ODESolver - SLICOT Library Routine Documentation

## ODESolver

### Solver for Ordinary Differential Equations (driver)

[Specification][Arguments][Method][References][Comments][Example]

Purpose

      Interface for using a common entry point, DSblock compatible for
      defining Differential Algebraic Equations using several packages.

      The equations follow the form:

                   dx/dt = f(x(t), (t), p, t)
                    y(t) = g(x(t), (t), p, t)

      The user must define only the subroutines ODEDER and ODEOUT and
      the Jacobians (JACFX, JACFU, JACFP) if used, and the interface
      adapts the structure to fit all the codes

Specification

          SUBROUTINE ODESolver(ISOLVER, CODEDER_, CODEOUT_, CJACFX_,
         $                     CJACFU_, CJACFP_,
         $                     NX, NY, NU, TINI, TOUT, X, U, Y,
         $                     IPAR, DPAR, RTOL, ATOL,
         $                     IWORK, LIWORK, DWORK, LDWORK,
         $                     IWARN, INFO)
          .. Scalar Arguments ..
          DOUBLE PRECISION   RTOL, TINI, TOUT
          INTEGER            ISOLVER, IWARN, INFO, NX, NY, NU,
         $                   LDWORK, LIWORK
          CHARACTER*9        CODEDER_, CODEOUT_, CJACFX_, CJACFU_, CJACFP_
          .. Array Arguments ..
          DOUBLE PRECISION   ATOL(*), DWORK(LDWORK), DPAR(*),
         $                   X(NX), Y(NY), U(NU)
          INTEGER            IWORK(LIWORK), IPAR(*)

Arguments

Mode Parameters

      ISOLVER     (input) INTEGER
                 Indicates the nonlinear solver package to be used:
                 = 1: LSODE,
                 = 2: LSODA,
                 = 3: LSODES,
                 = 4: RADAU5,
                 = 5: DASSL,
                 = 6: DASPK,
                 = 7: DGELDA.

Input/Output Parameters

      ODEDER  (input) EXTERNAL
              Evaluates the right hand side f of the ODE.

      ODEOUT  (input) EXTERNAL
              Evaluates the output signals function g.

      JACFX   (input) EXTERNAL
              Evaluates the jacobian matrix with respect to X.

      JACFU   (input) EXTERNAL
              Evaluates the jacobian matrix with respect to U.

      JACFP   (input) EXTERNAL
              Evaluates the jacobian matrix with respect to P.

      NX      (input) INTEGER
              Dimension of the state vector.

      NY      (input) INTEGER
              Dimension of the output vector.

      NU      (input) INTEGER
              Dimension of the input vector.

      TINI    (input) DOUBLE PRECISION
              Initial value of time.

      TOUT    (input) DOUBLE PRECISION
              Final value of time.

      X       (input/output) DOUBLE PRECISION array, dimension (NX)
              On entry, array containing the initial state variables.
              On exit, it has the last value of the state variables.

      U       (input) DOUBLE PRECISION array, dimension (NU)
              Array containing the input initial values.

      Y       (input/output) DOUBLE PRECISION array, dimension (NY)
              On entry, array containing the initial values of Y.
              On exit, it has the results of the system.

      IPAR    (input/output) INTEGER array, dimension (230)
              INPUT:
                 1..15   General
                16..25   ODEPACK
                 26..35   RADAU5
                36..50   DASSL/PK
                51..60   GELDA
                61..100  Reserved
              OUTPUT:
               101..110  General
               111..125  ODEPACK
               126..135  RADAU5
               136..145  DASSL/PK
               146..155  GELDA
               156..200  Reserved
              Any Mode:
               201..     User Available

              Common integer parameters for SOLVERS:
                 IPAR(1), Tolerance mode
                     0 : both RTOL and ATOL are scalars
                     1 : RTOL is scalar and ATOL is vector
                     2 : both RTOL and ATOL are vectors
                 IPAR(2), Compute Output Values, must be 1
                 IPAR(3), mf, Method flag
                     0 : No jacobian used (non-stiff method).
                     1 : User supplied full jacobian (stiff).
                     2 : User supplied banded jacobian (stiff).
                     3 : User supplied sparse jacobian (stiff).
                    10 : internally generated full jacobian (stiff).
                    11 : internally generated banded jacobian (stiff).
                    12 : internally generated sparse jacobian (stiff).
                 IPAR(5), Maximum number of steps allowed during one
                    call to the solver.
                 IPAR(6), ml, lower half-bandwithds of the banded
                    jacobian, excluding tne main diagonal.
                 IPAR(7), mu, upper half-bandwithds of the banded
                    jacobian, excluding tne main diagonal.
                 IPAR(8), Flag to generate extra printing at method
                    switches:
                        0 means no extra printing
                        1 for minimal printing
                        2 for full printing
                 IPAR(101) = Number of steps taken for the problem.
                 IPAR(102) = Number of f evaluations.
                 IPAR(103) = Number of jacobian evaluations.

              Common parameters for ODEPACK, DASSL and DASPK solvers:
                 IPAR(111) = The method order last used(successfully).
                 IPAR(112) = The order to be attempted on the next step.

              Common parameters for ODEPACK solver:
                 IPAR(16), Status Flag
                 IPAR(17), Optional inputs
                 IPAR(18), Maximum number of messages printed,
                    default value is 10.
                 IPAR(113) = Index of the component of largest in the
                    weighted local error vector ( e(i)/ewt(i) ).
                 IPAR(114) = Length of rwork actually required.
                 IPAR(115) = Length of iwork actually required.

               - LSODE and LSODES
                 IPAR(19), Maximum order to be allowed.
                    12 if meth = 1
                     5 if meth = 2
                    If exceds the default value, it will be reduced
                    to the default value.

               - LSODES
                 IPAR(118), Number of nonzero elements in the jacobian
                    matrix, including the diagonal (miter = 1 or 2).
                 IPAR(119), Number of groups of column indices, used in
                    difference quotient jacobian aproximations if
                    miter = 2.
                 IPAR(120), Number of sparse LU decompositions.
                 IPAR(121), Base address in rwork of the history array.
                 IPAR(122), Base address of the structure descriptor
                    array ian.
                 IPAR(123), Base address of the structure descriptor
                    array jan.
                 IPAR(124), Number of nonzero elements in the strict
                    lower triangle of the LU factorization.
                 IPAR(125),  Number of nonzero elements in the strict
                    upper triangle of the LU factorization.

               - LSODA
                 IPAR(22), Maximum order to be allowed for the
                    nonstiff method, default value is 12.
                    If exceds the default value, it will be reduced
                    to the default value.
                 IPAR(23), Maximum order to be allowed for the stiff
                    method, default value is 5.
                    If exceds the default value, it will be reduced
                    to the default value.
                 IPAR(116), Method indicator for the last successful
                    step  1 adams (nonstiff)
                          2 bdf (stiff)
                 IPAR(117), Current method indicator
                    1 adams (nonstiff)
                    2 bdf (stiff)

              Parameters for RADAU5 solver:
                 IPAR(26) Transforms the Jacobian matrix to
                           Hessenberg form.
                 IPAR(27) Maximum number of Newton iterations.
                 IPAR(28) Starting values for Newton's method
                       if 0 then is taken the extrapolated collocation
                            solution
                       if not equal 0 zero values are used.
                 IPAR(29) Dimension of the index 1 variables.
                 IPAR(30) Dimension of the index 2 variables.
                 IPAR(31) Dimension of the index 3 variables.
                 IPAR(32) Switch for step size strategy
                       0,1 Mod. Predictive controller(Gustafsson)
                       2   Classical step size control
                 IPAR(33) Value of M1.
                 IPAR(34) Value of M2.
                 IPAR(126), Number of accepted steps.
                 IPAR(127), Number of rejected steps.
                 IPAR(128), Number of LU-Decompositions of both
                     matrices
                 IPAR(129), Number of forward-backward substitutions,
                     of both systems.

              Common parameters for DASSL and DASPK solvers:
                 IPAR(36),  this parameter enables the code to
                      initialize itself. Must set to 0 to indicate the
                      start of every new problem.
                           0: Yes. (On each new problem)
                           1: No. (Allows 500 new steps)
                 IPAR(37),  code solve the problem without invoking
                      any special non negativity contraints:
                           0: Yes
                           1: To have constraint checking only in the
                             initial condition calculation.
                           2: To enforce nonnegativity in X during the
                             integration.
                           3: To enforce both options 1 and 2.

                 IPAR(38), Solver try to compute the initial T, X
                       and XPRIME:
                           0: The initial T, X and XPRIME are
                             consistent.
                           1: Given X_d calculate X_a and X'_d
                           2: Given X' calculate X.
                           ( X_d differential variables in X
                             X_a algebrac variables in X )
                 IPAR(136), Total number of error test failures so far.
                 IPAR(137), Total number of convergence test failures.

               -Parameters for DASPK
                 IPAR(39), DASPK use:
                           0: direct methods (compatible with DASSL)
                           1: Krylov method
                           2: Krylov method + Jac
                 IPAR(40),  DASPK uses scalars MAXLm KMP, NRMAX and EPLI
                         when uses Krylov method.
                           0: uses default values.
                           1: uses user values.
                 IPAR(41), Proceed to the integration after the initial
                        condition calculation is done. Used when INFOV(11)>0

                           0: Yes
                           1: No
                 IPAR(42), Errors are controled localy on all the variables.
                           0: Yes
                           1: No
                 IPAR(43), Use default values for initial condition heuristic
                          controls.
                           0: Yes
                           1: No and provide MXNIT, MXNJ, MXNH, LSOFF, STPTOL,
                             EPINIT.
                 IPAR(138), number of convergence failures of the linear
                          iteration
                 IPAR(139), length of IWORK actually required.
                 IPAR(140), length of RWORK actually required.
                 IPAR(141), total number of nonlinear iterations.
                 IPAR(142), total number of linear (Krylov) iterations
                 IPAR(143), number of PSOL calls.

      DPAR    (input/output) DOUBLE PRECISION array, dimension (202)
              INPUT:
                 1..15   General
                16..25   ODEPACK
                26..35   RADAU5
                36..50   DASSL/PK
                51..60   GELDA
                61..100  Reserved
              OUTPUT:
               101..110  General
               111..125  ODEPACK
               126..135  RADAU5
               136..145  DASSL/PK
               146..155  GELDA
               156..200  Reserved
              Any Mode:
               201..     User Available

              Common real parameters for SOLVERS:
                  DPAR(1), Initial step size guess. Optional in:
                            ODEPACK, DASSL, ..
                  DPAR(2), Maximum absolute step size allowed.

              Common parameters for ODEPACK and DASSL:
                  DPAR(111), Step size in t last used (successfully).
                  DPAR(113), Current value of the independent variable
                     which the solver has actually reached

              Common parameters for ODEPACK solvers:
                  DPAR(16), Critical value of t which the solver is not
                    overshoot.
                  DPAR(17), Minimum absolute step size allowed.
                  DPAR(112), Step size to be attempted on the next step.
                  DPAR(18), Tolerance scale factor, greater than 1.0.

                - LSODA
                  DPAR(115) Value of t at the time of the last method
                     switch, if any.

                - LSODA
                  DPAR(115) Value of t at the time of the last method
                     switch, if any.

                - LSODES
                  DPAR(19), The element threshhold for sparsity
                    determination when moss = 1 or 2.

              Parameters for RADAU5 solver:
                  DPAR(26), The rounding unit, default 1E-16.
                  DPAR(27), The safety factor in step size prediction,
                    default 0.9D0.
                  DPAR(28), Decides whether the jacobian should be
                    recomputed, default 0.001D0.
                  DPAR(29), Stopping criterion for Newton's method,
                    default MIN(0.03D0, RTOL(1)**0.5D0)
                  DPAR(30), DPAR(31): This saves, together with a
                    large DPAR(28), LU-decompositions and computing
                    time for large systems.
                  DPAR(32), DPAR(33), Parameters for step size
                    selection.

              Parameters for DASSL and DASPK solvers:
                  DPAR(36), Stopping point (Tstop)

Tolerances

       RTOL    DOUBLE PREISION
               Relative Tolerance.

       ATOL    DOUBLE PREISION
               Absolute Tolerance.

Workspace

       IWORK   INTEGER array, dimension (LIWORK)

       LIWORK  INTEGER
               Size of IWORK, depending on solver:
               - LSODE
                   20        for mf = 10,
                   20 + neq  for mf = 21, 22, 24, or 25.
                   if mf = 24, 25, input in iwork(1),iwork(2) the lower
                      and upper half-bandwidths ml,mu.
               -LSODA
                   20+NX
               -LSODES
                   30
               -DASSL
                   20+NEQ

       DWORK   DOUBLE PREISION array, dimension (LDWORK)

       LDWORK  INTEGER
               Size of DWORK, depending on solver:
               - LSODE
                  20+16*NX                       ,  IPAR(3) = 10,
                  22+ 9*NX+NX**2                 ,  IPAR(3) = 21 or 22,
                  22+10*NX+(2*IPAR(4)+IPAR(9))*NX,  IPAR(3) = 24 or 25.
               - LSODA
                  22+NX*max(16,NX+9)
               - LSODES
                  20+16*NX                                , mf=10
                  20+(2+1./lenrat)*nnz + (11+9./lenrat)*NX, mf=121,222
               - DASSL
                  >= 40 LRW .GE. 40+(MAXORD+4)*NEQ+NEQ**2, IPAR(3) = 1 or 10
                  >= 40+(MAXORD+4)*NEQ+(2*ML+MU+1)*NEQ,    IPAR(3) = 2
                  >= 40+(MAXORD+4)*NEQ+(2*ML+MU+1)*NEQ
                                 +2*(NEQ/(ML+MU+1)+1),     IPAR(3) = 11

Warning Indicator

        IWARN   INTEGER
                = 0:  no warning;
                = 1: LSODE/LSODA/LSODES/RADAU5 do not use
                      the input vector as argument;
                = 2:  Only the 1st element of RTOL is used;
                = 3:  Method (IPAR(3)) not allowed with
                      LSODE/LSODA/LSODES/RADAU5/DASSL/DASPK;
                = 4:  Only the 1st element of ATOL is used;
                = 5:  Option not allowed for IPAR(37);
                = 6:  Option not allowed for IPAR(38).

Error Indicator

         INFO    INTEGER
                = 0:  Successful exit;
                < 0:  If INFO = -i, the i-th argument had an illegal
                      value;
                = 1:  Wrong tolerance mode;
                = 2:  Sparse storage (IPAR(4)=1) incompatible with
                      LSODE/LSODA/RADAU5;
                = 3:  Dense storage (IPAR(4)=0) incompatible with LSODES
                = 100+ERROR: ODEDER returned ERROR
                = 200+ERROR: RADAU5 returned -ERROR
                = 300+ERROR: DDASSL returned -ERROR
                = 400+ERROR: DDASPK returned -ERROR
                = 500+ERROR: DGELDA returned -ERROR

Method

    Since the package integrates 9 different solvers, it is possible to solve differential
    equations by means of Backward Differential Formulas, Runge-Kutta, using direct or
    iterative methods (including preconditioning) for the linear system associated, differential
    equations with time-varying coefficients or of order higher than one. The interface facilitates
    the user the work of changing the integrator and testing the results, thus leading a more robust
    and efficient integrated package.

References

      [1]  A.C. Hindmarsh, Brief Description of ODEPACK: A Systematized Collection
           of ODE Solvers, http://www.netlib.org/odepack/doc

      [2]  L.R. Petzold DASSL Library Documentation, http://www.netlib.org/ode/

      [3]  P.N. Brown, A.C. Hindmarsh, L.R. Petzold, DASPK Package 1995 Revision

      [4]  R.S. Maier, Using DASPK on the TMC CM5. Experiences with Two Programming
           Models, Minesota Supercomputer Center, Technical Report.

      [5]  E. Hairer, G. Wanner, Solving Ordinary Dirential Equations II. Stiánd
           Dirential- Algebraic Problems., Springer Seried in Computational
           Mathermatics 14, Springer-Verlag 1991, Second Edition 1996.

      [6]  P. Kunkel, V. Mehrmann, W. Rath und J. Weickert, `GELDA: A Software
           Package for the Solution of General Linear Dirential Algebraic
           equations', SIAM Journal Scienti^Lc Computing, Vol. 18, 1997, pp.
           115 - 138.

      [7]  M. Otter, DSblock: A neutral description of dynamic systems.
           Version 3.3, http://www.netlib.org/odepack/doc

      [8]  M. Otter, H. Elmqvist, The DSblock model interface for exchanging model
           components, Proceedings of EUROSIM 95, ed. F.Brenenecker, Vienna, Sep.
           11-15, 1995

      [9]  M. Otter, The DSblock model interface, version 4.0, Incomplete Draft,
           http://dv.op.dlr.de/~otter7dsblock/dsblock4.0a.html

      [10] Ch. Lubich, U. Novak, U. Pohle, Ch. Engstler, MEXX - Numerical
           Software for the Integration of Constrained Mechanical Multibody
           Systems, http://www.netlib.org/odepack/doc

      [11] Working Group on Software (WGS), SLICOT Implementation and Documentation
           Standards (version 1.0), WGS-Report 90-1, Eindhoven University of
           Technology, May 1990.

      [12] P. Kunkel and V. Mehrmann, Canonical forms for linear differential-
           algebraic equations with variable coeÆcients., J. Comput. Appl.
           Math., 56:225{259, 1994.

      [13] Working Group on Software (WGS), SLICOT Implementation and Documentation
           Standards, WGS-Report 96-1, Eindhoven University of Technology, updated:
           Feb. 1998, ../../REPORTS/rep96-1.ps.Z.

      [14] A. Varga, Standarization of Interface for Nonlinear Systems Software
           in SLICOT, Deutsches Zentrum ur Luft un Raumfahrt, DLR. SLICOT-Working
           Note 1998-4, 1998, Available at
           ../../REPORTS/SLWN1998-4.ps.Z.

      [15] D. Kirk, Optimal Control Theory: An Introduction, Prentice-Hall.
           Englewood Cli, NJ, 1970.

      [16] F.L. Lewis and V.L. Syrmos, Optimal Control, Addison-Wesley.
           New York, 1995.

      [17] W.M.Lioen, J.J.B de Swart, Test Set for Initial Value Problem Solvers,
           Technical Report NM-R9615, CWI, Amsterdam, 1996.
           http://www.cwi.nl/cwi/projects/IVPTestset/.

      [18] V.Hernandez, I.Blanquer, E.Arias, and P.Ruiz,
           Definition and Implementation of a SLICOT Standard Interface and the
           associated MATLAB Gateway for the Solution of Nonlinear Control Systems
           by using ODE and DAE Packages}, Universidad Politecnica de Valencia,
           DSIC. SLICOT Working Note 2000-3: July 2000. Available at
           ../../REPORTS/SLWN2000-3.ps.Z.

      [19] J.J.B. de Swart, W.M. Lioen, W.A. van der Veen, SIDE, November 25,
           1998. Available at http://www.cwi.nl/cwi/projects/PSIDE/.

      [20] Kim, H.Young, F.L.Lewis, D.M.Dawson, Intelligent optimal control of
           robotic manipulators using neural networks.

      [21] J.C.Fernandez, E.Arias, V.Hernandez, L.Penalver, High Performance
           Algorithm for Tracking Trajectories of Robot Manipulators,
           Preprints of the Proceedings of the 6th IFAC International Workshop on
           Algorithms and Architectures for Real-Time Control (AARTC-2000),
           pages 127-134.

Numerical Aspects

      The numerical aspects of the routine lie on the features of the
      different packages integrated. Several packages are more robust
      than others, and other packages simply cannot deal with problems
      that others do. For a detailed description of the numerical aspects
      of each method is recommended to check the references above.

Further Comments

      Several packages (LSODES, LSOIBT) deal only with sparse matrices.
      The interface checks the suitability of the methods to the
      parameters and show a warning message if problems could arise.

Example

Program Text

`* ODESOLVER EXAMPLE PROGRAM TEXT FOR LSODEX PROBLEM`
`*`
`* .. Parameters ..`
`INTEGER NIN, NOUT`
`PARAMETER ( NIN = 5, NOUT = 6 )`
`INTEGER LSODE_, LSODA_, LSODES_, RADAU5_, DASSL_, DASPK_, DGELDA_`
`PARAMETER (LSODE_ = 1, LSODA_ = 2, LSODES_ = 3)`
`PARAMETER (RADAU5_ = 4, DASSL_ = 5, DASPK_ = 6)`
`PARAMETER (DGELDA_ = 7)`
`*`
`EXTERNAL IARGC_`
`INTEGER IARGC_`
`INTEGER NUMARGS`
`CHARACTER*80 NAME`
`CHARACTER*80 SOLVER`
`*`
`* .. Executable Statements ..`
`*`
`WRITE ( NOUT, FMT = 99999 )`
`*`
`NUMARGS = IARGC_()`
`*`
`CALL GETARG_(0, NAME)`
`IF (NUMARGS .NE. 1) THEN`
`WRITE (*,*) 'Syntax Error: ',NAME(1:8),' <solver>'`
`WRITE (*,*) 'Solvers : LSODE, LSODA, LSODES, RADAU5, DASSL, DASP`
`&K'`
`ELSE`
`*`
`CALL GETARG_(1, SOLVER)`
`*`
`WRITE (*,*) 'Problem: LSODEX Solver: ',SOLVER(1:7)`
`*`
`IF (SOLVER(1:5) .EQ. 'LSODE') THEN`
`CALL TEST(LSODE_)`
`ELSEIF (SOLVER(1:5) .EQ. 'LSODA') THEN`
`CALL TEST(LSODA_)`
`ELSEIF (SOLVER(1:6) .EQ. 'LSODES') THEN`
`CALL TEST(LSODES_)`
`ELSEIF (SOLVER(1:6) .EQ. 'RADAU5') THEN`
`CALL TEST(RADAU5_)`
`ELSEIF (SOLVER(1:5) .EQ. 'DASSL') THEN`
`CALL TEST(DASSL_)`
`ELSEIF (SOLVER(1:5) .EQ. 'DASPK') THEN`
`CALL TEST(DASPK_)`
`ELSE`
`WRITE (*,*) 'Error: Solver: ', SOLVER,' unknown'`
`ENDIF`
`ENDIF`
`*`
`99999 FORMAT (' ODESOLVER EXAMPLE PROGRAM RESULTS FOR LSODEX PROBLEM'`
`. ,/1X)`
`END`
*
*
*
*
`SUBROUTINE TEST( ISOLVER )`
`*`
`*`
`* PURPOSE`
`*`
`* Testing subroutine ODESolver`
`*`
`* ARGUMENTS`
`*`
`* Input/Output Parameters`
`*`
`* ISOLVER (input) INTEGER`
`* Indicates the nonlinear solver package to be used:`
`* = 1: LSODE,`
`* = 2: LSODA,`
`* = 3: LSODES,`
`* = 4: RADAU5,`
`* = 5: DASSL,`
`* = 6: DASPK,`
`* = 7: DGELDA.`
`*`
`* METHOD`
`*`
`* REFERENCES`
`*`
`* CONTRIBUTORS`
`*`
`* REVISIONS`
`*`
`* -`
`*`
`* KEYWORDS`
`*`
`*`
`* ******************************************************************`
`* .. Parameters ..`
`INTEGER LSODE_, LSODA_, LSODES_, RADAU5_, DASSL_, DASPK_, DGELDA_`
`PARAMETER (LSODE_ = 1, LSODA_ = 2, LSODES_ = 3)`
`PARAMETER (RADAU5_ = 4, DASSL_ = 5, DASPK_ = 6)`
`PARAMETER (DGELDA_ = 7)`
`INTEGER NIN, NOUT`
`PARAMETER ( NIN = 5, NOUT = 6 )`
`INTEGER MD, ND, LPAR, LWORK`
`PARAMETER ( MD = 400, ND = 100, LPAR = 250,`
`$ LWORK = 650000 )`
`* .. Scalar Arguments ..`
`INTEGER ISOLVER`
`* .. Local Scalars ..`
`INTEGER NEQN, NDISC, MLJAC, MUJAC, MLMAS, MUMAS`
`INTEGER IWARN, INFO`
`DOUBLE PRECISION ATOL(MD), RTOL, NORM`
`LOGICAL NUMJAC, NUMMAS, CONSIS`
`* .. Local Arrays ..`
`CHARACTER FULLNM*40, PROBLM*8, TYPE*3`
`CHARACTER*9 ODEDER, ODEOUT, JACFX, JACFU, JACFP`
`INTEGER IND(MD), IPAR(LPAR), IWORK(LWORK)`
`DOUBLE PRECISION T(0:ND), RPAR(LPAR), DWORK(LWORK)`
`DOUBLE PRECISION X(MD), XPRIME(MD), U(MD), Y(MD)`
`* .. External Functions ..`
`DOUBLE PRECISION DNRM2`
`EXTERNAL DNRM2`
`* .. External Subroutines ..`
`EXTERNAL PLSODEX,ILSODEX,SLSODEX`
`EXTERNAL DAXPY`
`* .. Executable Statements ..`
`*`
`DO 20 I=1,NEQN`
`U(I)=0D0`
`Y(I)=0D0`
`20 CONTINUE`
`DO 40 I=1,LPAR`
`IPAR(I)=0`
`RPAR(I)=0D0`
`40 CONTINUE`
`DO 60 I=1,LWORK`
`IWORK(I)=0`
`DWORK(I)=0D0`
`60 CONTINUE`
`IPAR(2)=1`
`* Get the problem dependent parameters.`
`RPAR(1)=1D-3`
`IPAR(1)=0`
`ATOL(1)=1D-6`
`ATOL(2)=1D-10`
`ATOL(3)=1D-6`
`RTOL=1D-4`
`CALL PLSODEX(FULLNM,PROBLM,TYPE,NEQN,NDISC,T,NUMJAC,MLJAC,`
`$ MUJAC,NUMMAS,MLMAS,MUMAS,IND)`
`CALL ILSODEX(NEQN,T(0),X,XPRIME,CONSIS)`
`CALL SLSODEX(NEQN,T(1),XPRIME)`

`IF ( TYPE.NE.'ODE' ) THEN`
`WRITE ( NOUT, FMT = 99998 )`
`ELSE`
`WRITE ( NOUT, FMT = 99997 ) FULLNM, PROBLM, TYPE, ISOLVER`
`IF ( NUMJAC ) THEN`
`IPAR(3)=0`
`ELSE`
`IPAR(3)=1`
`END IF`
`IPAR(6)=MLJAC`
`IPAR(7)=MUJAC`
`ODEDER=''`
`ODEOUT=''`
`JACFX=''`
`JACFU=''`
`JACFP=''`

`CALL ODESolver( ISOLVER, ODEDER, ODEOUT, JACFX, JACFU, JACFP,`
`$ NEQN, NEQN, NEQN, T(0), T(1), X, U, Y,`
`$ IPAR, RPAR, RTOL, ATOL,`
`$ IWORK, LWORK, DWORK, LWORK, IWARN, INFO )`

`IF ( INFO.NE.0 ) THEN`
`WRITE ( NOUT, FMT = 99996 ) INFO`
`ELSE`
`IF ( IWARN.NE.0 ) THEN`
`WRITE ( NOUT, FMT = 99995 ) IWARN`
`ENDIF`
`IF ( NEQN .LE. 10 ) THEN`
`WRITE ( NOUT, FMT = 99994 )`
`DO 80 I=1,NEQN`
`WRITE ( NOUT, FMT = 99993 ) X(I), XPRIME(I)`
`80 CONTINUE`
`END IF`
`NORM=DNRM2(NEQN,XPRIME,1)`
`IF ( NORM.EQ.0D0 ) THEN`
`NORM=1D0`
`END IF`
`CALL DAXPY(NEQN,-1D0,X,1,XPRIME,1)`
`NORM=DNRM2(NEQN,XPRIME,1)/NORM`
`WRITE ( NOUT, FMT = 99992 ) NORM`
`END IF`
`END IF`
`*`
`99998 FORMAT (' ERROR: This test is only intended for ODE problems')`
`99997 FORMAT (' ',A,' (',A,' , ',A,') with SOLVER ',I2)`
`99996 FORMAT (' INFO on exit from ODESolver = ',I3)`
`99995 FORMAT (' IWARN on exit from ODESolver = ',I3)`
`99994 FORMAT (' Solution: (calculated) (reference)')`
`99993 FORMAT (F,F)`
`99992 FORMAT (' Relative error comparing with the reference solution:'`
`$ ,E,/1X)`
`* *** Last line of TEST ***`
`END`

`SUBROUTINE ODEDER_( NX, NU, T, X, U, RPAR, IPAR, F, INFO )`
`*`
`*`
`* PURPOSE`
`*`
`* Interface routine between ODESolver and the problem function FEVAL`
`*`
`* ARGUMENTS`
`*`
`* Input/Output Parameters`
`*`
`* NX (input) INTEGER`
`* Dimension of the state vector.`
`*`
`* NU (input) INTEGER`
`* Dimension of the input vector.`
`*`
`* T (input) INTEGER`
`* The time point where the function is evaluated.`
`*`
`* X (input) DOUBLE PRECISION array, dimension (NX)`
`* Array containing the state variables.`
`*`
`* U (input) DOUBLE PRECISION array, dimension (NU)`
`* Array containing the input values.`
`*`
`* RPAR (input/output) DOUBLE PRECISION array`
`* Array for communication between the driver and FEVAL.`
`*`
`* IPAR (input/output) INTEGER array`
`* Array for communication between the driver and FEVAL.`
`*`
`* F (output) DOUBLE PRECISION array, dimension (NX)`
`* The resulting function value f(T,X).`
`*`
`* Error Indicator`
`*`
`* INFO INTEGER`
`* Return values of error from FEVAL or 100 in case`
`* a bad problem was choosen.`
`*`
`* METHOD`
`*`
`* REFERENCES`
`*`
`* CONTRIBUTORS`
`*`
`* REVISIONS`
`*`
`* -`
`*`
`* KEYWORDS`
`*`
`*`
`* ******************************************************************`
`*`
`* .. Scalar Arguments ..`
`INTEGER NX, NU, INFO`
`DOUBLE PRECISION T`
`* .. Array Arguments ..`
`INTEGER IPAR(*)`
`DOUBLE PRECISION X(NX), U(NU), RPAR(*), F(NX)`
`* .. External Subroutines ..`
`EXTERNAL FLSODEX`
`* .. Executable Statements ..`
`CALL FLSODEX(NX,T,X,X,F,INFO,RPAR,IPAR)`
`* *** Last line of ODEDER_ ***`
`END`

`SUBROUTINE JACFX_( NX, DUMMY, LDFX, T, X, DUMMY2, RPAR, IPAR, FX,`
`$ INFO )`
`*`
`*`
`* PURPOSE`
`*`
`* Interface routine between ODESolver and the problem function JEVAL`
`*`
`* ARGUMENTS`
`*`
`* Input/Output Parameters`
`*`
`* NX (input) INTEGER`
`* Dimension of the state vector.`
`*`
`* DUMMY (input) INTEGER`
`*`
`* LDFX (input) INTEGER`
`* The leading dimension of the array FX.`
`*`
`* T (input) INTEGER`
`* The time point where the derivative is evaluated.`
`*`
`* X (input) DOUBLE PRECISION array, dimension (NX)`
`* Array containing the state variables.`
`*`
`* DUMMY2 (input) DOUBLE PRECISION`
`*`
`* RPAR (input/output) DOUBLE PRECISION array`
`* Array for communication between the driver and FEVAL.`
`*`
`* IPAR (input/output) INTEGER array`
`* Array for communication between the driver and FEVAL.`
`*`
`* FX (output) DOUBLE PRECISION array, dimension (LDFX,NX)`
`* The array with the resulting Jacobian matrix.`
`*`
`* Error Indicator`
`*`
`* INFO INTEGER`
`* Return values of error from JEVAL or 100 in case`
`* a bad problem was choosen.`
`*`
`* METHOD`
`*`
`* REFERENCES`
`*`
`* CONTRIBUTORS`
`*`
`* REVISIONS`
`*`
`* -`
`*`
`* KEYWORDS`
`*`
`*`
`* ******************************************************************`
`*`
`* .. Scalar Arguments ..`
`INTEGER NX, DUMMY, LDFX, INFO`
`DOUBLE PRECISION T`
`* .. Array Arguments ..`
`INTEGER IPAR(*)`
`DOUBLE PRECISION X(NX), DUMMY2(*), RPAR(*), FX(NX)`
`* .. External Subroutines ..`
`EXTERNAL JLSODEX`
`* .. Executable Statements ..`
`CALL JLSODEX(LDFX,NX,T,X,X,FX,INFO,RPAR,IPAR)`
`* *** Last line of JACFX_ ***`
`END`

Program Data

    No data required

Program Results

     ODESOLVER EXAMPLE PROGRAM RESULTS

     Problem: LSODEX   Solver: LSODE
     IWARN on exit from ODESolver =   1
     Solution: (calculated)
     8.287534436182735E-08
     3.329129749822125E-13
     1.118553835127275E-07

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