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);
 
  

See Also

ode ,   dasrt ,   impl ,   fort ,   link ,   external ,