### 23.1 Ordinary Differential Equations

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

         dx
         -- = f (x, t)
         dt

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 that names the function 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, 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 intial 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`.

— Loadable Function: lsode_options (opt, val)

> 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 specifing the method of integration to use to solve the ODE system. Valid values are
>   "adams"
>   "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`.