## 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
```  [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

*     DAESOLVER EXAMPLE PROGRAM TEXT FOR LSODIX PROBLEM
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. 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 )
*
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     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 )
*
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     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 )
*
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     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 )
*
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     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 )
*
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     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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