ode

Arguments

y0: a real vector or matrix: initial state, at t0.
t0: a real scalar, the initial time.
t: a real vector, the times at which the solution is computed.
f: a function, external, string or list, the right hand side of the differential equation.
type: a string, the solver to use. The available solvers are "adams", "stiff", "rk", "rkf", "fix", "discrete", "root".
atol, rtol: absolute and relative tolerances on the final solution y (decimal numbers). If each is a single value, it applies to each component of y. Otherwise, it must be a vector of same size as size(y), and is applied element-wise to y.
jac: a function, external, string or list, the Jacobian of the function f.
ng: an integer.
g: an external (function or character string or list).
k0: an integer (initial time).
kvect: an integer vector.
y: a real vector or matrix. The solution.
rd: a real vector
w, iw: real vectors. (INPUT/OUTPUT). See ode() optional output

Description

ode solves explicit Ordinary Different Equations defined by:

$\begin{eqnarray} \begin{array}{l} \dfrac{dy}{dt} = f(t,y)\\ y(t_0)=y_0 \end{array} \end{eqnarray}$

It is an interface to various solvers, in particular to ODEPACK.

In this help, we only describe the use of ode for standard explicit ODE systems.

The simplest call of ode is: y = ode(y0,t0,t,f) where y0 is the vector of initial conditions, t0 is the initial time, t is the vector of times at which the solution y is computed and y is matrix of solution vectors y=[y(t(1)),y(t(2)),...].

The input argument f defines the right hand side of the first order differential equation. This argument is a function with a specific header.

If f is a Scilab function, its syntax must be
```
ydot = f(t,y)
 
```
where t is a real scalar (the time) and y is a real vector (the state) and ydot is a real vector (the first order derivative dy/dt).
If f is a string, it is the name of a Fortran subroutine or a C compiled function. For example, if we call ode(y0,t0,t,"fex"), then the subroutine fex is called.

The Fortran routine must have the header:
```
fex(n,t,y,ydot)
 
```
where n is an integer, t is a double precision scalar, y and ydot are double precision vectors.

The C function must have the header:
```
fex(int *n,double *t,double *y,double *ydot)
 
```
where t is the time, y the state and ydot is the state derivative (dy/dt).

This external can be build in a OS independent way using ilib_for_link and dynamically linked to Scilab by the link function.
It may happen that the simulator f needs extra arguments. In this case, we can use the following feature. The f argument can also be a list lst=list(simuf,u1,u2,...un) where simuf is a Scilab function with syntax: ydot = f(t,y,u1,u2,...,un) and u1, u2, ..., un are extra arguments which are automatically passed to the simulator simuf.

The function f can return a p-by-q matrix instead of a vector. With this matrix notation, we solve the n=p+q ODE's system dY/dt=F(t,Y) where Y is a p x q matrix. Then initial conditions, Y0, must also be a p x q matrix and the result of ode is the p-by-q(T+1) matrix [Y(t_0),Y(t_1),...,Y(t_T)].

The tolerances rtol and atol are thresholds for relative and absolute estimated errors. The estimated error on y(i) is: rtol(i)*abs(y(i))+atol(i) and integration is carried out as far as this error is small for all components of the state. If rtol and/or atol is a constant rtol(i) and/or atol(i) are set to this constant value. Default values for rtol and atol are respectively rtol=1.d-5 and atol=1.d-7 for most solvers and rtol=1.d-3 and atol=1.d-4 for "rfk" and "fix".

For stiff problems, it is better to give the Jacobian of the RHS function as the optional argument jac. The Jacobian is an external i.e. a function with specified syntax, or the name of a Fortran subroutine or a C function (character string) with specified calling sequence or a list.

If jac is a function the syntax should be J=jac(t,y) where t is a real scalar (time) and y a real vector (state). The result matrix J must evaluate to df/dx i.e. J(k,i) = dfk/dxi where fk is the k-th component of f.
If jac is a character string it refers to the name of a Fortran subroutine or a C function.

The Fortran routine must have the header:
```
subroutine fex(n,t,y,ml,mu,J,nrpd)
integer n,ml,mu,nrpd
double precision t,y(*),J(*)
 
```
The C function must have the header:
```
void fex(int *n,double *t,double *y,int *ml,int *mu,double *J,int *nrpd,)
 
```
In most cases you have not to refer ml, mu and nrpd.
If jac is a list the same conventions as for f apply.

Optional arguments w and iw are vectors for storing information returned by the integration routine (see ode_optional_output for details). When these vectors are provided in RHS of ode the integration re-starts with the same parameters as in its previous stop.

More options can be given to ODEPACK solvers by using %ODEOPTIONS variable. See odeoptions.

The solvers

The type of problem solved and the method used depend on the value of the first optional argument type which can be one of the following strings:

<not given>:: lsoda solver of package ODEPACK is called by default. It automatically selects between nonstiff predictor-corrector Adams method and stiff Backward Differentiation Formula (BDF) method. It uses nonstiff method initially and dynamically monitors data in order to decide which method to use.
"adams":: This is for nonstiff problems. lsode solver of package ODEPACK is called and it uses the Adams method.
"stiff":: This is for stiff problems. lsode solver of package ODEPACK is called and it uses the BDF method.
"rk":: Adaptive Runge-Kutta of order 4 (RK4) method.
"rkf":: The Shampine and Watts program based on Fehlberg's Runge-Kutta pair of order 4 and 5 (RKF45) method is used. This is for non-stiff and mildly stiff problems when derivative evaluations are inexpensive. This method should generally not be used when the user is demanding high accuracy.
"fix":: Same solver as "rkf", but the user interface is very simple, i.e. only rtol and atol parameters can be passed to the solver. This is the simplest method to try.
"root":: ODE solver with rootfinding capabilities. The lsodar solver of package ODEPACK is used. It is a variant of the lsoda solver where it finds the roots of a given vector function. See help on ode_root for more details.
"discrete":: Discrete time simulation. See help on ode_discrete for more details.

Examples

In the following example, we solve the Ordinary Differential Equation dy/dt=y^2-y*sin(t)+cos(t) with the initial condition y(0)=0. We use the default solver.

function ydot=f(t, y)
    ydot=y^2-y*sin(t)+cos(t)
endfunction
y0=0;
t0=0;
t=0:0.1:%pi;
y = ode(y0,t0,t,f);
plot(t,y)

In the following example, we solve the equation dy/dt=A*y. The exact solution is y(t)=expm(A*t)*y(0), where expm is the matrix exponential. The unknown is the 2-by-1 matrix y(t).

function ydot=f(t, y)
    ydot=A*y
endfunction
function J=Jacobian(t, y)
    J=A
endfunction
A=[10,0;0,-1];
y0=[0;1];
t0=0;
t=1;
ode("stiff",y0,t0,t,f,Jacobian)
// Compare with exact solution:
expm(A*t)*y0

In the following example, we solve the ODE dx/dt = A x(t) + B u(t) with u(t)=sin(omega*t). Notice the extra arguments of f: A, u, B, omega are passed to function f in a list.

function xdot=linear(t, x, A, u, B, omega)
    xdot=A*x+B*u(t,omega)
endfunction
function ut=u(t, omega)
    ut=sin(omega*t)
endfunction
A=[1 1;0 2];
B=[1;1];
omega=5;
y0=[1;0];
t0=0;
t=[0.1,0.2,0.5,1];
ode(y0,t0,t,list(linear,A,u,B,omega))

In the following example, we solve the Riccati differential equation dX/dt=A'*X + X*A - X'*B*X + C where initial condition X(0) is the 2-by-2 identity matrix.

function Xdot=ric(t, X, A, B, C)
    Xdot=A'*X+X*A-X'*B*X+C
endfunction
A=[1,1;0,2];
B=[1,0;0,1];
C=[1,0;0,1];
y0=eye(A);
t0=0;
t=0:0.1:%pi;
X = ode(y0,t0,t,list(ric,A,B,C))

In the following example, we solve the differential equation dY/dt=A*Y where the unknown Y(t) is a 2-by-2 matrix. The exact solution is Y(t)=expm(A*t), where expm is the matrix exponential.

function ydot=f(t, y, A)
    ydot=A*y;
endfunction
A=[1,1;0,2];
y0=eye(A);
t0=0;
t=1;
ode(y0,t0,t,list(f,A))
// Compare with the exact solution:
expm(A*t)
ode("adams",y0,t0,t,f)

With a compiler

The following example requires a C compiler.

// ---------- Simple one dimension ODE (C coded external)
ccode=['#include <math.h>'
       'void myode(int *n,double *t,double *y,double *ydot)'
       '{'
       '  ydot[0]=y[0]*y[0]-y[0]*sin(*t)+cos(*t);'
       '}']
mputl(ccode,TMPDIR+'/myode.c') //create the C file
// Compile
cd TMPDIR
ilib_for_link('myode','myode.c',[],'c',[],'loader.sce');
exec('loader.sce') //incremental linking
y0=0;
t0=0;
t=0:0.1:%pi;
y = ode(y0,t0,t,'myode');

Bibliography

Alan C. Hindmarsh, "lsode and lsodi, two new initial value ordinary differential equation solvers", ACM-Signum newsletter, vol. 15, no. 4 (1980), pp. 10-11.

Used Functions

The associated routines can be found in SCI/modules/differential_equations/src/fortran directory: lsode.f lsoda.f lsodar.f

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