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*DECK SLSODE
SUBROUTINE SLSODE (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
REAL Y, T, TOUT, RTOL, ATOL, RWORK
DIMENSION NEQ(*), Y(*), RTOL(*), ATOL(*), RWORK(LRW), IWORK(LIW)
C***BEGIN PROLOGUE SLSODE
C***PURPOSE Livermore Solver for Ordinary Differential Equations.
C SLSODE solves the initial-value problem for stiff or
C nonstiff systems of first-order ODE's,
C dy/dt = f(t,y), or, in component form,
C dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(N)), i=1,...,N.
C***CATEGORY I1A
C***TYPE SINGLE PRECISION (SLSODE-S, DLSODE-D)
C***KEYWORDS ORDINARY DIFFERENTIAL EQUATIONS, INITIAL VALUE PROBLEM,
C STIFF, NONSTIFF
C***AUTHOR Hindmarsh, Alan C., (LLNL)
C Center for Applied Scientific Computing, L-561
C Lawrence Livermore National Laboratory
C Livermore, CA 94551.
C***DESCRIPTION
C
C NOTE: The "Usage" and "Arguments" sections treat only a subset of
C available options, in condensed fashion. The options
C covered and the information supplied will support most
C standard uses of SLSODE.
C
C For more sophisticated uses, full details on all options are
C given in the concluding section, headed "Long Description."
C A synopsis of the SLSODE Long Description is provided at the
C beginning of that section; general topics covered are:
C - Elements of the call sequence; optional input and output
C - Optional supplemental routines in the SLSODE package
C - internal COMMON block
C
C *Usage:
C Communication between the user and the SLSODE package, for normal
C situations, is summarized here. This summary describes a subset
C of the available options. See "Long Description" for complete
C details, including optional communication, nonstandard options,
C and instructions for special situations.
C
C A sample program is given in the "Examples" section.
C
C Refer to the argument descriptions for the definitions of the
C quantities that appear in the following sample declarations.
C
C For MF = 10,
C PARAMETER (LRW = 20 + 16*NEQ, LIW = 20)
C For MF = 21 or 22,
C PARAMETER (LRW = 22 + 9*NEQ + NEQ**2, LIW = 20 + NEQ)
C For MF = 24 or 25,
C PARAMETER (LRW = 22 + 10*NEQ + (2*ML+MU)*NEQ,
C * LIW = 20 + NEQ)
C
C EXTERNAL F, JAC
C INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK(LIW),
C * LIW, MF
C REAL Y(NEQ), T, TOUT, RTOL, ATOL(ntol), RWORK(LRW)
C
C CALL SLSODE (F, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
C * ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JAC, MF)
C
C *Arguments:
C F :EXT Name of subroutine for right-hand-side vector f.
C This name must be declared EXTERNAL in calling
C program. The form of F must be:
C
C SUBROUTINE F (NEQ, T, Y, YDOT)
C INTEGER NEQ
C REAL T, Y(*), YDOT(*)
C
C The inputs are NEQ, T, Y. F is to set
C
C YDOT(i) = f(i,T,Y(1),Y(2),...,Y(NEQ)),
C i = 1, ..., NEQ .
C
C NEQ :IN Number of first-order ODE's.
C
C Y :INOUT Array of values of the y(t) vector, of length NEQ.
C Input: For the first call, Y should contain the
C values of y(t) at t = T. (Y is an input
C variable only if ISTATE = 1.)
C Output: On return, Y will contain the values at the
C new t-value.
C
C T :INOUT Value of the independent variable. On return it
C will be the current value of t (normally TOUT).
C
C TOUT :IN Next point where output is desired (.NE. T).
C
C ITOL :IN 1 or 2 according as ATOL (below) is a scalar or
C an array.
C
C RTOL :IN Relative tolerance parameter (scalar).
C
C ATOL :IN Absolute tolerance parameter (scalar or array).
C If ITOL = 1, ATOL need not be dimensioned.
C If ITOL = 2, ATOL must be dimensioned at least NEQ.
C
C The estimated local error in Y(i) will be controlled
C so as to be roughly less (in magnitude) than
C
C EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
C EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
C
C Thus the local error test passes if, in each
C component, either the absolute error is less than
C ATOL (or ATOL(i)), or the relative error is less
C than RTOL.
C
C Use RTOL = 0.0 for pure absolute error control, and
C use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative
C error control. Caution: Actual (global) errors may
C exceed these local tolerances, so choose them
C conservatively.
C
C ITASK :IN Flag indicating the task SLSODE is to perform.
C Use ITASK = 1 for normal computation of output
C values of y at t = TOUT.
C
C ISTATE:INOUT Index used for input and output to specify the state
C of the calculation.
C Input:
C 1 This is the first call for a problem.
C 2 This is a subsequent call.
C Output:
C 1 Nothing was done, as TOUT was equal to T.
C 2 SLSODE was successful (otherwise, negative).
C Note that ISTATE need not be modified after a
C successful return.
C -1 Excess work done on this call (perhaps wrong
C MF).
C -2 Excess accuracy requested (tolerances too
C small).
C -3 Illegal input detected (see printed message).
C -4 Repeated error test failures (check all
C inputs).
C -5 Repeated convergence failures (perhaps bad
C Jacobian supplied or wrong choice of MF or
C tolerances).
C -6 Error weight became zero during problem
C (solution component i vanished, and ATOL or
C ATOL(i) = 0.).
C
C IOPT :IN Flag indicating whether optional inputs are used:
C 0 No.
C 1 Yes. (See "Optional inputs" under "Long
C Description," Part 1.)
C
C RWORK :WORK Real work array of length at least:
C 20 + 16*NEQ for MF = 10,
C 22 + 9*NEQ + NEQ**2 for MF = 21 or 22,
C 22 + 10*NEQ + (2*ML + MU)*NEQ for MF = 24 or 25.
C
C LRW :IN Declared length of RWORK (in user's DIMENSION
C statement).
C
C IWORK :WORK Integer work array of length at least:
C 20 for MF = 10,
C 20 + NEQ for MF = 21, 22, 24, or 25.
C
C If MF = 24 or 25, input in IWORK(1),IWORK(2) the
C lower and upper Jacobian half-bandwidths ML,MU.
C
C On return, IWORK contains information that may be
C of interest to the user:
C
C Name Location Meaning
C ----- --------- -----------------------------------------
C NST IWORK(11) Number of steps taken for the problem so
C far.
C NFE IWORK(12) Number of f evaluations for the problem
C so far.
C NJE IWORK(13) Number of Jacobian evaluations (and of
C matrix LU decompositions) for the problem
C so far.
C NQU IWORK(14) Method order last used (successfully).
C LENRW IWORK(17) Length of RWORK actually required. This
C is defined on normal returns and on an
C illegal input return for insufficient
C storage.
C LENIW IWORK(18) Length of IWORK actually required. This
C is defined on normal returns and on an
C illegal input return for insufficient
C storage.
C
C LIW :IN Declared length of IWORK (in user's DIMENSION
C statement).
C
C JAC :EXT Name of subroutine for Jacobian matrix (MF =
C 21 or 24). If used, this name must be declared
C EXTERNAL in calling program. If not used, pass a
C dummy name. The form of JAC must be:
C
C SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD)
C INTEGER NEQ, ML, MU, NROWPD
C REAL T, Y(*), PD(NROWPD,*)
C
C See item c, under "Description" below for more
C information about JAC.
C
C MF :IN Method flag. Standard values are:
C 10 Nonstiff (Adams) method, no Jacobian used.
C 21 Stiff (BDF) method, user-supplied full Jacobian.
C 22 Stiff method, internally generated full
C Jacobian.
C 24 Stiff method, user-supplied banded Jacobian.
C 25 Stiff method, internally generated banded
C Jacobian.
C
C *Description:
C SLSODE solves the initial value problem for stiff or nonstiff
C systems of first-order ODE's,
C
C dy/dt = f(t,y) ,
C
C or, in component form,
C
C dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(NEQ))
C (i = 1, ..., NEQ) .
C
C SLSODE is a package based on the GEAR and GEARB packages, and on
C the October 23, 1978, version of the tentative ODEPACK user
C interface standard, with minor modifications.
C
C The steps in solving such a problem are as follows.
C
C a. First write a subroutine of the form
C
C SUBROUTINE F (NEQ, T, Y, YDOT)
C INTEGER NEQ
C REAL T, Y(*), YDOT(*)
C
C which supplies the vector function f by loading YDOT(i) with
C f(i).
C
C b. Next determine (or guess) whether or not the problem is stiff.
C Stiffness occurs when the Jacobian matrix df/dy has an
C eigenvalue whose real part is negative and large in magnitude
C compared to the reciprocal of the t span of interest. If the
C problem is nonstiff, use method flag MF = 10. If it is stiff,
C there are four standard choices for MF, and SLSODE requires the
C Jacobian matrix in some form. This matrix is regarded either
C as full (MF = 21 or 22), or banded (MF = 24 or 25). In the
C banded case, SLSODE requires two half-bandwidth parameters ML
C and MU. These are, respectively, the widths of the lower and
C upper parts of the band, excluding the main diagonal. Thus the
C band consists of the locations (i,j) with
C
C i - ML <= j <= i + MU ,
C
C and the full bandwidth is ML + MU + 1 .
C
C c. If the problem is stiff, you are encouraged to supply the
C Jacobian directly (MF = 21 or 24), but if this is not feasible,
C SLSODE will compute it internally by difference quotients (MF =
C 22 or 25). If you are supplying the Jacobian, write a
C subroutine of the form
C
C SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD)
C INTEGER NEQ, ML, MU, NRWOPD
C REAL T, Y(*), PD(NROWPD,*)
C
C which provides df/dy by loading PD as follows:
C - For a full Jacobian (MF = 21), load PD(i,j) with df(i)/dy(j),
C the partial derivative of f(i) with respect to y(j). (Ignore
C the ML and MU arguments in this case.)
C - For a banded Jacobian (MF = 24), load PD(i-j+MU+1,j) with
C df(i)/dy(j); i.e., load the diagonal lines of df/dy into the
C rows of PD from the top down.
C - In either case, only nonzero elements need be loaded.
C
C d. Write a main program that calls subroutine SLSODE once for each
C point at which answers are desired. This should also provide
C for possible use of logical unit 6 for output of error messages
C by SLSODE.
C
C Before the first call to SLSODE, set ISTATE = 1, set Y and T to
C the initial values, and set TOUT to the first output point. To
C continue the integration after a successful return, simply
C reset TOUT and call SLSODE again. No other parameters need be
C reset.
C
C *Examples:
C The following is a simple example problem, with the coding needed
C for its solution by SLSODE. The problem is from chemical kinetics,
C and consists of the following three rate equations:
C
C dy1/dt = -.04*y1 + 1.E4*y2*y3
C dy2/dt = .04*y1 - 1.E4*y2*y3 - 3.E7*y2**2
C dy3/dt = 3.E7*y2**2
C
C on the interval from t = 0.0 to t = 4.E10, with initial conditions
C y1 = 1.0, y2 = y3 = 0. The problem is stiff.
C
C The following coding solves this problem with SLSODE, using
C MF = 21 and printing results at t = .4, 4., ..., 4.E10. It uses
C ITOL = 2 and ATOL much smaller for y2 than for y1 or y3 because y2
C has much smaller values. At the end of the run, statistical
C quantities of interest are printed.
C
C EXTERNAL FEX, JEX
C INTEGER IOPT, IOUT, ISTATE, ITASK, ITOL, IWORK(23), LIW, LRW,
C * MF, NEQ
C REAL ATOL(3), RTOL, RWORK(58), T, TOUT, Y(3)
C NEQ = 3
C Y(1) = 1.
C Y(2) = 0.
C Y(3) = 0.
C T = 0.
C TOUT = .4
C ITOL = 2
C RTOL = 1.E-4
C ATOL(1) = 1.E-6
C ATOL(2) = 1.E-10
C ATOL(3) = 1.E-6
C ITASK = 1
C ISTATE = 1
C IOPT = 0
C LRW = 58
C LIW = 23
C MF = 21
C DO 40 IOUT = 1,12
C CALL SLSODE (FEX, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
C * ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JEX, MF)
C WRITE(6,20) T, Y(1), Y(2), Y(3)
C 20 FORMAT(' At t =',E12.4,' y =',3E14.6)
C IF (ISTATE .LT. 0) GO TO 80
C 40 TOUT = TOUT*10.
C WRITE(6,60) IWORK(11), IWORK(12), IWORK(13)
C 60 FORMAT(/' No. steps =',i4,', No. f-s =',i4,', No. J-s =',i4)
C STOP
C 80 WRITE(6,90) ISTATE
C 90 FORMAT(///' Error halt.. ISTATE =',I3)
C STOP
C END
C
C SUBROUTINE FEX (NEQ, T, Y, YDOT)
C INTEGER NEQ
C REAL T, Y(3), YDOT(3)
C YDOT(1) = -.04*Y(1) + 1.E4*Y(2)*Y(3)
C YDOT(3) = 3.E7*Y(2)*Y(2)
C YDOT(2) = -YDOT(1) - YDOT(3)
C RETURN
C END
C
C SUBROUTINE JEX (NEQ, T, Y, ML, MU, PD, NRPD)
C INTEGER NEQ, ML, MU, NRPD
C REAL T, Y(3), PD(NRPD,3)
C PD(1,1) = -.04
C PD(1,2) = 1.E4*Y(3)
C PD(1,3) = 1.E4*Y(2)
C PD(2,1) = .04
C PD(2,3) = -PD(1,3)
C PD(3,2) = 6.E7*Y(2)
C PD(2,2) = -PD(1,2) - PD(3,2)
C RETURN
C END
C
C The output from this program (on a Cray-1 in single precision)
C is as follows.
C
C At t = 4.0000e-01 y = 9.851726e-01 3.386406e-05 1.479357e-02
C At t = 4.0000e+00 y = 9.055142e-01 2.240418e-05 9.446344e-02
C At t = 4.0000e+01 y = 7.158050e-01 9.184616e-06 2.841858e-01
C At t = 4.0000e+02 y = 4.504846e-01 3.222434e-06 5.495122e-01
C At t = 4.0000e+03 y = 1.831701e-01 8.940379e-07 8.168290e-01
C At t = 4.0000e+04 y = 3.897016e-02 1.621193e-07 9.610297e-01
C At t = 4.0000e+05 y = 4.935213e-03 1.983756e-08 9.950648e-01
C At t = 4.0000e+06 y = 5.159269e-04 2.064759e-09 9.994841e-01
C At t = 4.0000e+07 y = 5.306413e-05 2.122677e-10 9.999469e-01
C At t = 4.0000e+08 y = 5.494530e-06 2.197825e-11 9.999945e-01
C At t = 4.0000e+09 y = 5.129458e-07 2.051784e-12 9.999995e-01
C At t = 4.0000e+10 y = -7.170603e-08 -2.868241e-13 1.000000e+00
C
C No. steps = 330, No. f-s = 405, No. J-s = 69
C
C *Accuracy:
C The accuracy of the solution depends on the choice of tolerances
C RTOL and ATOL. Actual (global) errors may exceed these local
C tolerances, so choose them conservatively.
C
C *Cautions:
C The work arrays should not be altered between calls to SLSODE for
C the same problem, except possibly for the conditional and optional
C inputs.
C
C *Portability:
C Since NEQ is dimensioned inside SLSODE, some compilers may object
C to a call to SLSODE with NEQ a scalar variable. In this event,
C use DIMENSION NEQ(1). Similar remarks apply to RTOL and ATOL.
C
C Note to Cray users:
C For maximum efficiency, use the CFT77 compiler. Appropriate
C compiler optimization directives have been inserted for CFT77.
C
C *Reference:
C Alan C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE
C Solvers," in Scientific Computing, R. S. Stepleman, et al., Eds.
C (North-Holland, Amsterdam, 1983), pp. 55-64.
C
C *Long Description:
C The following complete description of the user interface to
C SLSODE consists of four parts:
C
C 1. The call sequence to subroutine SLSODE, which is a driver
C routine for the solver. This includes descriptions of both
C the call sequence arguments and user-supplied routines.
C Following these descriptions is a description of optional
C inputs available through the call sequence, and then a
C description of optional outputs in the work arrays.
C
C 2. Descriptions of other routines in the SLSODE package that may
C be (optionally) called by the user. These provide the ability
C to alter error message handling, save and restore the internal
C COMMON, and obtain specified derivatives of the solution y(t).
C
C 3. Descriptions of COMMON block to be declared in overlay or
C similar environments, or to be saved when doing an interrupt
C of the problem and continued solution later.
C
C 4. Description of two routines in the SLSODE package, either of
C which the user may replace with his own version, if desired.
C These relate to the measurement of errors.
C
C
C Part 1. Call Sequence
C ----------------------
C
C Arguments
C ---------
C The call sequence parameters used for input only are
C
C F, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK, IOPT, LRW, LIW, JAC, MF,
C
C and those used for both input and output are
C
C Y, T, ISTATE.
C
C The work arrays RWORK and IWORK are also used for conditional and
C optional inputs and optional outputs. (The term output here
C refers to the return from subroutine SLSODE to the user's calling
C program.)
C
C The legality of input parameters will be thoroughly checked on the
C initial call for the problem, but not checked thereafter unless a
C change in input parameters is flagged by ISTATE = 3 on input.
C
C The descriptions of the call arguments are as follows.
C
C F The name of the user-supplied subroutine defining the ODE
C system. The system must be put in the first-order form
C dy/dt = f(t,y), where f is a vector-valued function of
C the scalar t and the vector y. Subroutine F is to compute
C the function f. It is to have the form
C
C SUBROUTINE F (NEQ, T, Y, YDOT)
C REAL T, Y(*), YDOT(*)
C
C where NEQ, T, and Y are input, and the array YDOT =
C f(T,Y) is output. Y and YDOT are arrays of length NEQ.
C Subroutine F should not alter Y(1),...,Y(NEQ). F must be
C declared EXTERNAL in the calling program.
C
C Subroutine F may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),..., if NEQ is an array
C (dimensioned in F) and/or Y has length exceeding NEQ(1).
C See the descriptions of NEQ and Y below.
C
C If quantities computed in the F routine are needed
C externally to SLSODE, 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 SINTDY
C instead.
C
C NEQ The size of the ODE system (number of first-order
C ordinary differential equations). Used only for input.
C NEQ may be decreased, but not increased, during the
C problem. If NEQ is decreased (with ISTATE = 3 on input),
C the remaining components of Y should be left undisturbed,
C if these are to be accessed in F and/or JAC.
C
C Normally, NEQ is a scalar, and it is generally referred
C to as a scalar in this user interface description.
C However, NEQ may be an array, with NEQ(1) set to the
C system size. (The SLSODE package accesses only NEQ(1).)
C In either case, this parameter is passed as the NEQ
C argument in all calls to F and JAC. Hence, if it is an
C array, locations NEQ(2),... may be used to store other
C integer data and pass it to F and/or JAC. Subroutines
C F and/or JAC must include NEQ in a DIMENSION statement
C in that case.
C
C Y A real array for the vector of dependent variables, of
C length NEQ or more. Used for both input and output on
C the first call (ISTATE = 1), and only for output on
C other calls. On the first call, Y must contain the
C vector of initial values. On output, Y contains the
C computed solution vector, evaluated at T. If desired,
C the Y array may be used for other purposes between
C calls to the solver.
C
C This array is passed as the Y argument in all calls to F
C and JAC. Hence its length may exceed NEQ, and locations
C Y(NEQ+1),... may be used to store other real data and
C pass it to F and/or JAC. (The SLSODE package accesses
C only Y(1),...,Y(NEQ).)
C
C T The independent variable. On input, T is used only on
C the first call, as the initial point of the integration.
C On output, after each call, T is the value at which a
C computed solution Y is evaluated (usually the same as
C TOUT). On an error return, T is the farthest point
C reached.
C
C TOUT The next value of T at which a computed solution is
C desired. Used only for input.
C
C When starting the problem (ISTATE = 1), TOUT may be equal
C to T for one call, then should not equal T for the next
C call. For the initial T, an input value of TOUT .NE. T
C is used in order to determine the direction of the
C integration (i.e., the algebraic sign of the step sizes)
C and the rough scale of the problem. Integration in
C either direction (forward or backward in T) is permitted.
C
C If ITASK = 2 or 5 (one-step modes), TOUT is ignored
C after the first call (i.e., the first call with
C TOUT .NE. T). Otherwise, TOUT is required on every call.
C
C If ITASK = 1, 3, or 4, the values of TOUT need not be
C monotone, but a value of TOUT which backs up is limited
C to the current internal T interval, whose endpoints are
C TCUR - HU and TCUR. (See "Optional Outputs" below for
C TCUR and HU.)
C
C
C ITOL An indicator for the type of error control. See
C description below under ATOL. Used only for input.
C
C RTOL A relative error tolerance parameter, either a scalar or
C an array of length NEQ. See description below under
C ATOL. Input only.
C
C ATOL An absolute error tolerance parameter, either a scalar or
C an array of length NEQ. Input only.
C
C The input parameters ITOL, RTOL, and ATOL determine the
C error control performed by the solver. The solver will
C control the vector e = (e(i)) of estimated local errors
C in Y, according to an inequality of the form
C
C rms-norm of ( e(i)/EWT(i) ) <= 1,
C
C where
C
C EWT(i) = RTOL(i)*ABS(Y(i)) + ATOL(i),
C
C and the rms-norm (root-mean-square norm) here is
C
C rms-norm(v) = SQRT(sum v(i)**2 / NEQ).
C
C Here EWT = (EWT(i)) is a vector of weights which must
C always be positive, and the values of RTOL and ATOL
C should all be nonnegative. The following table gives the
C types (scalar/array) of RTOL and ATOL, and the
C corresponding form of EWT(i).
C
C ITOL RTOL ATOL EWT(i)
C ---- ------ ------ -----------------------------
C 1 scalar scalar RTOL*ABS(Y(i)) + ATOL
C 2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
C 3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
C 4 array array RTOL(i)*ABS(Y(i)) + ATOL(i)
C
C When either of these parameters is a scalar, it need not
C be dimensioned in the user's calling program.
C
C If none of the above choices (with ITOL, RTOL, and ATOL
C fixed throughout the problem) is suitable, more general
C error controls can be obtained by substituting
C user-supplied routines for the setting of EWT and/or for
C the norm calculation. See Part 4 below.
C
C If global errors are to be estimated by making a repeated
C run on the same problem with smaller tolerances, then all
C components of RTOL and ATOL (i.e., of EWT) should be
C scaled down uniformly.
C
C ITASK An index specifying the task to be performed. Input
C only. ITASK has the following values and meanings:
C 1 Normal computation of output values of y(t) at
C t = TOUT (by overshooting and interpolating).
C 2 Take one step only and return.
C 3 Stop at the first internal mesh point at or beyond
C t = TOUT and return.
C 4 Normal computation of output values of y(t) at
C t = TOUT but without overshooting t = TCRIT. TCRIT
C must be input as RWORK(1). TCRIT may be equal to or
C beyond TOUT, but not behind it in the direction of
C integration. This option is useful if the problem
C has a singularity at or beyond t = TCRIT.
C 5 Take one step, without passing TCRIT, and return.
C TCRIT must be input as RWORK(1).
C
C Note: If ITASK = 4 or 5 and the solver reaches TCRIT
C (within roundoff), it will return T = TCRIT (exactly) to
C indicate this (unless ITASK = 4 and TOUT comes before
C TCRIT, in which case answers at T = TOUT are returned
C first).
C
C ISTATE An index used for input and output to specify the state
C of the calculation.
C
C On input, the values of ISTATE are as follows:
C 1 This is the first call for the problem
C (initializations will be done). See "Note" below.
C 2 This is not the first call, and the calculation is to
C continue normally, with no change in any input
C parameters except possibly TOUT and ITASK. (If ITOL,
C RTOL, and/or ATOL are changed between calls with
C ISTATE = 2, the new values will be used but not
C tested for legality.)
C 3 This is not the first call, and the calculation is to
C continue normally, but with a change in input
C parameters other than TOUT and ITASK. Changes are
C allowed in NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, MF,
C ML, MU, and any of the optional inputs except H0.
C (See IWORK description for ML and MU.)
C
C Note: A preliminary call with TOUT = T is not counted as
C a first call here, as no initialization or checking of
C input is done. (Such a call is sometimes useful for the
C purpose of outputting the initial conditions.) Thus the
C first call for which TOUT .NE. T requires ISTATE = 1 on
C input.
C
C On output, ISTATE has the following values and meanings:
C 1 Nothing was done, as TOUT was equal to T with
C ISTATE = 1 on input.
C 2 The integration was performed successfully.
C -1 An excessive amount of work (more than MXSTEP steps)
C was done on this call, before completing the
C requested task, but the integration was otherwise
C successful as far as T. (MXSTEP is an optional input
C and is normally 500.) To continue, the user may
C simply reset ISTATE to a value >1 and call again (the
C excess work step counter will be reset to 0). In
C addition, the user may increase MXSTEP to avoid this
C error return; see "Optional Inputs" below.
C -2 Too much accuracy was requested for the precision of
C the machine being used. This was detected before
C completing the requested task, but the integration
C was successful as far as T. To continue, the
C tolerance parameters must be reset, and ISTATE must
C be set to 3. The optional output TOLSF may be used
C for this purpose. (Note: If this condition is
C detected before taking any steps, then an illegal
C input return (ISTATE = -3) occurs instead.)
C -3 Illegal input was detected, before taking any
C integration steps. See written message for details.
C (Note: If the solver detects an infinite loop of
C calls to the solver with illegal input, it will cause
C the run to stop.)
C -4 There were repeated error-test failures on one
C attempted step, before completing the requested task,
C but the integration was successful as far as T. The
C problem may have a singularity, or the input may be
C inappropriate.
C -5 There were repeated convergence-test failures on one
C attempted step, before completing the requested task,
C but the integration was successful as far as T. This
C may be caused by an inaccurate Jacobian matrix, if
C one is being used.
C -6 EWT(i) became zero for some i during the integration.
C Pure relative error control (ATOL(i)=0.0) was
C requested on a variable which has now vanished. The
C integration was successful as far as T.
C
C Note: Since the normal output value of ISTATE is 2, it
C does not need to be reset for normal continuation. Also,
C since a negative input value of ISTATE will be regarded
C as illegal, a negative output value requires the user to
C change it, and possibly other inputs, before calling the
C solver again.
C
C IOPT An integer flag to specify whether any optional inputs
C are being used on this call. Input only. The optional
C inputs are listed under a separate heading below.
C 0 No optional inputs are being used. Default values
C will be used in all cases.
C 1 One or more optional inputs are being used.
C
C RWORK A real working array (single precision). The length of
C RWORK must be at least
C
C 20 + NYH*(MAXORD + 1) + 3*NEQ + LWM
C
C where
C NYH = the initial value of NEQ,
C MAXORD = 12 (if METH = 1) or 5 (if METH = 2) (unless a
C smaller value is given as an optional input),
C LWM = 0 if MITER = 0,
C LWM = NEQ**2 + 2 if MITER = 1 or 2,
C LWM = NEQ + 2 if MITER = 3, and
C LWM = (2*ML + MU + 1)*NEQ + 2
C if MITER = 4 or 5.
C (See the MF description below for METH and MITER.)
C
C Thus if MAXORD has its default value and NEQ is constant,
C this length is:
C 20 + 16*NEQ for MF = 10,
C 22 + 16*NEQ + NEQ**2 for MF = 11 or 12,
C 22 + 17*NEQ for MF = 13,
C 22 + 17*NEQ + (2*ML + MU)*NEQ for MF = 14 or 15,
C 20 + 9*NEQ for MF = 20,
C 22 + 9*NEQ + NEQ**2 for MF = 21 or 22,
C 22 + 10*NEQ for MF = 23,
C 22 + 10*NEQ + (2*ML + MU)*NEQ for MF = 24 or 25.
C
C The first 20 words of RWORK are reserved for conditional
C and optional inputs and optional outputs.
C
C The following word in RWORK is a conditional input:
C RWORK(1) = TCRIT, the critical value of t which the
C solver is not to overshoot. Required if ITASK
C is 4 or 5, and ignored otherwise. See ITASK.
C
C LRW The length of the array RWORK, as declared by the user.
C (This will be checked by the solver.)
C
C IWORK An integer work array. Its length must be at least
C 20 if MITER = 0 or 3 (MF = 10, 13, 20, 23), or
C 20 + NEQ otherwise (MF = 11, 12, 14, 15, 21, 22, 24, 25).
C (See the MF description below for MITER.) The first few
C words of IWORK are used for conditional and optional
C inputs and optional outputs.
C
C The following two words in IWORK are conditional inputs:
C IWORK(1) = ML These are the lower and upper half-
C IWORK(2) = MU bandwidths, respectively, of the banded
C Jacobian, excluding the main diagonal.
C The band is defined by the matrix locations
C (i,j) with i - ML <= j <= i + MU. ML and MU
C must satisfy 0 <= ML,MU <= NEQ - 1. These are
C required if MITER is 4 or 5, and ignored
C otherwise. ML and MU may in fact be the band
C parameters for a matrix to which df/dy is only
C approximately equal.
C
C LIW The length of the array IWORK, as declared by the user.
C (This will be checked by the solver.)
C
C Note: The work arrays must not be altered between calls to SLSODE
C for the same problem, except possibly for the conditional and
C optional inputs, and except for the last 3*NEQ words of RWORK.
C The latter space is used for internal scratch space, and so is
C available for use by the user outside SLSODE between calls, if
C desired (but not for use by F or JAC).
C
C JAC The name of the user-supplied routine (MITER = 1 or 4) to
C compute the Jacobian matrix, df/dy, as a function of the
C scalar t and the vector y. (See the MF description below
C for MITER.) It is to have the form
C
C SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD)
C REAL T, Y(*), PD(NROWPD,*)
C
C where NEQ, T, Y, ML, MU, and NROWPD are input and the
C array PD is to be loaded with partial derivatives
C (elements of the Jacobian matrix) on output. PD must be
C given a first dimension of NROWPD. T and Y have the same
C meaning as in subroutine F.
C
C In the full matrix case (MITER = 1), ML and MU are
C ignored, and the Jacobian is to be loaded into PD in
C columnwise manner, with df(i)/dy(j) loaded into PD(i,j).
C
C In the band matrix case (MITER = 4), the elements within
C the band are to be loaded into PD in columnwise manner,
C with diagonal lines of df/dy loaded into the rows of PD.
C Thus df(i)/dy(j) is to be loaded into PD(i-j+MU+1,j). ML
C and MU are the half-bandwidth parameters (see IWORK).
C The locations in PD in the two triangular areas which
C correspond to nonexistent matrix elements can be ignored
C or loaded arbitrarily, as they are overwritten by SLSODE.
C
C JAC need not provide df/dy exactly. A crude approximation
C (possibly with a smaller bandwidth) will do.
C
C In either case, PD is preset to zero by the solver, so
C that only the nonzero elements need be loaded by JAC.
C Each call to JAC is preceded by a call to F with the same
C arguments NEQ, T, and Y. Thus to gain some efficiency,
C intermediate quantities shared by both calculations may
C be saved in a user COMMON block by F and not recomputed
C by JAC, if desired. Also, JAC may alter the Y array, if
C desired. JAC must be declared EXTERNAL in the calling
C program.
C
C Subroutine JAC may access user-defined quantities in
C NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
C (dimensioned in JAC) and/or Y has length exceeding
C NEQ(1). See the descriptions of NEQ and Y above.
C
C MF The method flag. Used only for input. The legal values
C of MF are 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24,
C and 25. MF has decimal digits METH and MITER:
C MF = 10*METH + MITER .
C
C METH indicates the basic linear multistep method:
C 1 Implicit Adams method.
C 2 Method based on backward differentiation formulas
C (BDF's).
C
C MITER indicates the corrector iteration method:
C 0 Functional iteration (no Jacobian matrix is
C involved).
C 1 Chord iteration with a user-supplied full (NEQ by
C NEQ) Jacobian.
C 2 Chord iteration with an internally generated
C (difference quotient) full Jacobian (using NEQ
C extra calls to F per df/dy value).
C 3 Chord iteration with an internally generated
C diagonal Jacobian approximation (using one extra call
C to F per df/dy evaluation).
C 4 Chord iteration with a user-supplied banded Jacobian.
C 5 Chord iteration with an internally generated banded
C Jacobian (using ML + MU + 1 extra calls to F per
C df/dy evaluation).
C
C If MITER = 1 or 4, the user must supply a subroutine JAC
C (the name is arbitrary) as described above under JAC.
C For other values of MITER, a dummy argument can be used.
C
C Optional Inputs
C ---------------
C The following is a list of the optional inputs provided for in the
C call sequence. (See also Part 2.) For each such input variable,
C this table lists its name as used in this documentation, its
C location in the call sequence, its meaning, and the default value.
C The use of any of these inputs requires IOPT = 1, and in that case
C all of these inputs are examined. A value of zero for any of
C these optional inputs will cause the default value to be used.
C Thus to use a subset of the optional inputs, simply preload
C locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively,
C and then set those of interest to nonzero values.
C
C Name Location Meaning and default value
C ------ --------- -----------------------------------------------
C H0 RWORK(5) Step size to be attempted on the first step.
C The default value is determined by the solver.
C HMAX RWORK(6) Maximum absolute step size allowed. The
C default value is infinite.
C HMIN RWORK(7) Minimum absolute step size allowed. The
C default value is 0. (This lower bound is not
C enforced on the final step before reaching
C TCRIT when ITASK = 4 or 5.)
C MAXORD IWORK(5) Maximum order to be allowed. The default value
C is 12 if METH = 1, and 5 if METH = 2. (See the
C MF description above for METH.) If MAXORD
C exceeds the default value, it will be reduced
C to the default value. If MAXORD is changed
C during the problem, it may cause the current
C order to be reduced.
C MXSTEP IWORK(6) Maximum number of (internally defined) steps
C allowed during one call to the solver. The
C default value is 500.
C MXHNIL IWORK(7) Maximum number of messages printed (per
C problem) warning that T + H = T on a step
C (H = step size). This must be positive to
C result in a nondefault value. The default
C value is 10.
C
C Optional Outputs
C ----------------
C As optional additional output from SLSODE, the variables listed
C below are quantities related to the performance of SLSODE which
C are available to the user. These are communicated by way of the
C work arrays, but also have internal mnemonic names as shown.
C Except where stated otherwise, all of these outputs are defined on
C any successful return from SLSODE, and on any return with ISTATE =
C -1, -2, -4, -5, or -6. On an illegal input return (ISTATE = -3),
C they will be unchanged from their existing values (if any), except
C possibly for TOLSF, LENRW, and LENIW. On any error return,
C outputs relevant to the error will be defined, as noted below.
C
C Name Location Meaning
C ----- --------- ------------------------------------------------
C HU RWORK(11) Step size in t last used (successfully).
C HCUR RWORK(12) Step size to be attempted on the next step.
C TCUR RWORK(13) Current value of the independent variable which
C the solver has actually reached, i.e., the
C current internal mesh point in t. On output,
C TCUR will always be at least as far as the
C argument T, but may be farther (if interpolation
C was done).
C TOLSF RWORK(14) Tolerance scale factor, greater than 1.0,
C computed when a request for too much accuracy
C was detected (ISTATE = -3 if detected at the
C start of the problem, ISTATE = -2 otherwise).
C If ITOL is left unaltered but RTOL and ATOL are
C uniformly scaled up by a factor of TOLSF for the
C next call, then the solver is deemed likely to
C succeed. (The user may also ignore TOLSF and
C alter the tolerance parameters in any other way
C appropriate.)
C NST IWORK(11) Number of steps taken for the problem so far.
C NFE IWORK(12) Number of F evaluations for the problem so far.
C NJE IWORK(13) Number of Jacobian evaluations (and of matrix LU
C decompositions) for the problem so far.
C NQU IWORK(14) Method order last used (successfully).
C NQCUR IWORK(15) Order to be attempted on the next step.
C IMXER IWORK(16) Index of the component of largest magnitude in
C the weighted local error vector ( e(i)/EWT(i) ),
C on an error return with ISTATE = -4 or -5.
C LENRW IWORK(17) Length of RWORK actually required. This is
C defined on normal returns and on an illegal
C input return for insufficient storage.
C LENIW IWORK(18) Length of IWORK actually required. This is
C defined on normal returns and on an illegal
C input return for insufficient storage.
C
C The following two arrays are segments of the RWORK array which may
C also be of interest to the user as optional outputs. For each
C array, the table below gives its internal name, its base address
C in RWORK, and its description.
C
C Name Base address Description
C ---- ------------ ----------------------------------------------
C YH 21 The Nordsieck history array, of size NYH by
C (NQCUR + 1), where NYH is the initial value of
C NEQ. For j = 0,1,...,NQCUR, column j + 1 of
C YH contains HCUR**j/factorial(j) times the jth
C derivative of the interpolating polynomial
C currently representing the solution, evaluated
C at t = TCUR.
C ACOR LENRW-NEQ+1 Array of size NEQ used for the accumulated
C corrections on each step, scaled on output to
C represent the estimated local error in Y on
C the last step. This is the vector e in the
C description of the error control. It is
C defined only on successful return from SLSODE.
C
C
C Part 2. Other Callable Routines
C --------------------------------
C
C The following are optional calls which the user may make to gain
C additional capabilities in conjunction with SLSODE.
C
C Form of call Function
C ------------------------ ----------------------------------------
C CALL XSETUN(LUN) Set the logical unit number, LUN, for
C output of messages from SLSODE, if the
C default is not desired. The default
C value of LUN is 6. This call may be made
C at any time and will take effect
C immediately.
C CALL XSETF(MFLAG) Set a flag to control the printing of
C messages by SLSODE. MFLAG = 0 means do
C not print. (Danger: this risks losing
C valuable information.) MFLAG = 1 means
C print (the default). This call may be
C made at any time and will take effect
C immediately.
C CALL SSRCOM(RSAV,ISAV,JOB) Saves and restores the contents of the
C internal COMMON blocks used by SLSODE
C (see Part 3 below). RSAV must be a
C real array of length 218 or more, and
C ISAV must be an integer array of length
C 37 or more. JOB = 1 means save COMMON
C into RSAV/ISAV. JOB = 2 means restore
C COMMON from same. SSRCOM is useful if
C one is interrupting a run and restarting
C later, or alternating between two or
C more problems solved with SLSODE.
C CALL SINTDY(,,,,,) Provide derivatives of y, of various
C (see below) orders, at a specified point t, if
C desired. It may be called only after a
C successful return from SLSODE. Detailed
C instructions follow.
C
C Detailed instructions for using SINTDY
C --------------------------------------
C The form of the CALL is:
C
C CALL SINTDY (T, K, RWORK(21), NYH, DKY, IFLAG)
C
C The input parameters are:
C
C T Value of independent variable where answers are
C desired (normally the same as the T last returned by
C SLSODE). For valid results, T must lie between
C TCUR - HU and TCUR. (See "Optional Outputs" above
C for TCUR and HU.)
C K Integer order of the derivative desired. K must
C satisfy 0 <= K <= NQCUR, where NQCUR is the current