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');

## See also

- ode
- dasrt
- daskr
- impl
- call
- link
- external