Scilab Function

dassl - differential algebraic equation

### Calling Sequence

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

### Parameters

• 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 : real column vector of initial conditions.
• ydot0 : real column vector of the time derivative of y at t0 (may be an estimate).
• t0 : real number is the initial instant.
• t : 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. 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).
• function : Its calling sequence 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.
• list : it must be as follows:
• ```list(res,x1,x2,...)
```

where the calling sequence 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,...).

• string : it must refer to the name of a fortran subroutine (see source code of of Ex-dassl.f in routines/defaut/).
• jac : external (function or list or string). Computes the value of dg/dy+cj*dg/dydot for a given value of parameter cj
• function : Its calling sequence 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
• list : it must be as follows
• ```list(jac,x1,x2,...)
```

where the calling sequence of the function jac is now

```r=jac(t,y,ydot,x1,x2,...)
```

jac still returns dg/dy+cj*dg/dydot as a function of (t,y,ydot,cj,x1,x2,...).

• character string : it must refer to the name of a fortran subroutine (see source code of Ex-dassl.f in routines/defaut/).
• 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(4)=[] 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
```

Detailed examples are given in SCIDIR/tests/dassldasrt.tst

### Examples

``` deff('[r,ires]=chemres(t,y,yd)',[
'r(1)=-0.04*y(1)+1d4*y(2)*y(3)-yd(1);';
'r(2)=0.04*y(1)-1d4*y(2)*y(3)-3d7*y(2)*y(2)-yd(2);'
'r(3)=y(1)+y(2)+y(3)-1;'
'ires=0']);
deff('[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       ]'])

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;
y=dassl([y0,yd0],0,4d10,chemres,info);
y=dassl([y0,yd0],0,4d10,chemres,chemjac,info);
```