dlsode Subroutine

public subroutine dlsode(f, Neq, Y, T, Tout, Itol, Rtol, Atol, Itask, Istate, Iopt, Rwork, Lrw, Iwork, Liw, jac, Mf)

Synopsis

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.

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 (external)

; 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 (intent IN)

Number of first-order ODE’s.

Y (intent 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 (intent INOUT)

Value of the independent variable. On return it will be the current value of t (normally TOUT).

TOUT (intent IN)

Next point where output is desired (.NE. T).

ITOL (intent IN)

1 or 2 according as ATOL (below) is a scalar or an array.

RTOL (intent IN)

Relative tolerance parameter (scalar).

ATOL (intent 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 (intent IN)

Flag indicating the task DLSODE is to perform. Use ITASK = 1 for normal computation of output values of y at t = TOUT.

ISTATE (intent INOUT)

Index used for input and output to specify the state of the calculation.

Input:

Output:

IOPT (intent IN)

Flag indicating whether optional inputs are used:

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 (intent 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:

LIW (intent IN)

Declared length of IWORK (in user’s DIMENSION statement).

JAC (external)

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 (intent IN)

Method flag. Standard values are:

Long 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.

program dlsode_ex
use m_odepack
implicit none
external fex
external jex

integer,parameter            ::  dp=kind(0.0d0)
real(kind=dp),dimension(3)   ::  atol,y
integer                      ::  iopt,iout,istate,itask,itol,liw,lrw,mf,neq
integer,dimension(23)        ::  iwork
real(kind=dp)                ::  rtol,t,tout
real(kind=dp),dimension(58)  ::  rwork

   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 iout = 1,12
      call dlsode(fex,[neq],y,t,tout,itol,[rtol],atol,itask,istate,iopt,   &
                & rwork,lrw,iwork,liw,jex,mf)
      write (6,99010) t,y(1),y(2),y(3)
   99010 format (' At t =',d12.4,'   y =',3D14.6)
      if ( istate<0 ) then
         write (6,99020) istate
   99020 format (///' Error halt.. ISTATE =',i3)
         stop 1
      else
         tout = tout*10.D0
      endif
   enddo
   write (6,99030) iwork(11),iwork(12),iwork(13)
   99030 format (/' No. steps =',i4,',  No. f-s =',i4,',  No. J-s =',i4)

end program dlsode_ex

subroutine fex(Neq,T,Y,Ydot)
implicit none
integer,parameter                         ::  dp=kind(0.0d0)

integer                                   ::  Neq
real(kind=dp)                             ::  T
real(kind=dp),intent(in),dimension(3)     ::  Y
real(kind=dp),intent(inout),dimension(3)  ::  Ydot

   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)
end subroutine fex

subroutine jex(Neq,T,Y,Ml,Mu,Pd,Nrpd)
implicit none

integer,parameter                              ::  dp=kind(0.0d0)
integer                                        ::  Neq
real(kind=dp)                                  ::  T
real(kind=dp),intent(in),dimension(3)          ::  Y
integer                                        ::  Ml
integer                                        ::  Mu
real(kind=dp),intent(inout),dimension(Nrpd,3)  ::  Pd
integer,intent(in)                             ::  Nrpd

   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)
end subroutine jex

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

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

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

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

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

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

indicator for the type of error control. See description below under ATOL. Used only for input.

RTOL

relative error tolerance parameter, either a scalar or an array of length NEQ. See description below under ATOL. Input only.

ATOL

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).

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

index specifying the task to be performed. Input only. ITASK has the following values and meanings:

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

index used for input and output to specify the state of the calculation.

On input, the values of ISTATE are as follows:

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:

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

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

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

length of the array RWORK, as declared by the user. (This will be checked by the solver.)

IWORK

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:

MITER indicates the corrector iteration method:

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.

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.

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.

Part 2. Other Callable Routines

The following are optional calls which the user may make to gain additional capabilities in conjunction with DLSODE.

Detailed instructions for using DINTDY

The form of the CALL is:

    CALL DINTDY (T, K, RWORK(21), NYH, DKY, IFLAG)

The input parameters are:

The output parameters are:

Part 3. Save and Restore Current State

If the solution of a given problem by DLSODE is to be interrupted and then later continued, as when restarting an interrupted run or alternating between two or more problems, the user should save, following the return from the last DLSODE call prior to the interruption, the contents of the call sequence variables and the internal state values, and later restore these values before the next DLSODE call for that problem. In addition, if XSETUN and/or XSETF was called for non-default handling of error messages, then these calls must be repeated. To save and restore the current state use subroutine DSRCOM (see Part 2 above).

Part 4. Optionally Replaceable Solver Routines

Below are descriptions of two routines in the DLSODE package which relate to the measurement of errors. Either routine can be replaced by a user-supplied version, if desired. However, since such a replacement may have a major impact on performance, it should be done only when absolutely necessary, and only with great caution. (Note: The means by which the package version of a routine is superseded by the user’s version may be system- dependent.)

DEWSET()

The following subroutine is called just before each internal integration step, and sets the array of error weights, EWT, as described under ITOL/RTOL/ATOL above:

       SUBROUTINE DEWSET (NEQ, ITOL, RTOL, ATOL, YCUR, EWT)

where NEQ, ITOL, RTOL, and ATOL are as in the DLSODE call sequence, YCUR contains the current dependent variable vector, and EWT is the array of weights set by DEWSET.

If the user supplies this subroutine, it must return in EWT(i) (i = 1,…,NEQ) a positive quantity suitable for comparing errors in Y(i) to. The EWT array returned by DEWSET is passed to the DVNORM routine (see below), and also used by DLSODE in the computation of the optional output IMXER, the diagonal Jacobian approximation, and the increments for difference quotient Jacobians.

In the user-supplied version of DEWSET, it may be desirable to use the current values of derivatives of y. Derivatives up to order NQ are available from the history array YH, described above under optional outputs. In DEWSET, YH is identical to the YCUR array, extended to NQ + 1 columns with a column length of NYH and scale factors of H**j/factorial(j). On the first call for the problem, given by NST = 0, NQ is 1 and H is temporarily set to 1.0. NYH is the initial value of NEQ. The quantities NQ, H, and NST can be obtained by including in SEWSET the statements:

       DOUBLE PRECISION RLS
       COMMON /DLS001/ RLS(218),ILS(37)
       NQ = ILS(33)
       NST = ILS(34)
       H = RLS(212)

Thus, for example, the current value of dy/dt can be obtained as YCUR(NYH+i)/H (i=1,…,NEQ) (and the division by H is unnecessary when NST = 0).

DVNORM()

DVNORM is a real function routine which computes the weighted root-mean-square norm of a vector v:

        d = DVNORM (n, v, w)

     where:
     n = the length of the vector,
     v = real array of length n containing the vector,
     w = real array of length n containing weights,
     d = SQRT( (1/n) * sum(v(i)*w(i))**2 ).

DVNORM is called with n = NEQ and with w(i) = 1.0/EWT(i), where EWT is as set by subroutine DEWSET.

If the user supplies this function, it should return a nonnegative value of DVNORM suitable for use in the error control in DLSODE. None of the arguments should be altered by DVNORM. For example, a user-supplied DVNORM routine might:

• Substitute a max-norm of (v(i)*w(i)) for the rms-norm, or
• Ignore some components of v in the norm, with the effect of suppressing the error control on those components of Y.

Pedigree:

DLSODE is derived from the Livermore Solver for Ordinary Differential Equations package ODEPACK,

AUTHOR

      Hindmarsh, Alan C., (LLNL)
      Center for Applied Scientific Computing, L-561
      Lawrence Livermore National Laboratory
      Livermore, CA 94551.

Arguments