## DAESolver

### Solver for Algebraic 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 (CASE A):

          F(dx(t)/dt, x(t), u(t), p, t) = 0
          y(t) = g(dx(t)/dx, x(t), u(t), p, t)

      for the most general model which can only be solved by DASSL and
      DASSPK.

      A restricted case can be solved with RADAU5, LSODI, LSOIBT, if
      the system is expressed as (CASE B):

          F(x(t), u(t), p, t)*dx(t)/dt = A(x(t), u(t), p, t)
          y(t) = g(dx(t)/dx, x(t), u(t), p, t)

      And finally, the GELDA package is able to solve DAEs with the
      expression (CASE C):

          F(u(t), p, t)*dx(t)/dt = A(u(t), p, t)*x(t) + E(u(t), p, t)

      The user must define the subroutines:
         DAEDF:    F(dx(t)/dt, x(t), u(t), p, t)  for CASES: A, B and C
         DAEDA:    A(x(t), u(t), p, t)           for CASES: B and C
         DAEDE:    E(u(t), p, t)                  for CASES: C
         DAEOUT:   g(dx(t)/dx, x(t), u(t), p, t)
      and the Jacobians (JACFX, JACFU, JACFP) if used. The interface
      adapts the structure to fit all the codes

Specification

         SUBROUTINE DAESolver(ISOLVER,CDAEDF_,CDAEDA_,CDAEDE_,CDAEOUT_,
       $                      CJACFX_,CJACFU_,CJACFP_,CJACFXDOT_,
       $                      NX, NY, NU, NP, TINI, TOUT,
       $                      X, XDOTI, Y, U, P,
       $                      IPAR, RPAR, RTOL, ATOL,
       $                      IWORK, LIWORK, DWORK, LDWORK,
       $                      IWARN, INFO)
        .. Scalar Arguments ..
        DOUBLE PRECISION    TINI, TOUT
        INTEGER             ISOLVER, IWARN, INFO,
       $                    NX, NY, NU, NP,
       $                    LDWORK, LIWORK
        CHARACTER*9         CDAEDF_, CDAEDA_,CDAEDE_, CDAEOUT_,
       $                    CJACFX_, CJACFU_, CJACFP_, CJACFXDOT_,
       $                    CDAEDF, CDAEDA,CDAEDE, CDAEOUT,
       $                    CJACFX, CJACFU, CJACFP, CJACFXDOT
        .. Array Arguments ..
        DOUBLE PRECISION    DWORK(LDWORK), RPAR(*), ATOL(*), RTOL(*)
       $                    X(NX), XDOTI(NX), Y(NY), U(NU), P(NP)
        INTEGER             IWORK(LIWORK), IPAR(*)

Arguments

Mode Parameters

       ISOLVER INTEGER
               Indicates the nonlinear solver packages to be used
                 = 1: LSODI,
                 = 2: LSOIBT,
                 = 3: RADAU5,
                 = 4: DASSL,
                 = 5: DASPK,
                 = 6: DGELDA.

Input/Output Parameters

         DAEDF   (input) EXTERNAL
                 Evaluates the F(dx(t)/dt, x(t), u(t), p, t).

         DAEDA   (input) EXTERNAL
                 Evaluates the A(x(t), u(t), p, t).

         DAEDE   (input) EXTERNAL
                 Evaluates the E(u(t), p, t).

         DAEOUT  (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.

         NP      (input) INTEGER
                 Dimension of the parameter 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.

         XDOTI   (input) DOUBLE PRECISION array, dimension (NX)
                 Array containing dx(t)/dt at initial point.

         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.

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

         P       (input) DOUBLE PRECISION array, dimension (NP)
                 Array containing the parameter variables.

         IPAR    (input/output) INTEGER array, dimension (201)
                 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 a scalar and atol is a vector
                        2 : both rtol and atol are vectors
                    IPAR(2), Compute Output Values only at TOUT (and not
                        at the intermediate step). (1:Yes, 0:No)
                    IPAR(3), mfjac, Method flag for jacobian
                        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(6), ml, lower half-bandwithds of the banded
                       jacobian, excluding tne main diagonal.
                    IPAR(7), mu, upper half-bandwithds of the banded
                       jacobian, excluding the main diagonal.
             (Note: IPAR(6) and IPAR(7) are obligatories only if the
              jacobian matrix is banded)
                    IPAR(101) = Number of steps taken for the problem.
                    IPAR(102) = Number of residual evaluations.
                    IPAR(103) = Number of jacobian evaluations.

                 Common parameters for RADAU5, ODEPACK and DGELDA:
                    IPAR(9), mfmass, Method flag for mass-matrix
                        0 : No mass-matrix used (non-stiff method).
                        1 : User supplied full mass-matrix (stiff).
                        2 : User supplied banded mass-matrix (stiff).
                       10 : Identity mass-matrix is used (stiff).
                    IPAR(10), mlmass, lower half-bandwithds of the banded
                       mass matrix, excluding the main diagonal.
                    IPAR(11), mumass, upper half-bandwithds of the banded
                       mass matrix, excluding the main diagonal.
                    IPAR(12), Maximum number of steps allowed during one
                       call to the solver.

                 Common parameters for ODEPACK, DASSL, DASPK and DGELDA:
                    IPAR(13), Maximum order to be allowed.
                       default values : 12 if meth = 1
                                         5 if meth = 2
                       If exceds the default value, it will be reduced
                       to the default value.
                       In DASSL, DASPK and DGELDA : (1 .LE. MAXORD .LE. 5)
                    IPAR(111) = The method order last used(successfully).
                    IPAR(112) = The order to be attempted on the next step.

                 Common parameters for ODEPACK package:
                    IPAR(16), Status Flag
                    IPAR(17), Optional inputs, must be 0
                    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.

                 - LSOIBIT
                    IPAR(24), mb, block size.
                       (mb .GE. 1) and mb*IPAR(28) = NX
                    IPAR(25), nb, number of blocks in the main diagonal.
                       (nb .ge. 4) and nb*IPAR(27) = NX

                 - RADAU5
                    IPAR(26) Transforms the Jacobian matrix to Hessenberg
                        form.(Only if IPAR(9)=1 and IPAR(3)=1 or 10)
                    IPAR(27) Maximum number of Newton iterations in
                        each step.
                    IPAR(28) Starting values for Newton's method
                          .EQ. 0 -> is taken the extrapolated collocation
                               solution
                          .NE. 0 -> zero values are used.
                    IPAR(29) Dimension of the index 1 variables( >0 ).
                    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 (default 0).
                    IPAR(34) Value of M2 (default(M2=M1).
                    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, DASPK and DGELDA 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(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.

                 Common parameters for DASSL and DASPK solvers:
                    IPAR(37),  code solve the problem without invoking
                         any special non negativity constraints:
                              0: Yes
                              1: To have constraint checking only in the
                                initial condition calculation.
                              2: To enforze nonnegativity in X during the
                                integration.
                              3: To enforce both options 1 and 2.
                   IPAR(137), Total number of convergence test failures.

                 - DASPK
                   IPAR(39), DASPK use:
                             0: direct methods (dense or band)
                             1: Krylov method  (iterative)
                             2: Krylov method + Jac (iterative)
                   IPAR(41), Proceed to the integration after the initial
                           condition calculation is done. Used when
                           IPAR(38) > 0:    0: Yes
                                            1: No
                   IPAR(42), Errors are controled localy on all the
                         variables:      0:Yes
                                         1: No
                   IPAR(8), Extra printing
                             0, no printing
                             1, for minimal printing
                             2, for full printing
                   IPAR(44), maximum number of iterations in the SPIGMR
                        algorithm. (.LE. NX)
                   IPAR(45), number of vectors on which orthogonalization
                        is done in the SPIGMR algorithm. (.LE. IPAR(44))
                   IPAR(46), maximum number of restarts of the SPIGMR
                        algorithm per nonlinear iteration. (.GE. 0)
                   IPAR(47), maximum number of Newton iterations per
                        Jacobian or preconditioner evaluation. (> 0)
                   IPAR(48), maximum number of Jacobian or preconditioner
                        evaluations. (> 0)
                   IPAR(49), maximum number of values of the artificial
                        stepsize parameter H to be tried if IPAR(38) = 1.
                        (> 0).
                   IPAR(50), flag to turn off the linesearch algorithm.
                            0 : ON
                            1 : OFF (default)
                   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.

                 - DGELDA
                   IPAR(51), contains the strangeness index.
                   IPAR(52), number of differential components
                   IPAR(53), number of algebraic components
                   IPAR(54), number of undetermined components
                   IPAR(55), method used:
                            if 1 then uses the BDF solver
                               2 then uses the Runge-Kutta solver
                   IPAR(56), E(t) and A(t) are: 1  time dependent
                                                0  constants
                   IPAR(57), Maximum index of the problem. ( .GE. 0 )
                   IPAR(58), Step size strategy:
                        0, Mod. predictive controlled of Gustafsson(safer)
                        1, classical step size control(faster)

         RPAR    (input/output) DOUBLE PRECISION array, dimension (201)
                 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 parameters for solvers:
                    RPAR(1), Initial step size guess.Obligatory in RADAU5.
                    RPAR(2), Maximum absolute step size allowed.

                 Common parameters for ODEPACK, DASSL, DASPK and DGELDA:
                    RPAR(111), Step size in t last used (successfully).
                    RPAR(112), Step size to be attempted on the next step.
                    RPAR(113), Current value of the independent variable
                       which the solver has actually reached

                 Common parameters for ODEPACK solver:
                    RPAR(16), Critical value of t which the solver is not
                       overshoot.
                    RPAR(17), Minimum absolute step size allowed.
                    RPAR(18), Tolerance scale factor, greater than 1.0.

                 Parameters for RADAU5 solver:
                    RPAR(26), The rounding unit, default 1E-16.
                    RPAR(27), The safety factor in step size prediction,
                      default 0.9D0.
                    RPAR(28), Decides whether the jacobian should be
                      recomputed, default 0.001D0.
                       Increase when jacobian evaluations are costly
                       For small systems should be smaller.
                    RPAR(29), Stopping criterion for Newton's method,
                      default MIN(0.03D0, RTOL(1)**0.5D0).
                    RPAR(30), RPAR(31): This saves, together with a
                      large RPAR(28), LU-decompositions and computing
                      time for large systems.
                      Small systems: RPAR(30)=1.D0, RPAR(31)=1.2D0
                      Large full systems: RPAR(30)=0.99D0, RPAR(31)=2.D0
                      might be good.
                    RPAR(32), RPAR(33), Parameters for step size
                      selection.Condition: RPAR(32)<=HNEW/HOLD<=RPAR(33)

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

                   - DASPK
                    RPAR(37), convergence test constant in SPIGMR
                        algorithm. (0 .LT. RPAR(37) .LT. 1.0)
                    RPAR(38), minimum scaled step in linesearch algorithm.
                        The default is  = (unit roundoff)**(2/3). (> 0)
                    RPAR(39), swing factor in the Newton iteration
                        convergence test. (default 0.1) (> 0)

                   - DASPK
                    RPAR(40), safety factor used in step size prediction.
                    RPAR(41) and RPAR(42) restric the relation between the
                        new and old stepsize in step size selection.
                          1/RPAR(41) .LE. Hnew/Hold .LE. 1/RPAR(42)
                    RPAR(43), RPAR(44) QUOT1 and QUOT2 repectively.
                        If QUOT1 < Hnew/Hold < QUOT2 and A and E are
                        constants, the work can be saved by setting
                        Hnew=Hold and using the system matrix of the
                        previous step.

Tolerances

         RTOL    DOUBLE PRECISION
                 Relative Tolerance.

         ATOL    DOUBLE PRECISION
                 Absolute Tolerance.

Workspace

         IWORK   INTEGER array, dimension (LIWORK)

         LIWORK  INTEGER
                 Minimum size of DWORK, depending on solver:
                 - LSODI, LSOIBT, DASSL
                    20 + NX
                 - RADAU5
                    3*N+20

         DWORK   DOUBLE PRECISION array, dimension (LDWORK)

         LDWORK  INTEGER
                 Size of DWORK, depending on solver:
                 - LSODI
                    22 +  9*NX + NX**2             , IPAR(3) = 1 or 10
                    22 + 10*NX + (2*ML + MU)*NX    , IPAR(3) = 2 or 11
                 - LSOIBT
                    20 + nyh*(maxord + 1) + 3*NX + lenw     where
                      nyh    = Initial value of NX
                      maxord = Maximum order allowed(default or IPAR(13)
                      lenw   = 3*mb*mb*nb + 2
                 - RADAU5
                    N*(LJAC+LMAS+3*LE+12)+20
                     where  LJAC=N               if (full jacobian)
                            LJAC=MLJAC+MUJAC+1   if (banded jacobian)
                       and  LMAS=0               if (IPAR(9) = 10 or 11)
                            LMAS=N               if (IPAR(9) = 1)
                            LMAS=MLMAS+MUMAS+1   if (IPAR(9) = 2)
                       and  LE=N                 if (IPAR(9) = 1 or 10)
                            LE=2*MLJAC+MUJAC+1   if (IPAR(9) = 2 or 11)
                 - 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:  LSODI/LSOIBT/RADAU5 do not use the input vector as argument;
                 = 2:  LSODI/LSOIBT do not use the param vector as argument;
                 = 3:  RTOL and ATOL are used as scalars;

Error Indicator

         INFO    INTEGER
                 = 0:  Successful exit;
                 < 0:  If INFO = -i, the i-th argument had an illegal
                       value;
                 = 1:  Wrong tolerance mode;
                 = 2:  Method (IPAR(9)) is not allowed for ODEPACK/RADAU5;
                 = 3:  Method (IPAR(3)) is not allowed for LSODE/RADAU5/DASSL;
                 = 4:  Option not allowed for IPAR(37);
                 = 5:  Option not allowed for IPAR(38);
                 = 100+ERROR: RADAU5 returned -ERROR;
                 = 200+ERROR: DASSL returned -ERROR;
                 = 300+ERROR: DASPK returned -ERROR;
                 = 400+ERROR: DGELDA returned -ERROR.

Method

    Since the package integrates 8 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

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           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.

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           Feb. 1998, ../../REPORTS/rep96-1.ps.Z.

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           ../../REPORTS/SLWN1998-4.ps.Z.

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           Englewood Cli, NJ, 1970.

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           New York, 1995.

      [17] W.M.Lioen, J.J.B de Swart, Test Set for Initial Value Problem Solvers,
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           http://www.cwi.nl/cwi/projects/IVPTestset/.

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           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

`* DAESOLVER EXAMPLE PROGRAM TEXT FOR LSODIX PROBLEM`
`*`
`* .. Parameters ..`
`INTEGER NIN, NOUT`
`PARAMETER ( NIN = 5, NOUT = 6 )`
`INTEGER LSODI_, LSOIBT_, RADAU5_, DASSL_, DASPK_, GELDA_`
`PARAMETER (LSODI_ = 1, LSOIBT_ = 2)`
`PARAMETER (RADAU5_ = 3, DASSL_ = 4, DASPK_ = 5)`
`PARAMETER (GELDA_ = 6)`
`* .. Executable Statements ..`
`*`
`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 : LSODI, LSOIBT, RADAU5, DASSL, DASPK, GELD`
`&A'`
`ELSE`
`*`
`CALL GETARG_(1, SOLVER)`
`*`
`WRITE (*,*) 'Problem: LSODIX Solver: ',SOLVER(1:7)`
`*`
`IF (SOLVER(1:5) .EQ. 'LSODI') THEN`
`CALL TEST(LSODI_)`
`ELSEIF (SOLVER(1:6) .EQ. 'LSOIBT') THEN`
`CALL TEST(LSOIBT_)`
`ELSEIF (SOLVER(1:6) .EQ. 'RADAU5') THEN`
`CALL TEST(RADAU5_)`
`ELSEIF (SOLVER(1:5) .EQ. 'GELDA') THEN`
`CALL TEST(GELDA_)`
`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 (' DAESOLVER EXAMPLE PROGRAM RESULTS FOR LSODIX PROBLEM'`
`. ,/1X)`
`END`

`SUBROUTINE TEST( ISOLVER )`
`*`
`*`
`* PURPOSE`
`*`
`* Testing subroutine DAESolver`
`*`
`* ARGUMENTS`
`*`
`* Input/Output Parameters`
`*`
`* ISOLVER (input) INTEGER`
`* Indicates the nonlinear solver package to be used:`
`* = 1: LSODI,`
`* = 2: LSOIBT,`
`* = 3: RADAU5,`
`* = 4: DASSL,`
`* = 5: DASPK,`
`* = 6: DGELDA.`
`*`
`* METHOD`
`*`
`* REFERENCES`
`*`
`* CONTRIBUTORS`
`*`
`* REVISIONS`
`*`
`* -`
`*`
`* KEYWORDS`
`*`
`*`
`* ******************************************************************`
`* .. Parameters ..`
`INTEGER LSODI_, LSOIBT_, RADAU5_, DASSL_, DASPK_, GELDA_`
`PARAMETER (LSODI_ = 1, LSOIBT_ = 2)`
`PARAMETER (RADAU5_ = 3, DASSL_ = 4, DASPK_ = 5)`
`PARAMETER (GELDA_ = 6)`
`INTEGER NIN, NOUT`
`PARAMETER ( NIN = 5, NOUT = 6 )`
`INTEGER MD, ND, LPAR, LWORK`
`PARAMETER ( MD = 400, ND = 100, LPAR = 201,`
`$ LWORK = 10000 )`
`* .. Common variables ..`
`COMMON /TESTING/ ISOLVER2`
`INTEGER ISOLVER2`
`* .. Scalar Arguments ..`
`INTEGER ISOLVER`
`* .. Local Scalars ..`
`INTEGER NEQN, NDISC, MLJAC, MUJAC, MLMAS, MUMAS`
`INTEGER IWARN, INFO`
`DOUBLE PRECISION ATOL, RTOL, NORM`
`LOGICAL NUMJAC, NUMMAS, CONSIS`
`* .. Local Arrays ..`
`CHARACTER FULLNM*40, PROBLM*8, TYPE*3`
`CHARACTER*9 CDAEDF,CDAEDA,CDAEDE,CDAEOUT,`
`$ CJACFX,CJACFU,CJACFP,CJACFXDOT`
`INTEGER IND(MD), IPAR(LPAR), IWORK(LWORK)`
`DOUBLE PRECISION T(0:ND), RPAR(LPAR), DWORK(LWORK)`
`DOUBLE PRECISION X(MD), XPRIME(MD), Y(MD), U(MD), P(MD), SOLU(MD)`
`* .. External Functions ..`
`DOUBLE PRECISION DNRM2`
`EXTERNAL DNRM2`
`* .. External Subroutines ..`
`EXTERNAL PLSODIX, ILSODIX, SLSODIX`
`EXTERNAL DAXPY`
`* .. Executable Statements ..`
`*`
`ISOLVER2 = ISOLVER`
`DO 20 I=1,NEQN`
`Y(I)=0D0`
`U(I)=0D0`
`P(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`
`* Get the problem dependent parameters.`
`RTOL=1D-4`
`ATOL=1D-6`
`IPAR(1)=0`
`IPAR(2)=1`
`IPAR(3)=1`
`IPAR(12)= 10000`
`IF (ISOLVER .EQ. LSODI_ .OR. ISOLVER .EQ. RADAU5_) THEN`
`IPAR(9)=1`
`IPAR(16)=1`
`C IPAR(17)=0`
`RPAR(1)=1D-3`
`ELSE`
`C (ISOLVER .EQ. DASSL_ .OR. ISOLVER .EQ. DASPK_)`
`C IPAR(36)=0`
`C IPAR(37)=0`
`C IPAR(38)=0`
`IPAR(39)=1`
`END IF`
`CALL PLSODIX(FULLNM,PROBLM,TYPE,NEQN,NDISC,T,NUMJAC,MLJAC,`
`$ MUJAC,NUMMAS,MLMAS,MUMAS,IND)`
`CALL ILSODIX(NEQN,T(0),X,XPRIME,CONSIS)`
`x(1) = 1.0d0`
`x(2) = 0.0d0`
`x(3) = 0.0d0`
`xprime(1) = -0.04D0`
`xprime(2) = 0.04D0`
`xprime(3) = 0.0D0`
`CALL SLSODIX(NEQN,T(1),SOLU)`

`IF ( TYPE.NE.'DAE' ) THEN`
`WRITE ( NOUT, FMT = 99998 )`
`ELSE`
`WRITE ( NOUT, FMT = 99997 ) FULLNM, PROBLM, TYPE, ISOLVER`
`CDAEDF=''`
`CDAEDA=''`
`CDAEDE=''`
`CDAEOUT=''`
`CJACFX=''`
`CJACFU=''`
`CJACFP=''`
`CJACFXDOT=''`

`CALL DAESolver( ISOLVER, CDAEDF, CDAEDA, CDAEDE, CDAEOUT,`
`$ CJACFX, CJACFU, CJACFP, CJACFXDOT,`
`$ NEQN, NEQN, NEQN, NEQN, T(0), T(1),`
`$ X, XPRIME, Y, U, P,`
`$ 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. 30 ) THEN`
`WRITE ( NOUT, FMT = 99994 )`
`DO 80 I=1,NEQN`
`WRITE ( NOUT, FMT = 99993 ) I, X(I), SOLU(I)`
`80 CONTINUE`
`END IF`
`NORM=DNRM2(NEQN,SOLU,1)`
`IF ( NORM.EQ.0D0 ) THEN`
`NORM=1D0`
`END IF`
`CALL DAXPY(NEQN,-1D0,X,1,SOLU,1)`
`NORM=DNRM2(NEQN,SOLU,1)/NORM`
`WRITE ( NOUT, FMT = 99992 ) NORM`
`END IF`
`END IF`
`*`
`99998 FORMAT (' ERROR: This test is only intended for DAE problems')`
`99997 FORMAT (' ',A,' (',A,' , ',A,') with SOLVER ',I2)`
`99996 FORMAT (' INFO on exit from DAESolver = ',I3)`
`99995 FORMAT (' IWARN on exit from DAESolver = ',I3)`
`99994 FORMAT (' Solution: (calculated) (reference)')`
`99993 FORMAT (I,F,F)`
`99992 FORMAT (' Relative error comparing with the reference solution:'`
`$ ,E,/1X)`
`* *** Last line of TEST ***`
`END`

`SUBROUTINE DAEDA_( RPAR, NRP, IPAR, NIP, X, NX, U, NU, P, NP,`
`$ F, LDF, T, INFO )`
`*`
`*`
`* PURPOSE`
`*`
`* Interface routine between DAESolver and the problem function FEVAL`
`*`
`* ARGUMENTS`
`*`
`* Input/Output Parameters`
`*`
`* RPAR (input/output) DOUBLE PRECISION array, dimension (NRP)`
`* Array for communication between the driver and FEVAL.`
`*`
`* NRP (input) INTEGER`
`* Dimension of RPAR array.`
`*`
`* IPAR (input/output) INTEGER array, dimension (NIP)`
`* Array for communication between the driver and FEVAL.`
`*`
`* NIP (input) INTEGER`
`* Dimension of IPAR array.`
`*`
`* X (input) DOUBLE PRECISION array, dimension (NX)`
`* Array containing the state variables.`
`*`
`* NX (input) INTEGER`
`* Dimension of the state vector.`
`*`
`* U (input) DOUBLE PRECISION array, dimension (NU)`
`* Array containing the input values.`
`*`
`* NU (input) INTEGER`
`* Dimension of the input vector.`
`*`
`* P (input) DOUBLE PRECISION array, dimension (NP)`
`* Array containing the parameter values.`
`*`
`* NP (input) INTEGER`
`* Dimension of the parameter vector.`
`*`
`* F (output) DOUBLE PRECISION array, dimension (LDF,NX)`
`* The resulting function value f(T,X).`
`*`
`* LDF (input) INTEGER`
`* The leading dimension of F.`
`*`
`* T (input) INTEGER`
`* The time point where the function is evaluated.`
`*`
`* Error Indicator`
`*`
`* INFO INTEGER`
`* Returns values of error from FEVAL or 100 in case`
`* a bad problem was choosen.`
`*`
`* METHOD`
`*`
`* REFERENCES`
`*`
`* CONTRIBUTORS`
`*`
`* REVISIONS`
`*`
`* -`
`*`
`* KEYWORDS`
`*`
`*`
`* ******************************************************************`
`*`
`* .. Common variables ..`
`COMMON /TESTING/ ISOLVER`
`INTEGER LSODI_, LSOIBT_, RADAU5_, DASSL_, DASPK_, GELDA_`
`PARAMETER (LSODI_ = 1, LSOIBT_ = 2)`
`PARAMETER (RADAU5_ = 3, DASSL_ = 4, DASPK_ = 5)`
`PARAMETER (GELDA_ = 6)`
`* .. Scalar Arguments ..`
`INTEGER NRP, NIP, NX, NU, NP, LDF, INFO`
`DOUBLE PRECISION T`
`* .. Array Arguments ..`
`INTEGER IPAR(NIP)`
`DOUBLE PRECISION RPAR(NRP), X(NX), U(NU), P(NP),`
`$ F(LDF,NX)`
`* .. External Subroutines ..`
`EXTERNAL FLSODIX`
`* .. Executable Statements ..`
`CALL FLSODIX(NX,T,X,X,F,INFO,RPAR,IPAR)`
`* *** Last line of DAEDA_ ***`
`END`

`SUBROUTINE DAEDF_( RPAR, NRP, IPAR, NIP, X, XPRIME, NX,`
`$ U, NU, P, NP, T, F, LDF, INFO )`
`*`
`*`
`* PURPOSE`
`*`
`* Interface routine between DAESolver and the problem function MEVAL`
`*`
`* ARGUMENTS`
`*`
`* Input/Output Parameters`
`*`
`* RPAR (input/output) DOUBLE PRECISION array, dimension (NRP)`
`* Array for communication between the driver and MEVAL.`
`*`
`* NRP (input) INTEGER`
`* Dimension of RPAR array.`
`*`
`* IPAR (input/output) INTEGER array, dimension (NIP)`
`* Array for communication between the driver and MEVAL.`
`*`
`* NIP (input) INTEGER`
`* Dimension of IPAR array.`
`*`
`* X (input) DOUBLE PRECISION array, dimension (NX)`
`* Array containing the state variables.`
`*`
`* XPRIME (input) DOUBLE PRECISION array, dimension (NX)`
`* Array containing the state variables derivative.`
`*`
`* NX (input) INTEGER`
`* Dimension of the state vector.`
`*`
`* U (input) DOUBLE PRECISION array, dimension (NU)`
`* Array containing the input values.`
`*`
`* NU (input) INTEGER`
`* Dimension of the input vector.`
`*`
`* P (input) DOUBLE PRECISION array, dimension (NP)`
`* Array containing the parameter values.`
`*`
`* NP (input) INTEGER`
`* Dimension of the parameter vector.`
`*`
`* T (input) INTEGER`
`* The time point where the function is evaluated.`
`*`
`* F (output) DOUBLE PRECISION array, dimension (LDF,NX)`
`* The resulting function value f(T,X).`
`*`
`* LDF (input) INTEGER`
`* The leading dimension of F.`
`*`
`* Error Indicator`
`*`
`* INFO INTEGER`
`* Returns values of error from MEVAL or 100 in case`
`* a bad problem was choosen.`
`*`
`* METHOD`
`*`
`* REFERENCES`
`*`
`* CONTRIBUTORS`
`*`
`* REVISIONS`
`*`
`* -`
`*`
`* KEYWORDS`
`*`
`*`
`* ******************************************************************`
`*`
`* .. Common variables ..`
`COMMON /TESTING/ ISOLVER`
`INTEGER ISOLVER`
`INTEGER LSODI_, LSOIBT_, RADAU5_, DASSL_, DASPK_, GELDA_`
`PARAMETER (LSODI_ = 1, LSOIBT_ = 2)`
`PARAMETER (RADAU5_ = 3, DASSL_ = 4, DASPK_ = 5)`
`PARAMETER (GELDA_ = 6)`
`* .. Scalar Arguments ..`
`INTEGER NRP, NIP, NX, NU, NP, LDF, INFO`
`DOUBLE PRECISION T`
`* .. Array Arguments ..`
`INTEGER IPAR(NIP)`
`DOUBLE PRECISION RPAR(NRP), X(NX), XPRIME(NX), U(NU), P(NP),`
`$ F(LDF,NX)`
`* .. Local Scalars ..`
`INTEGER I`
`* .. External Subroutines ..`
`EXTERNAL MLSODIX, RLSODIX`
`* .. Executable Statements ..`
`IF (ISOLVER .EQ. DASSL_ .OR. ISOLVER .EQ. DASPK_) THEN`
`CALL RLSODIX(LDF,NX,T,X,XPRIME,F,INFO,RPAR,IPAR)`
`ELSE`
`CALL MLSODIX(LDF,NX,T,X,XPRIME,F,INFO,RPAR,IPAR)`
`ENDIF`
`* *** Last line of DAEDF_ ***`
`END`

`SUBROUTINE JACFX_( NRP, NIP, RPAR, IPAR, NX, NU,`
`$ NP, X, U, P, T, FX, LDFX,`
`$ INFO )`
`*`
`*`
`* PURPOSE`
`*`
`* Interface routine between DAESolver and the problem function JEVAL`
`*`
`* ARGUMENTS`
`*`
`* Input/Output Parameters`
`*`
`* NRP (input) INTEGER`
`* Dimension of RPAR array.`
`*`
`* NIP (input) INTEGER`
`* Dimension of IPAR array.`
`*`
`* RPAR (input/output) DOUBLE PRECISION array`
`* Array for communication between the driver and JEVAL.`
`*`
`* IPAR (input/output) INTEGER array`
`* Array for communication between the driver and JEVAL.`
`*`
`* NX (input) INTEGER`
`* Dimension of the state vector.`
`*`
`* NU (input) INTEGER`
`* Dimension of the input vector.`
`*`
`* NP (input) INTEGER`
`* Dimension of the parameter vector.`
`*`
`* X (input) DOUBLE PRECISION array, dimension (NX)`
`* Array containing the state variables.`
`*`
`* U (input) DOUBLE PRECISION array, dimension (NU)`
`* Array containing the input values.`
`*`
`* P (input) DOUBLE PRECISION array, dimension (NP)`
`* Array containing the parameter values.`
`*`
`* T (input) INTEGER`
`* The time point where the derivative is evaluated.`
`*`
`* FX (output) DOUBLE PRECISION array, dimension (LDFX,NX)`
`* The array with the resulting Jacobian matrix.`
`*`
`* LDFX (input) INTEGER`
`* The leading dimension of the array FX.`
`*`
`* Error Indicator`
`*`
`* INFO INTEGER`
`* Returns values of error from JEVAL or 100 in case`
`* a bad problem was choosen.`
`*`
`* METHOD`
`*`
`* REFERENCES`
`*`
`* CONTRIBUTORS`
`*`
`* REVISIONS`
`*`
`* -`
`*`
`* KEYWORDS`
`*`
`*`
`* ******************************************************************`
`*`
`* .. Common variables ..`
`COMMON /TESTING/ ISOLVER`
`INTEGER LSODI_, LSOIBT_, RADAU5_, DASSL_, DASPK_, GELDA_`
`PARAMETER (LSODI_ = 1, LSOIBT_ = 2)`
`PARAMETER (RADAU5_ = 3, DASSL_ = 4, DASPK_ = 5)`
`PARAMETER (GELDA_ = 6)`
`* .. Scalar Arguments ..`
`INTEGER NRP, NIP, NX, NU, NP, LDFX, INFO`
`DOUBLE PRECISION T`
`* .. Array Arguments ..`
`INTEGER IPAR(NIP)`
`DOUBLE PRECISION X(NX), U(NU), P(NP), RPAR(NRP), FX(LDFX,NX)`
`* .. External Subroutines ..`
`EXTERNAL JLSODIX`
`* .. Executable Statements ..`
`CALL JLSODIX(LDFX,NX,T,X,X,FX,INFO,RPAR,IPAR)`
`* *** Last line of JACFX_ ***`
`END`

`SUBROUTINE JACFXDOT_( NRP, NIP, RPAR, IPAR,`
`$ NX, NU, NP, XPRIME, U, P, T, J, LDJ, INFO )`
`*`
`*`
`* PURPOSE`
`*`
`* MATJACFXDOT routine for TRANSAMP problem`
`*`
`* ARGUMENTS`
`*`
`* Input/Output Parameters`
`*`
`* NRP (input) INTEGER`
`* Dimension of RPAR array.`
`*`
`* NIP (input) INTEGER`
`* Dimension of IPAR array.`
`*`
`* RPAR (input/output) DOUBLE PRECISION array`
`* Array for communication with the driver.`
`*`
`* IPAR (input/output) INTEGER array`
`* Array for communication with the driver.`
`*`
`* NX (input) INTEGER`
`* Dimension of the state vector.`
`*`
`*`
`* NU (input) INTEGER`
`* Dimension of the input vector.`
`*`
`* NP (input) INTEGER`
`* Dimension of the parameter vector.`
`*`
`* XPRIME (input) DOUBLE PRECISION array, dimension (NX)`
`* Array containing the derivative of the state variables.`
`*`
`* U (input) DOUBLE PRECISION array, dimension (NU)`
`* Array containing the input values.`
`*`
`* P (input) DOUBLE PRECISION array, dimension (NP)`
`* Array containing the parameter values.`
`*`
`* T (input) INTEGER`
`* The time point where the derivative is evaluated.`
`*`
`* J (output) DOUBLE PRECISION array, dimension (LDJ,NX)`
`* The array with the resulting derivative matrix.`
`*`
`* LDJ (input) INTEGER`
`* The leading dimension of the array J.`
`*`
`* Error Indicator`
`*`
`* INFO INTEGER`
`* Returns 1 in case a bad problem was choosen.`
`*`
`* METHOD`
`*`
`* REFERENCES`
`*`
`* CONTRIBUTORS`
`*`
`* REVISIONS`
`*`
`* -`
`*`
`* KEYWORDS`
`*`
`*`
`* ******************************************************************`
`*`
`* .. Common variables ..`
`COMMON /TESTING/ ISOLVER`
`INTEGER LSODI_, LSOIBT_, RADAU5_, DASSL_, DASPK_, GELDA_`
`PARAMETER (LSODI_ = 1, LSOIBT_ = 2)`
`PARAMETER (RADAU5_ = 3, DASSL_ = 4, DASPK_ = 5)`
`PARAMETER (GELDA_ = 6)`
`* .. Scalar Arguments ..`
`INTEGER NRP, NIP, NX, NU, NP, LDJ, INFO`
`DOUBLE PRECISION T`
`* .. Array Arguments ..`
`INTEGER IPAR(NIP)`
`DOUBLE PRECISION XPRIME(NX), U(NU), P(NP), RPAR(NRP), J(LDJ,NX)`
`* .. Executable Statements ..`
`*`
`CALL JDOTLSODIX(LDJ,NX,T,XPRIME,XPRIME,J,INFO,RPAR,IPAR)`
`ENDIF`
`* *** Last line of JACFXDOT_ ***`
`END`
``

Program Data

    No data required

Program Results

     DAESOLVER EXAMPLE PROGRAM RESULTS

     Problem: LSODIX   Solver: LSODI
     lsodix                                   (lsodix   , DAE) with SOLVER  1
     IWARN on exit from DAESolver =   2
     Solution: (calculated) (reference)
     6.462112224297606E-07
     1.255974374648338E-10
     6.117680951077711E-07
     Relative error comparing with the reference solution:    .8898590685949503E-06

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