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<a name="Ordinary-Differential-Equations"></a>
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Up: <a rel="up" accesskey="u" href="Differential-Equations.html#Differential-Equations">Differential Equations</a>
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<h3 class="section">23.1 Ordinary Differential Equations</h3>
<p>The function <code>lsode</code> can be used to solve ODEs of the form
<pre class="example"> dx
-- = f (x, t)
dt
</pre>
<p class="noindent">using Hindmarsh's ODE solver <span class="sc">Lsode</span>.
<!-- ./DLD-FUNCTIONS/lsode.cc -->
<p><a name="doc_002dlsode"></a>
<div class="defun">
— Loadable Function: [<var>x</var>, <var>istate</var>, <var>msg</var>] = <b>lsode</b> (<var>fcn, x_0, t, t_crit</var>)<var><a name="index-lsode-1786"></a></var><br>
<blockquote><p>Solve the set of differential equations
<pre class="example"> dx
-- = f(x, t)
dt
</pre>
<p>with
<pre class="example"> x(t_0) = x_0
</pre>
<p>The solution is returned in the matrix <var>x</var>, with each row
corresponding to an element of the vector <var>t</var>. The first element
of <var>t</var> should be t_0 and should correspond to the initial
state of the system <var>x_0</var>, so that the first row of the output
is <var>x_0</var>.
<p>The first argument, <var>fcn</var>, 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
<pre class="example"> <var>xdot</var> = f (<var>x</var>, <var>t</var>)
</pre>
<p class="noindent">in which <var>xdot</var> and <var>x</var> are vectors and <var>t</var> is a scalar.
<p>If <var>fcn</var> 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
<pre class="example"> <var>jac</var> = j (<var>x</var>, <var>t</var>)
</pre>
<p>in which <var>jac</var> is the matrix of partial derivatives
<pre class="example"> | 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 |
</pre>
<p>The second and third arguments specify the initial state of the system,
x_0, and the initial value of the independent variable t_0.
<p>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.
<p>After a successful computation, the value of <var>istate</var> will be 2
(consistent with the Fortran version of <span class="sc">Lsode</span>).
<p>If the computation is not successful, <var>istate</var> will be something
other than 2 and <var>msg</var> will contain additional information.
<p>You can use the function <code>lsode_options</code> to set optional
parameters for <code>lsode</code>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002ddaspk.html#doc_002ddaspk">daspk</a>, <a href="doc_002ddassl.html#doc_002ddassl">dassl</a>, <a href="doc_002ddasrt.html#doc_002ddasrt">dasrt</a>.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/lsode.cc -->
<p><a name="doc_002dlsode_005foptions"></a>
<div class="defun">
— Loadable Function: <b>lsode_options</b> (<var>opt, val</var>)<var><a name="index-lsode_005foptions-1787"></a></var><br>
<blockquote><p>When called with two arguments, this function
allows you set options parameters for the function <code>lsode</code>.
Given one argument, <code>lsode_options</code> 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.
<p>Options include
<dl>
<dt><code>"absolute tolerance"</code><dd>Absolute tolerance. May be either vector or scalar. If a vector, it
must match the dimension of the state vector.
<br><dt><code>"relative tolerance"</code><dd>Relative tolerance parameter. Unlike the absolute tolerance, this
parameter may only be a scalar.
<p>The local error test applied at each integration step is
<pre class="example"> abs (local error in x(i)) <= ...
rtol * abs (y(i)) + atol(i)
</pre>
<br><dt><code>"integration method"</code><dd>A string specifying the method of integration to use to solve the ODE
system. Valid values are
<dl>
<dt>"adams"<dt>"non-stiff"<dd>No Jacobian used (even if it is available).
<br><dt>"bdf"<br><dt>"stiff"<dd>Use stiff backward differentiation formula (BDF) method. If a
function to compute the Jacobian is not supplied, <code>lsode</code> will
compute a finite difference approximation of the Jacobian matrix.
</dl>
<br><dt><code>"initial step size"</code><dd>The step size to be attempted on the first step (default is determined
automatically).
<br><dt><code>"maximum order"</code><dd>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.
<br><dt><code>"maximum step size"</code><dd>Setting the maximum stepsize will avoid passing over very large
regions (default is not specified).
<br><dt><code>"minimum step size"</code><dd>The minimum absolute step size allowed (default is 0).
<br><dt><code>"step limit"</code><dd>Maximum number of steps allowed (default is 100000).
</dl>
</p></blockquote></div>
<p>Here is an example of solving a set of three differential equations using
<code>lsode</code>. Given the function
<p><a name="index-oregonator-1788"></a>
<pre class="example"> 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
</pre>
<p class="noindent">and the initial condition <code>x0 = [ 4; 1.1; 4 ]</code>, the set of
equations can be integrated using the command
<pre class="example"> t = linspace (0, 500, 1000);
y = lsode ("f", x0, t);
</pre>
<p>If you try this, you will see that the value of the result changes
dramatically between <var>t</var> = 0 and 5, and again around <var>t</var> = 305.
A more efficient set of output points might be
<pre class="example"> t = [0, logspace (-1, log10(303), 150), \
logspace (log10(304), log10(500), 150)];
</pre>
<p>See Alan C. Hindmarsh, <cite>ODEPACK, A Systematized Collection of ODE
Solvers</cite>, in Scientific Computing, R. S. Stepleman, editor, (1983) for
more information about the inner workings of <code>lsode</code>.
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