! MF The method flag. Used only for input. The legal values
! of MF are 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24,
! and 25. MF has decimal digits METH and MITER:
! MF = 10*METH + MITER .
!
! METH indicates the basic linear multistep method:
! 1 Implicit Adams method.
! 2 Method based on backward differentiation formulas
! (BDF's).
!
! MITER indicates the corrector iteration method:
! 0 Functional iteration (no Jacobian matrix is
! involved).
! 1 Chord iteration with a user-supplied full (NEQ by
! NEQ) Jacobian.
! 2 Chord iteration with an internally generated
! (difference quotient) full Jacobian (using NEQ
! extra calls to F per df/dy value).
! 3 Chord iteration with an internally generated
! diagonal Jacobian approximation (using one extra call
! to F per df/dy evaluation).
! 4 Chord iteration with a user-supplied banded Jacobian.
! 5 Chord iteration with an internally generated banded
! Jacobian (using ML + MU + 1 extra calls to F per
! df/dy evaluation).
!
! If MITER = 1 or 4, the user must supply a subroutine JAC
! (the name is arbitrary) as described above under JAC.
! For other values of MITER, a dummy argument can be used.
!
! Optional Inputs
! ---------------
! The following is a list of the optional inputs provided for in the
! call sequence. (See also Part 2.) For each such input variable,
! this table lists its name as used in this documentation, its
! location in the call sequence, its meaning, and the default value.
! The use of any of these inputs requires IOPT = 1, and in that case
! all of these inputs are examined. A value of zero for any of
! these optional inputs will cause the default value to be used.
! Thus to use a subset of the optional inputs, simply preload
! locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively,
! and then set those of interest to nonzero values.
!
! Name Location Meaning and default value
! ------ --------- -----------------------------------------------
! H0 RWORK(5) Step size to be attempted on the first step.
! The default value is determined by the solver.
! HMAX RWORK(6) Maximum absolute step size allowed. The
! default value is infinite.
! HMIN RWORK(7) Minimum absolute step size allowed. The
! default value is 0. (This lower bound is not
! enforced on the final step before reaching
! TCRIT when ITASK = 4 or 5.)
! MAXORD IWORK(5) Maximum order to be allowed. The default value
! is 12 if METH = 1, and 5 if METH = 2. (See the
! MF description above for METH.) If MAXORD
! exceeds the default value, it will be reduced
! to the default value. If MAXORD is changed
! during the problem, it may cause the current
! order to be reduced.
! MXSTEP IWORK(6) Maximum number of (internally defined) steps
! allowed during one call to the solver. The
! default value is 500.
! MXHNIL IWORK(7) Maximum number of messages printed (per
! problem) warning that T + H = T on a step
! (H = step size). This must be positive to
! result in a nondefault value. The default
! value is 10.
!
! Optional Outputs
! ----------------
! As optional additional output from DLSODE, the variables listed
! below are quantities related to the performance of DLSODE which
! are available to the user. These are communicated by way of the
! work arrays, but also have internal mnemonic names as shown.
! Except where stated otherwise, all of these outputs are defined on
! any successful return from DLSODE, and on any return with ISTATE =
! -1, -2, -4, -5, or -6. On an illegal input return (ISTATE = -3),
! they will be unchanged from their existing values (if any), except
! possibly for TOLSF, LENRW, and LENIW. On any error return,
! outputs relevant to the error will be defined, as noted below.
!
! Name Location Meaning
! ----- --------- ------------------------------------------------
! HU RWORK(11) Step size in t last used (successfully).
! HCUR RWORK(12) Step size to be attempted on the next step.
! TCUR RWORK(13) Current value of the independent variable which
! the solver has actually reached, i.e., the
! current internal mesh point in t. On output,
! TCUR will always be at least as far as the
! argument T, but may be farther (if interpolation
! was done).
! TOLSF RWORK(14) Tolerance scale factor, greater than 1.0,
! computed when a request for too much accuracy
! was detected (ISTATE = -3 if detected at the
! start of the problem, ISTATE = -2 otherwise).
! If ITOL is left unaltered but RTOL and ATOL are
! uniformly scaled up by a factor of TOLSF for the
! next call, then the solver is deemed likely to
! succeed. (The user may also ignore TOLSF and
! alter the tolerance parameters in any other way
! appropriate.)
! NST IWORK(11) Number of steps taken for the problem so far.
! NFE IWORK(12) Number of F evaluations for the problem so far.
! NJE IWORK(13) Number of Jacobian evaluations (and of matrix LU
! decompositions) for the problem so far.
! NQU IWORK(14) Method order last used (successfully).
! NQCUR IWORK(15) Order to be attempted on the next step.
! IMXER IWORK(16) Index of the component of largest magnitude in
! the weighted local error vector ( e(i)/EWT(i) ),
! on an error return with ISTATE = -4 or -5.
! LENRW IWORK(17) Length of RWORK actually required. This is
! defined on normal returns and on an illegal
! input return for insufficient storage.
! LENIW IWORK(18) Length of IWORK actually required. This is
! defined on normal returns and on an illegal
! input return for insufficient storage.
!
! The following two arrays are segments of the RWORK array which may
! also be of interest to the user as optional outputs. For each
! array, the table below gives its internal name, its base address
! in RWORK, and its description.
!
! Name Base address Description
! ---- ------------ ----------------------------------------------
! YH 21 The Nordsieck history array, of size NYH by
! (NQCUR + 1), where NYH is the initial value of
! NEQ. For j = 0,1,...,NQCUR, column j + 1 of
! YH contains HCUR**j/factorial(j) times the jth
! derivative of the interpolating polynomial
! currently representing the solution, evaluated
! at t = TCUR.
! ACOR LENRW-NEQ+1 Array of size NEQ used for the accumulated
! corrections on each step, scaled on output to
! represent the estimated local error in Y on
! the last step. This is the vector e in the
! description of the error control. It is
! defined only on successful return from DLSODE.
!
!
! Part 2. Other Callable Routines
! --------------------------------
!
! The following are optional calls which the user may make to gain
! additional capabilities in conjunction with DLSODE.
!
! Form of call Function
! ------------------------ ----------------------------------------
! CALL XSETUN(LUN) Set the logical unit number, LUN, for
! output of messages from DLSODE, if the
! default is not desired. The default
! value of LUN is 6. This call may be made
! at any time and will take effect
! immediately.
! CALL XSETF(MFLAG) Set a flag to control the printing of
! messages by DLSODE. MFLAG = 0 means do
! not print. (Danger: this risks losing
! valuable information.) MFLAG = 1 means
! print (the default). This call may be
! made at any time and will take effect
! immediately.
! CALL DSRCOM(RSAV,ISAV,JOB) Saves and restores the contents of the
! internal COMMON blocks used by DLSODE
! (see Part 3 below). RSAV must be a
! real array of length 218 or more, and
! ISAV must be an integer array of length
! 37 or more. JOB = 1 means save COMMON
! into RSAV/ISAV. JOB = 2 means restore
! COMMON from same. DSRCOM is useful if
! one is interrupting a run and restarting
! later, or alternating between two or
! more problems solved with DLSODE.
! CALL DINTDY(,,,,,) Provide derivatives of y, of various
! (see below) orders, at a specified point t, if
! desired. It may be called only after a
! successful return from DLSODE. Detailed
! instructions follow.
!
! Detailed instructions for using DINTDY
! --------------------------------------
! The form of the CALL is:
!
! CALL DINTDY (T, K, RWORK(21), NYH, DKY, IFLAG)
!
! The input parameters are:
!
! T Value of independent variable where answers are
! desired (normally the same as the T last returned by
! DLSODE). For valid results, T must lie between
! TCUR - HU and TCUR. (See "Optional Outputs" above
! for TCUR and HU.)
! K Integer order of the derivative desired. K must
! satisfy 0 <= K <= NQCUR, where NQCUR is the current