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<a name="Ordinary-Differential-Equations"></a>
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<a name="Ordinary-Differential-Equations-1"></a>
<h3 class="section">24.1 Ordinary Differential Equations</h3>

<p>The function <code>lsode</code> can be used to solve ODEs of the form
</p>
<div class="example">
<pre class="example">dx
-- = f (x, t)
dt
</pre></div>


<p>using Hindmarsh&rsquo;s ODE solver <small>LSODE</small>.
</p>


<a name="XREFlsode"></a><dl>
<dt><a name="index-lsode"></a>: <em>[<var>x</var>, <var>istate</var>, <var>msg</var>] =</em> <strong>lsode</strong> <em>(<var>fcn</var>, <var>x_0</var>, <var>t</var>)</em></dt>
<dt><a name="index-lsode-1"></a>: <em>[<var>x</var>, <var>istate</var>, <var>msg</var>] =</em> <strong>lsode</strong> <em>(<var>fcn</var>, <var>x_0</var>, <var>t</var>, <var>t_crit</var>)</em></dt>
<dd><p>Ordinary Differential Equation (ODE) solver.
</p>
<p>The set of differential equations to solve is
</p>
<div class="example">
<pre class="example">dx
-- = f (x, t)
dt
</pre></div>

<p>with
</p>
<div class="example">
<pre class="example">x(t_0) = x_0
</pre></div>

<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 <em>t_0</em> 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>
<p>The first argument, <var>fcn</var>, is a string, inline, or function handle
that names the function <em>f</em> to call to compute the vector of right
hand sides for the set of equations.  The function must have the form
</p>
<div class="example">
<pre class="example"><var>xdot</var> = f (<var>x</var>, <var>t</var>)
</pre></div>

<p>in which <var>xdot</var> and <var>x</var> are vectors and <var>t</var> is a scalar.
</p>
<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 <em>f</em> described above, and the second element names a
function to compute the Jacobian of <em>f</em>.  The Jacobian function
must have the form
</p>
<div class="example">
<pre class="example"><var>jac</var> = j (<var>x</var>, <var>t</var>)
</pre></div>

<p>in which <var>jac</var> is the matrix of partial derivatives
</p>
<div class="example">
<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></div>


<p>The second argument specifies the initial state of the system <em>x_0</em>.  The
third argument is a vector, <var>t</var>, specifying the time values for which a
solution is sought.
</p>
<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>
<p>After a successful computation, the value of <var>istate</var> will be 2
(consistent with the Fortran version of <small>LSODE</small>).
</p>
<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>
<p>You can use the function <code>lsode_options</code> to set optional
parameters for <code>lsode</code>.
</p>
<p><strong>See also:</strong> <a href="Differential_002dAlgebraic-Equations.html#XREFdaspk">daspk</a>, <a href="Differential_002dAlgebraic-Equations.html#XREFdassl">dassl</a>, <a href="Differential_002dAlgebraic-Equations.html#XREFdasrt">dasrt</a>.
</p></dd></dl>


<a name="XREFlsode_005foptions"></a><dl>
<dt><a name="index-lsode_005foptions"></a>: <em></em> <strong>lsode_options</strong> <em>()</em></dt>
<dt><a name="index-lsode_005foptions-1"></a>: <em>val =</em> <strong>lsode_options</strong> <em>(<var>opt</var>)</em></dt>
<dt><a name="index-lsode_005foptions-2"></a>: <em></em> <strong>lsode_options</strong> <em>(<var>opt</var>, <var>val</var>)</em></dt>
<dd><p>Query or set options for the function <code>lsode</code>.
</p>
<p>When called with no arguments, the names of all available options and
their current values are displayed.
</p>
<p>Given one argument, return the value of the option <var>opt</var>.
</p>
<p>When called with two arguments, <code>lsode_options</code> sets the option
<var>opt</var> to value <var>val</var>.
</p>
<p>Options include
</p>
<dl compact="compact">
<dt><code>&quot;absolute tolerance&quot;</code></dt>
<dd><p>Absolute tolerance.  May be either vector or scalar.  If a vector, it
must match the dimension of the state vector.
</p>
</dd>
<dt><code>&quot;relative tolerance&quot;</code></dt>
<dd><p>Relative tolerance parameter.  Unlike the absolute tolerance, this
parameter may only be a scalar.
</p>
<p>The local error test applied at each integration step is
</p>
<div class="example">
<pre class="example">  abs (local error in x(i)) &lt;= ...
      rtol * abs (y(i)) + atol(i)
</pre></div>

</dd>
<dt><code>&quot;integration method&quot;</code></dt>
<dd><p>A string specifying the method of integration to use to solve the ODE
system.  Valid values are
</p>
<dl compact="compact">
<dt><code>&quot;adams&quot;</code></dt>
<dt><code>&quot;non-stiff&quot;</code></dt>
<dd><p>No Jacobian used (even if it is available).
</p>
</dd>
<dt><code>&quot;bdf&quot;</code></dt>
<dt><code>&quot;stiff&quot;</code></dt>
<dd><p>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.
</p></dd>
</dl>

</dd>
<dt><code>&quot;initial step size&quot;</code></dt>
<dd><p>The step size to be attempted on the first step (default is determined
automatically).
</p>
</dd>
<dt><code>&quot;maximum order&quot;</code></dt>
<dd><p>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.
</p>
</dd>
<dt><code>&quot;maximum step size&quot;</code></dt>
<dd><p>Setting the maximum stepsize will avoid passing over very large
regions  (default is not specified).
</p>
</dd>
<dt><code>&quot;minimum step size&quot;</code></dt>
<dd><p>The minimum absolute step size allowed (default is 0).
</p>
</dd>
<dt><code>&quot;step limit&quot;</code></dt>
<dd><p>Maximum number of steps allowed (default is 100000).
</p></dd>
</dl>
</dd></dl>


<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"></a>

<div class="example">
<pre class="example">## oregonator differential equation
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></div>

<p>and the initial condition <code>x0 = [ 4; 1.1; 4 ]</code>, the set of
equations can be integrated using the command
</p>
<div class="example">
<pre class="example">t = linspace (0, 500, 1000);

y = lsode (&quot;f&quot;, x0, t);
</pre></div>

<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
</p>
<div class="example">
<pre class="example">t = [0, logspace(-1, log10(303), 150), ...
        logspace(log10(304), log10(500), 150)];
</pre></div>

<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>.
</p>
<p>An m-file for the differential equation used above is included with the
Octave distribution in the examples directory under the name
<samp>oregonator.m</samp>.
</p>
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