dasrt DAE solver with zero crossing

## Syntax

    [r,nn,[,hd]]=dasrt(x0,t0,t [,atol,[rtol]],res [,jac],ng, surf [,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, it 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`.

nn

  a vector with two entries `[times num]` `times` is the value of the time at which the surface is crossed, `num` is the number of the crossed surface.

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

  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,...)`.

    Warning: this form must not be used if there is no extra argument to pass to the function.

  - 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

  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 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(*)

surf

  external (function or list or string). Computes the value of the column vector `surf(t,y)` with `ng` components. Each component defines a surface. It may be defined by:

  - A Scilab function.

    Its syntax must be `surf(t,y)`

  - A list.

    It must be as follows

        list(surf,x1,x2,...)

    where the syntax of the function `surf` is now

        r=surf(t,y,x1,x2,...)

  - A character string.

    It must refer to the name of a Fortran subroutine linked with scilab.

    In C the syntax must be:

        void surf(int *ny, double *t, double y[], int *ng, double gout[])

    In Fortran it must be:

        subroutine surf(ny,t,y,ng,gout)
        double precision t, y(*),gout(*)
        integer ny,ng

info

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

  info(1)

    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)

    flag which indicates if `dassl` returns its intermediate computed values (`flag=1`) or only the user specified time point values (`flag=0`).

  info(3)

    `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)

    real scalar which gives the maximum step size. Set `info(4)=[]` if no limitation.

  info(5)

    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

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

r

  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

Solution of the implicit differential equation.

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

Returns the surface crossing instants and the number of the surface reached in `nn`.

Detailed examples can be found in SCIDIR/tests/unit_tests/dassldasrt.tst

## Examples

    //dy/dt = ((2*log(y)+8)/t -5)*y,  y(1) = 1,  1<=t<=6
    //g1 = ((2*log(y)+8)/t - 5)*y
    //g2 = log(y) - 2.2491
    y0=1;t=2:6;t0=1;y0d=3;
    atol=1.d-6;rtol=0;ng=2;

    deff('[delta,ires]=res1(t,y,ydot)','ires=0;delta=ydot-((2*log(y)+8)/t-5)*y')
    deff('[rts]=gr1(t,y)','rts=[((2*log(y)+8)/t-5)*y;log(y)-2.2491]')

    [yy,nn]=dasrt([y0,y0d],t0,t,atol,rtol,res1,ng,gr1);
    //(Should return nn=[2.4698972 2])

## See also

- ode
- dassl
- daskr
- impl
- call
- link
- external