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# ode

ordinary differential equation solver

### Syntax

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
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:

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 beydot = 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.

### 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 ilib_for_link('myode','myode.c',[],'c',[],TMPDIR+'/loader.sce'); exec(TMPDIR+'/loader.sce') //incremental linking y0=0; t0=0; t=0:0.1:%pi; y = ode(y0,t0,t,'myode');

### See also

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

### 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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