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# ode

ordinary differential equation solver

### Calling Sequence

y=ode(y0,t0,t,f) [y,w,iw]=ode([type],y0,t0,t [,rtol [,atol]],f [,jac] [,w,iw]) [y,rd,w,iw]=ode("root",y0,t0,t [,rtol [,atol]],f [,jac],ng,g [,w,iw]) y=ode("discrete",y0,k0,kvect,f)

### Arguments

- y0
real vector or matrix (initial conditions).

- t0
real scalar (initial time).

- t
real vector (times at which the solution is computed).

- f
external (function or character string or list).

- type
one of the following character string:

`"adams" "stiff" "rk" "rkf" "fix" "discrete" "roots"`

- rtol,atol
real constants or real vectors of the same size as

`y`

.- jac
external (function or character string or list).

- w,iw
real vectors.

- ng
integer.

- g
external (function or character string or list).

- k0
integer (initial time).

- kvect
integer vector.

### Description

`ode`

is the standard function for solving explicit
ODE systems defined by: dy/dt=f(t,y) , y(t0)=y0. It is an interface to
various solvers, in particular to ODEPACK. 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.

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 RHS of the first order differential equation: dy/dt=f(t,y). It 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

`f`

is a Scilab function, its syntax must be`ydot = f(t,y)`

, where`t`

is a real scalar (time) and`y`

a real vector (state) and`ydot`

a real vector (dy/dt)If

`f`

is a character string, it refers to the name of a Fortran subroutine or a C function, i.e. if`ode(y0,t0,t,"fex")`

is the command, then the subroutine`fex`

is called.The Fortran routine must have the following calling sequence:

`fex(n,t,y,ydot)`

, with n an integer, t a double precision scalar, y and ydot double precision vectors.The C function must have the following prototype:

`fex(int *n,double *t,double *y,double *ydot)`

`t`

is the time,`y`

the state and`ydot`

the state derivative (dy/dt)This external can be build in a OS independant way using ilib_for_link and dynamically linked to Scilab by the link function.

The

`f`

argument can also be a list with the following structure:`lst=list(realf,u1,u2,...un)`

where`realf`

is a Scilab function with syntax:`ydot = f(t,y,u1,u2,...,un)`

This syntax allows to use parameters as the arguments of

`realf`

.

The function

`f`

can return a`p x 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 x q(T+1)`

matrix`[Y(t_0),Y(t_1),...,Y(t_T)]`

.Optional input parameters can be given for the error of the solution:

`rtol`

and`atol`

are threshold 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`

. It 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`

with`fk`

= kth component of f.If

`jac`

is a character string it refers to the name of a Fortran subroutine or a C function, with the following calling sequence:Fortran case:

subroutine fex(n,t,y,ml,mu,J,nrpd) integer n,ml,mu,nrpd double precision t,y(*),J(*)

C case:

`jac(n,t,y,ml,mu,J,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.

### Examples

// ---------- Simple one dimension ODE (Scilab function external) // dy/dt=y^2-y sin(t)+cos(t), y(0)=0 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) // ---------- 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 ilib_for_link('myode','myode.c',[],'c',TMPDIR+'/Makefile',TMPDIR+'/loader.sce');//compile exec(TMPDIR+'/loader.sce') //incremental linking y0=0;t0=0;t=0:0.1:%pi; y=ode(y0,t0,t,'myode'); // ---------- Simulation of dx/dt = A x(t) + B u(t) with u(t)=sin(omega*t), // x0=[1;0] // solution x(t) desired at t=0.1, 0.2, 0.5 ,1. // A and u function are passed to RHS function in a list. // B and omega are passed as global variables function xdot=linear(t, x, A, u),xdot=A*x+B*u(t),endfunction function ut=u(t),ut=sin(omega*t),endfunction A=[1 1;0 2];B=[1;1];omega=5; ode([1;0],0,[0.1,0.2,0.5,1],list(linear,A,u)) // ---------- Matrix notation Integration of the Riccati differential equation // Xdot=A'*X + X*A - X'*B*X + C , X(0)=Identity // Solution at t=[1,2] function Xdot=ric(t, X),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]; t0=0;t=0:0.1:%pi; X=ode(eye(A),0,t,ric) // ---------- Matrix notation, Computation of exp(A) A=[1,1;0,2]; function xdot=f(t, x),xdot=A*x;,endfunction ode(eye(A),0,1,f) ode("adams",eye(A),0,1,f) // ---------- Matrix notation, Computation of exp(A) with stiff matrix, Jacobian given A=[10,0;0,-1]; function xdot=f(t, x),xdot=A*x,endfunction function J=Jacobian(t, y),J=A,endfunction ode("stiff",[0;1],0,1,f,Jacobian)

### See Also

- ode_discrete — ordinary differential equation solver, discrete time simulation
- ode_root — ordinary differential equation solver with root finding
- dassl — differential algebraic equation
- impl — differential algebraic equation
- odedc — discrete/continuous ode solver
- odeoptions — set options for ode solvers
- csim — simulation (time response) of linear system
- ltitr — discrete time response (state space)
- rtitr — discrete time response (transfer matrix)

### Authors

- Alan C. Hindmarsh
, mathematics and statistics division, l-316 livermore, ca 94550.19

### 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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