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## 23.1 Ordinary Differential Equations

The function `lsode` can be used to solve ODEs of the form

using Hindmarsh's ODE solver LSODE.

Loadable Function: [x, istate, msg] = lsode (fcn, x_0, t, t_crit)

Solve the set of differential equations

 ```dx -- = f(x, t) dt ```

with

 ```x(t_0) = x_0 ```

The solution is returned in the matrix x, with each row corresponding to an element of the vector t. The first element of t should be t_0 and should correspond to the initial state of the system x_0, so that the first row of the output is x_0.

The first argument, fcn, is a string, inline, or function handle that names the function f to call to compute the vector of right hand sides for the set of equations. The function must have the form

 ```xdot = f (x, t) ```

in which xdot and x are vectors and t is a scalar.

If fcn is a two-element string array or a two-element cell array of strings, inline functions, or function handles, the first element names the function f described above, and the second element names a function to compute the Jacobian of f. The Jacobian function must have the form

 ```jac = j (x, t) ```

in which jac is the matrix of partial derivatives

 ``` | df_1 df_1 df_1 | | ---- ---- ... ---- | | dx_1 dx_2 dx_N | | | | df_2 df_2 df_2 | | ---- ---- ... ---- | df_i | dx_1 dx_2 dx_N | jac = ---- = | | dx_j | . . . . | | . . . . | | . . . . | | | | df_N df_N df_N | | ---- ---- ... ---- | | dx_1 dx_2 dx_N | ```

The second and third arguments specify the initial state of the system, x_0, and the initial value of the independent variable t_0.

The fourth argument is optional, and may be used to specify a set of times that the ODE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative.

After a successful computation, the value of istate will be 2 (consistent with the Fortran version of LSODE).

If the computation is not successful, istate will be something other than 2 and msg will contain additional information.

You can use the function `lsode_options` to set optional parameters for `lsode`.

When called with two arguments, this function allows you set options parameters for the function `lsode`. Given one argument, `lsode_options` returns the value of the corresponding option. If no arguments are supplied, the names of all the available options and their current values are displayed.

Options include

`"absolute tolerance"`

Absolute tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector.

`"relative tolerance"`

Relative tolerance parameter. Unlike the absolute tolerance, this parameter may only be a scalar.

The local error test applied at each integration step is

 ``` abs (local error in x(i)) <= ... rtol * abs (y(i)) + atol(i) ```
`"integration method"`

A string specifying the method of integration to use to solve the ODE system. Valid values are

"non-stiff"

No Jacobian used (even if it is available).

"bdf"
"stiff"

Use stiff backward differentiation formula (BDF) method. If a function to compute the Jacobian is not supplied, `lsode` will compute a finite difference approximation of the Jacobian matrix.

`"initial step size"`

The step size to be attempted on the first step (default is determined automatically).

`"maximum order"`

Restrict the maximum order of the solution method. If using the Adams method, this option must be between 1 and 12. Otherwise, it must be between 1 and 5, inclusive.

`"maximum step size"`

Setting the maximum stepsize will avoid passing over very large regions (default is not specified).

`"minimum step size"`

The minimum absolute step size allowed (default is 0).

`"step limit"`

Maximum number of steps allowed (default is 100000).

Here is an example of solving a set of three differential equations using `lsode`. Given the function

 ```function xdot = f (x, t) xdot = zeros (3,1); xdot(1) = 77.27 * (x(2) - x(1)*x(2) + x(1) \ - 8.375e-06*x(1)^2); xdot(2) = (x(3) - x(1)*x(2) - x(2)) / 77.27; xdot(3) = 0.161*(x(1) - x(3)); endfunction ```

and the initial condition `x0 = [ 4; 1.1; 4 ]`, the set of equations can be integrated using the command

 ```t = linspace (0, 500, 1000); y = lsode ("f", x0, t); ```

If you try this, you will see that the value of the result changes dramatically between t = 0 and 5, and again around t = 305. A more efficient set of output points might be

 ```t = [0, logspace (-1, log10(303), 150), \ logspace (log10(304), log10(500), 150)]; ```

See Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman, editor, (1983) for more information about the inner workings of `lsode`.

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