dassl

differential algebraic equation

Syntax

[r [,hd]] = dassl(x0,t0,t [,atol,[rtol]],res [,jac] [,info] [,hd])

Arguments

x0

is either y0 (ydot0 is estimated by dassl with zero as first estimate) or the matrix [y0 ydot0]. g(t,y0,ydot0) must be equal to zero. If you only know an estimate of ydot0 set info(7)=1.

y0: a real column vector of initial conditions.
ydot0: a real column vector of the time derivative of y at t0 (may be an estimate).

t0

a real number is the initial instant.

t

a real scalar or vector. Gives instants for which you want the solution. Note that you can get solution at each dassl's step point by setting info(2)=1.

atol, rtol

real scalars or column vectors of same size as y or both of size 1. atol, rtol give respectively absolute and relative error tolerances of solution. If vectors, the tolerances are specified for each component of y.

res

an external (function or list or string). Computes the value of g(t,y,ydot). It may be :

A Scilab function.

Its syntax must be [r,ires]=res(t,y,ydot) and res must return the residue r=g(t,y,ydot) and error flag ires. ires = 0 if res succeeds to compute r, =-1 if residue is locally not defined for (t,y,ydot), =-2 if parameters are out of admissible range.
A list.

This form allows to pass parameters other than t, y, ydot to the function. It must be as follows:
```
list(res,x1,x2,...)
```
where the syntax of the function res is now
```
r = res(t,y,ydot,x1,x2,...)
```
res still returns r=g(t,y,ydot) as a function of (t,y,ydot,x1,x2,...).
A string.

It must refer to the name of a C or Fortran subroutine linked with Scilab.

In C the syntax must be:
```
void res(double *t, double y[], double yd[], double r[],
         int *ires, double rpar[], int ipar[])
```
In Fortran it must be:
```
subroutine res(t,y,yd,r,ires,rpar,ipar)
double precision t, y(*),yd(*),r(*),rpar(*)
integer ires,ipar(*)
```
The rpar and ipar arrays must be present but cannot be used.

jac

an external (function or list or string). Computes the value of dg/dy+cj*dg/dydot for a given value of parameter cj.

A Scilab function.

Its syntax must be r=jac(t,y,ydot,cj) and the jac function must return r=dg(t,y,ydot)/dy+cj*dg(t,y,ydot)/dydot where cj is a real scalar.
A list.

It must be as follows
```
list(jac,x1,x2,...)
```
where the syntax of the function jac is now
```
r = jac(t,y,ydot,cj,x1,x2,...)
```
jac still returns dg/dy+cj*dg/dydot as a function of (t,y,ydot,cj,x1,x2,...).

A character string.

It must refer to the name of a C or Fortran subroutine linked with Scilab.

In C the syntax must be:

void jac(double *t, double y[], double yd[], double pd[],
         double *cj, double rpar[], int ipar[])

In Fortran it must be:

subroutine jac(t,y,yd,pd,cj,rpar,ipar)
double precision t, y(*),yd(*),pd(*),cj,rpar(*)
integer ipar(*)

info

optional list which contains 7 elements. Default value is list([],0,[],[],[],0,0).

info(1): a real scalar which gives the maximum time for which g is allowed to be evaluated or an empty matrix [] if no limits imposed for time.
info(2): a flag which indicates if dassl returns its intermediate computed values (flag=1) or only the user specified time point values (flag=0).
info(3): a 2 components vector which give the definition [ml,mu] of band matrix computed by jac; r(i - j + ml + mu + 1,j) = "dg(i)/dy(j)+cj*dg(i)/dydot(j)" .If jac returns a full matrix set info(3)=[].
info(4): a real scalar which gives the maximum step size. Set info(4)=[] if no limitation.
info(5): a real scalar which gives the initial step size. Set info(5)=[] if not specified.
info(6): set info(6)=1 if the solution is known to be non negative, else set info(6)=0.
info(7): set info(7)=1 if ydot0 is just an estimation, info(7)=0 if g(t0,y0,ydot0)=0.

hd

a real vector which allows to store the dassl context and to resume integration.

r

a real matrix. Each column is the vector [t;x(t);xdot(t)] where t is time index for which the solution had been computed.

Description

The dassl function integrate the differential algebraic equation and returns the evolution of y a given time points

g(t,y,ydot) = 0
y(t0) = y0  and   ydot(t0) = ydot0

Examples

function [r, ires]=chemres(t, y, yd)
   r=[-0.04*y(1)+1d4*y(2)*y(3)-yd(1)
       0.04*y(1)-1d4*y(2)*y(3)-3d7*y(2)*y(2)-yd(2)
       y(1)+y(2)+y(3)-1];
   ires=0
endfunction

function pd=chemjac(x, y, yd, cj)
    pd=[-0.04-cj , 1d4*y(3)               , 1d4*y(2);
         0.04    ,-1d4*y(3)-2*3d7*y(2)-cj ,-1d4*y(2);
         1       , 1                      , 1       ]
endfunction

y0 = [1;0;0];
yd0 = [-0.04;0.04;0];
t = [1.d-5:0.02:.4,0.41:.1:4,40,400,4000,40000,4d5,4d6,4d7,4d8,4d9,4d10];

y = dassl([y0,yd0],0,t,chemres);

info = list([],0,[],[],[],0,0);
info(2) = 1;
y1 = dassl([y0,yd0],0,4d10,chemres,info);
y2 = dassl([y0,yd0],0,4d10,chemres,chemjac,info);

//Using extra argument for parameters
//-----------------------------------
function [r, ires]=chemres(t, y, yd, a, b, c)
   r=[-a*y(1)+b*y(2)*y(3)-yd(1)
       a*y(1)-b*y(2)*y(3)-c*y(2)*y(2)-yd(2)
       y(1)+y(2)+y(3)-1];
   ires = 0
endfunction

function pd=chemjac(x, y, yd, cj, a, b, c)
    pd=[-a-cj , b*y(3)             , b*y(2);
         a    ,-b*y(3)-2*c*y(2)-cj ,-b*y(2);
         1    , 1                  , 1       ]
endfunction

y3 = dassl([y0,yd0], 0, t, list(chemres,0.04,1d4,3d7),list(chemjac,0.04,1d4,3d7));

// using C code
// ------------
// - create the C code
cd TMPDIR
rescode=['void chemres(double *t, double y[], double yd[], double r[], int *ires, double rpar[], int ipar[])'
         ' {'
         '   r[0] = -0.04*y[0]+1.0e4*y[1]*y[2]                -yd[0];'
         '   r[1] =  0.04*y[0]-1.0e4*y[1]*y[2]-3.0e7*y[1]*y[1]-yd[1];'
         '   r[2] =       y[0]+y[1]+y[2]-1;'
         '   *ires = 0;'
         ' }'];

jaccode=['void chemjac(double *t, double y[], double yd[], double pd[], double *cj, double rpar[], int ipar[])'
         ' {'
         '   /* first column*/'
         '   pd[0] = -0.04-*cj;'
         '   pd[1] =  0.04;'
         '   pd[2] =  1.0;'
         '    /* second column*/'
         '   pd[3] =  1.0e4*y[2];'
         '   pd[4] = -1.0e4*y[2]-2*3.0e7*y[1]-*cj;'
         '   pd[5] =  1.0;'
         '    /* third column*/'
         '   pd[6] =  1.0e4*y[1];'
         '   pd[7] = -1.0e4*y[1];'
         '   pd[8] =  1.0;'
         ' }'];
mputl([rescode;jaccode], 'mycode.c') //create the C file

// - compile it
ilib_for_link(['chemres','chemjac'],'mycode.c',[],'c','','loader.sce');//compile

// - link it with Scilab
exec('loader.sce') //incremental linking

// - call dassl
y4 = dassl([y0,yd0], 0, t, 'chemres', 'chemjac');

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dassl

Syntax

Arguments

Description

Examples

See also