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<a name="Matlab_002dcompatible-solvers"></a>
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<p>
Up: <a href="Ordinary-Differential-Equations.html#Ordinary-Differential-Equations" accesskey="u" rel="up">Ordinary Differential Equations</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Matlab_002dcompatible-solvers-1"></a>
<h4 class="subsection">24.1.1 Matlab-compatible solvers</h4>
<p>Octave also provides a set of solvers for initial value problems for Ordinary
Differential Equations that have a <small>MATLAB</small>-compatible interface.
The options for this class of methods are set using the functions.
</p>
<ul>
<li> <code>odeset</code>
</li><li> <code>odeget</code>
</li></ul>
<p>Currently implemented solvers are:
</p>
<ul>
<li> Runge-Kutta methods
<ul>
<li> <code>ode23</code> integrates a system of non-stiff ordinary differential
equations (ODEs) or index-1 differential-algebraic equations (DAEs). It
uses the third-order Bogacki-Shampine method and adapts the local
step size in order to satisfy a user-specified tolerance. The solver
requires three function evaluations per integration step.
</li><li> <code>ode45</code> integrates a system of non-stiff ODEs (or index-1 DAEs)
using the high-order, variable-step Dormand-Prince method. It
requires six function evaluations per integration step, but may
take larger steps on smooth problems than <code>ode23</code>: potentially
offering improved efficiency at smaller tolerances.
</li></ul>
</li><li> Linear multistep methods
<ul>
<li> <code>ode15s</code> integrates a system of stiff ODEs (or index-1 DAEs)
using a variable step, variable order method based on Backward Difference
Formulas (BDF).
</li><li> <code>ode15i</code> integrates a system of fully-implicit ODEs (or index-1
DAEs) using the same variable step, variable order method as <code>ode15s</code>.
The function <code>decic</code> can be used to compute consistent initial
conditions.
</li></ul>
</li></ul>
<a name="XREFode45"></a><dl>
<dt><a name="index-ode45"></a><em>[<var>t</var>, <var>y</var>] =</em> <strong>ode45</strong> <em>(<var>fun</var>, <var>trange</var>, <var>init</var>)</em></dt>
<dt><a name="index-ode45-1"></a><em>[<var>t</var>, <var>y</var>] =</em> <strong>ode45</strong> <em>(<var>fun</var>, <var>trange</var>, <var>init</var>, <var>ode_opt</var>)</em></dt>
<dt><a name="index-ode45-2"></a><em>[<var>t</var>, <var>y</var>, <var>te</var>, <var>ye</var>, <var>ie</var>] =</em> <strong>ode45</strong> <em>(…)</em></dt>
<dt><a name="index-ode45-3"></a><em><var>solution</var> =</em> <strong>ode45</strong> <em>(…)</em></dt>
<dt><a name="index-ode45-4"></a><em></em> <strong>ode45</strong> <em>(…)</em></dt>
<dd>
<p>Solve a set of non-stiff Ordinary Differential Equations (non-stiff ODEs)
with the well known explicit Dormand-Prince method of order 4.
</p>
<p><var>fun</var> is a function handle, inline function, or string containing the
name of the function that defines the ODE: <code>y' = f(t,y)</code>. The function
must accept two inputs where the first is time <var>t</var> and the second is a
column vector of unknowns <var>y</var>.
</p>
<p><var>trange</var> specifies the time interval over which the ODE will be
evaluated. Typically, it is a two-element vector specifying the initial and
final times (<code>[tinit, tfinal]</code>). If there are more than two elements
then the solution will also be evaluated at these intermediate time
instances.
</p>
<p>By default, <code>ode45</code> uses an adaptive timestep with the
<code>integrate_adaptive</code> algorithm. The tolerance for the timestep
computation may be changed by using the options <code>"RelTol"</code> and
<code>"AbsTol"</code>.
</p>
<p><var>init</var> contains the initial value for the unknowns. If it is a row
vector then the solution <var>y</var> will be a matrix in which each column is
the solution for the corresponding initial value in <var>init</var>.
</p>
<p>The optional fourth argument <var>ode_opt</var> specifies non-default options to
the ODE solver. It is a structure generated by <code>odeset</code>.
</p>
<p>The function typically returns two outputs. Variable <var>t</var> is a
column vector and contains the times where the solution was found. The
output <var>y</var> is a matrix in which each column refers to a different
unknown of the problem and each row corresponds to a time in <var>t</var>.
</p>
<p>The output can also be returned as a structure <var>solution</var> which has a
field <var>x</var> containing a row vector of times where the solution was
evaluated and a field <var>y</var> containing the solution matrix such that each
column corresponds to a time in <var>x</var>. Use
<code>fieldnames (<var>solution</var>)</code><!-- /@w --> to see the other fields and
additional information returned.
</p>
<p>If no output arguments are requested, and no <code>OutputFcn</code> is specified
in <var>ode_opt</var>, then the <code>OutputFcn</code> is set to <code>odeplot</code> and the
results of the solver are plotted immediately.
</p>
<p>If using the <code>"Events"</code> option then three additional outputs may be
returned. <var>te</var> holds the time when an Event function returned a zero.
<var>ye</var> holds the value of the solution at time <var>te</var>. <var>ie</var>
contains an index indicating which Event function was triggered in the case
of multiple Event functions.
</p>
<p>Example: Solve the Van der Pol equation
</p>
<div class="example">
<pre class="example">fvdp = @(<var>t</var>,<var>y</var>) [<var>y</var>(2); (1 - <var>y</var>(1)^2) * <var>y</var>(2) - <var>y</var>(1)];
[<var>t</var>,<var>y</var>] = ode45 (fvdp, [0, 20], [2, 0]);
</pre></div>
<p><strong>See also:</strong> <a href="#XREFodeset">odeset</a>, <a href="#XREFodeget">odeget</a>, <a href="#XREFode23">ode23</a>, <a href="#XREFode15s">ode15s</a>.
</p></dd></dl>
<a name="XREFode23"></a><dl>
<dt><a name="index-ode23"></a><em>[<var>t</var>, <var>y</var>] =</em> <strong>ode23</strong> <em>(<var>fun</var>, <var>trange</var>, <var>init</var>)</em></dt>
<dt><a name="index-ode23-1"></a><em>[<var>t</var>, <var>y</var>] =</em> <strong>ode23</strong> <em>(<var>fun</var>, <var>trange</var>, <var>init</var>, <var>ode_opt</var>)</em></dt>
<dt><a name="index-ode23-2"></a><em>[<var>t</var>, <var>y</var>, <var>te</var>, <var>ye</var>, <var>ie</var>] =</em> <strong>ode23</strong> <em>(…)</em></dt>
<dt><a name="index-ode23-3"></a><em><var>solution</var> =</em> <strong>ode23</strong> <em>(…)</em></dt>
<dt><a name="index-ode23-4"></a><em></em> <strong>ode23</strong> <em>(…)</em></dt>
<dd>
<p>Solve a set of non-stiff Ordinary Differential Equations (non-stiff ODEs)
with the well known explicit Bogacki-Shampine method of order 3.
</p>
<p><var>fun</var> is a function handle, inline function, or string containing the
name of the function that defines the ODE: <code>y' = f(t,y)</code>. The function
must accept two inputs where the first is time <var>t</var> and the second is a
column vector of unknowns <var>y</var>.
</p>
<p><var>trange</var> specifies the time interval over which the ODE will be
evaluated. Typically, it is a two-element vector specifying the initial and
final times (<code>[tinit, tfinal]</code>). If there are more than two elements
then the solution will also be evaluated at these intermediate time
instances.
</p>
<p>By default, <code>ode23</code> uses an adaptive timestep with the
<code>integrate_adaptive</code> algorithm. The tolerance for the timestep
computation may be changed by using the options <code>"RelTol"</code> and
<code>"AbsTol"</code>.
</p>
<p><var>init</var> contains the initial value for the unknowns. If it is a row
vector then the solution <var>y</var> will be a matrix in which each column is
the solution for the corresponding initial value in <var>init</var>.
</p>
<p>The optional fourth argument <var>ode_opt</var> specifies non-default options to
the ODE solver. It is a structure generated by <code>odeset</code>.
</p>
<p>The function typically returns two outputs. Variable <var>t</var> is a
column vector and contains the times where the solution was found. The
output <var>y</var> is a matrix in which each column refers to a different
unknown of the problem and each row corresponds to a time in <var>t</var>.
</p>
<p>The output can also be returned as a structure <var>solution</var> which has a
field <var>x</var> containing a row vector of times where the solution was
evaluated and a field <var>y</var> containing the solution matrix such that each
column corresponds to a time in <var>x</var>. Use
<code>fieldnames (<var>solution</var>)</code><!-- /@w --> to see the other fields and
additional information returned.
</p>
<p>If no output arguments are requested, and no <code>OutputFcn</code> is specified
in <var>ode_opt</var>, then the <code>OutputFcn</code> is set to <code>odeplot</code> and the
results of the solver are plotted immediately.
</p>
<p>If using the <code>"Events"</code> option then three additional outputs may be
returned. <var>te</var> holds the time when an Event function returned a zero.
<var>ye</var> holds the value of the solution at time <var>te</var>. <var>ie</var>
contains an index indicating which Event function was triggered in the case
of multiple Event functions.
</p>
<p>Example: Solve the Van der Pol equation
</p>
<div class="example">
<pre class="example">fvdp = @(<var>t</var>,<var>y</var>) [<var>y</var>(2); (1 - <var>y</var>(1)^2) * <var>y</var>(2) - <var>y</var>(1)];
[<var>t</var>,<var>y</var>] = ode23 (fvdp, [0, 20], [2, 0]);
</pre></div>
<p>Reference: For the definition of this method see
<a href="https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods">https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods</a>.
</p>
<p><strong>See also:</strong> <a href="#XREFodeset">odeset</a>, <a href="#XREFodeget">odeget</a>, <a href="#XREFode45">ode45</a>, <a href="#XREFode15s">ode15s</a>.
</p></dd></dl>
<a name="XREFode15s"></a><dl>
<dt><a name="index-ode15s"></a><em>[<var>t</var>, <var>y</var>] =</em> <strong>ode15s</strong> <em>(<var>fun</var>, <var>trange</var>, <var>y0</var>)</em></dt>
<dt><a name="index-ode15s-1"></a><em>[<var>t</var>, <var>y</var>] =</em> <strong>ode15s</strong> <em>(<var>fun</var>, <var>trange</var>, <var>y0</var>, <var>ode_opt</var>)</em></dt>
<dt><a name="index-ode15s-2"></a><em>[<var>t</var>, <var>y</var>, <var>te</var>, <var>ye</var>, <var>ie</var>] =</em> <strong>ode15s</strong> <em>(…)</em></dt>
<dt><a name="index-ode15s-3"></a><em><var>solution</var> =</em> <strong>ode15s</strong> <em>(…)</em></dt>
<dt><a name="index-ode15s-4"></a><em></em> <strong>ode15s</strong> <em>(…)</em></dt>
<dd><p>Solve a set of stiff Ordinary Differential Equations (ODEs) or stiff
semi-explicit index 1 Differential Algebraic Equations (DAEs).
</p>
<p><code>ode15s</code> uses a variable step, variable order BDF (Backward
Differentiation Formula) method that ranges from order 1 to 5.
</p>
<p><var>fun</var> is a function handle, inline function, or string containing the
name of the function that defines the ODE: <code>y' = f(t,y)</code>. The function
must accept two inputs where the first is time <var>t</var> and the second is a
column vector of unknowns <var>y</var>.
</p>
<p><var>trange</var> specifies the time interval over which the ODE will be
evaluated. Typically, it is a two-element vector specifying the initial and
final times (<code>[tinit, tfinal]</code>). If there are more than two elements
then the solution will also be evaluated at these intermediate time
instances.
</p>
<p><var>init</var> contains the initial value for the unknowns. If it is a row
vector then the solution <var>y</var> will be a matrix in which each column is
the solution for the corresponding initial value in <var>init</var>.
</p>
<p>The optional fourth argument <var>ode_opt</var> specifies non-default options to
the ODE solver. It is a structure generated by <code>odeset</code>.
</p>
<p>The function typically returns two outputs. Variable <var>t</var> is a
column vector and contains the times where the solution was found. The
output <var>y</var> is a matrix in which each column refers to a different
unknown of the problem and each row corresponds to a time in <var>t</var>.
</p>
<p>The output can also be returned as a structure <var>solution</var> which has a
field <var>x</var> containing a row vector of times where the solution was
evaluated and a field <var>y</var> containing the solution matrix such that each
column corresponds to a time in <var>x</var>. Use
<code>fieldnames (<var>solution</var>)</code><!-- /@w --> to see the other fields and
additional information returned.
</p>
<p>If no output arguments are requested, and no <code>OutputFcn</code> is specified
in <var>ode_opt</var>, then the <code>OutputFcn</code> is set to <code>odeplot</code> and the
results of the solver are plotted immediately.
</p>
<p>If using the <code>"Events"</code> option then three additional outputs may be
returned. <var>te</var> holds the time when an Event function returned a zero.
<var>ye</var> holds the value of the solution at time <var>te</var>. <var>ie</var>
contains an index indicating which Event function was triggered in the case
of multiple Event functions.
</p>
<p>Example: Solve Robertson’s equations:
</p>
<div class="smallexample">
<pre class="smallexample">function r = robertson_dae (<var>t</var>, <var>y</var>)
r = [ -0.04*<var>y</var>(1) + 1e4*<var>y</var>(2)*<var>y</var>(3)
0.04*<var>y</var>(1) - 1e4*<var>y</var>(2)*<var>y</var>(3) - 3e7*<var>y</var>(2)^2
<var>y</var>(1) + <var>y</var>(2) + <var>y</var>(3) - 1 ];
endfunction
opt = odeset ("Mass", [1 0 0; 0 1 0; 0 0 0], "MStateDependence", "none");
[<var>t</var>,<var>y</var>] = ode15s (@robertson_dae, [0, 1e3], [1; 0; 0], opt);
</pre></div>
<p><strong>See also:</strong> <a href="#XREFdecic">decic</a>, <a href="#XREFodeset">odeset</a>, <a href="#XREFodeget">odeget</a>, <a href="#XREFode23">ode23</a>, <a href="#XREFode45">ode45</a>.
</p></dd></dl>
<a name="XREFode15i"></a><dl>
<dt><a name="index-ode15i"></a><em>[<var>t</var>, <var>y</var>] =</em> <strong>ode15i</strong> <em>(<var>fun</var>, <var>trange</var>, <var>y0</var>, <var>yp0</var>)</em></dt>
<dt><a name="index-ode15i-1"></a><em>[<var>t</var>, <var>y</var>] =</em> <strong>ode15i</strong> <em>(<var>fun</var>, <var>trange</var>, <var>y0</var>, <var>yp0</var>, <var>ode_opt</var>)</em></dt>
<dt><a name="index-ode15i-2"></a><em>[<var>t</var>, <var>y</var>, <var>te</var>, <var>ye</var>, <var>ie</var>] =</em> <strong>ode15i</strong> <em>(…)</em></dt>
<dt><a name="index-ode15i-3"></a><em><var>solution</var> =</em> <strong>ode15i</strong> <em>(…)</em></dt>
<dt><a name="index-ode15i-4"></a><em></em> <strong>ode15i</strong> <em>(…)</em></dt>
<dd><p>Solve a set of fully-implicit Ordinary Differential Equations (ODEs) or
index 1 Differential Algebraic Equations (DAEs).
</p>
<p><code>ode15i</code> uses a variable step, variable order BDF (Backward
Differentiation Formula) method that ranges from order 1 to 5.
</p>
<p><var>fun</var> is a function handle, inline function, or string containing the
name of the function that defines the ODE: <code>0 = f(t,y,yp)</code>. The
function must accept three inputs where the first is time <var>t</var>, the
second is the function value <var>y</var> (a column vector), and the third
is the derivative value <var>yp</var> (a column vector).
</p>
<p><var>trange</var> specifies the time interval over which the ODE will be
evaluated. Typically, it is a two-element vector specifying the initial and
final times (<code>[tinit, tfinal]</code>). If there are more than two elements
then the solution will also be evaluated at these intermediate time
instances.
</p>
<p><var>y0</var> and <var>yp0</var> contain the initial values for the unknowns <var>y</var>
and <var>yp</var>. If they are row vectors then the solution <var>y</var> will be a
matrix in which each column is the solution for the corresponding initial
value in <var>y0</var> and <var>yp0</var>.
</p>
<p><var>y0</var> and <var>yp0</var> must be consistent initial conditions, meaning that
<code>f(t,y0,yp0) = 0</code> is satisfied. The function <code>decic</code> may be used
to compute consistent initial conditions given initial guesses.
</p>
<p>The optional fifth argument <var>ode_opt</var> specifies non-default options to
the ODE solver. It is a structure generated by <code>odeset</code>.
</p>
<p>The function typically returns two outputs. Variable <var>t</var> is a
column vector and contains the times where the solution was found. The
output <var>y</var> is a matrix in which each column refers to a different
unknown of the problem and each row corresponds to a time in <var>t</var>.
</p>
<p>The output can also be returned as a structure <var>solution</var> which has a
field <var>x</var> containing a row vector of times where the solution was
evaluated and a field <var>y</var> containing the solution matrix such that each
column corresponds to a time in <var>x</var>. Use
<code>fieldnames (<var>solution</var>)</code><!-- /@w --> to see the other fields and
additional information returned.
</p>
<p>If no output arguments are requested, and no <code>OutputFcn</code> is specified
in <var>ode_opt</var>, then the <code>OutputFcn</code> is set to <code>odeplot</code> and the
results of the solver are plotted immediately.
</p>
<p>If using the <code>"Events"</code> option then three additional outputs may be
returned. <var>te</var> holds the time when an Event function returned a zero.
<var>ye</var> holds the value of the solution at time <var>te</var>. <var>ie</var>
contains an index indicating which Event function was triggered in the case
of multiple Event functions.
</p>
<p>Example: Solve Robertson’s equations:
</p>
<div class="smallexample">
<pre class="smallexample">function r = robertson_dae (<var>t</var>, <var>y</var>, <var>yp</var>)
r = [ -(<var>yp</var>(1) + 0.04*<var>y</var>(1) - 1e4*<var>y</var>(2)*<var>y</var>(3))
-(<var>yp</var>(2) - 0.04*<var>y</var>(1) + 1e4*<var>y</var>(2)*<var>y</var>(3) + 3e7*<var>y</var>(2)^2)
<var>y</var>(1) + <var>y</var>(2) + <var>y</var>(3) - 1 ];
endfunction
[<var>t</var>,<var>y</var>] = ode15i (@robertson_dae, [0, 1e3], [1; 0; 0], [-1e-4; 1e-4; 0]);
</pre></div>
<p><strong>See also:</strong> <a href="#XREFdecic">decic</a>, <a href="#XREFodeset">odeset</a>, <a href="#XREFodeget">odeget</a>.
</p></dd></dl>
<a name="XREFdecic"></a><dl>
<dt><a name="index-decic"></a><em>[<var>y0_new</var>, <var>yp0_new</var>] =</em> <strong>decic</strong> <em>(<var>fun</var>, <var>t0</var>, <var>y0</var>, <var>fixed_y0</var>, <var>yp0</var>, <var>fixed_yp0</var>)</em></dt>
<dt><a name="index-decic-1"></a><em>[<var>y0_new</var>, <var>yp0_new</var>] =</em> <strong>decic</strong> <em>(<var>fun</var>, <var>t0</var>, <var>y0</var>, <var>fixed_y0</var>, <var>yp0</var>, <var>fixed_yp0</var>, <var>options</var>)</em></dt>
<dt><a name="index-decic-2"></a><em>[<var>y0_new</var>, <var>yp0_new</var>, <var>resnorm</var>] =</em> <strong>decic</strong> <em>(…)</em></dt>
<dd>
<p>Compute consistent implicit ODE initial conditions <var>y0_new</var> and
<var>yp0_new</var> given initial guesses <var>y0</var> and <var>yp0</var>.
</p>
<p>A maximum of <code>length (<var>y0</var>)</code> components between <var>fixed_y0</var> and
<var>fixed_yp0</var> may be chosen as fixed values.
</p>
<p><var>fun</var> is a function handle. The function must accept three inputs where
the first is time <var>t</var>, the second is a column vector of unknowns
<var>y</var>, and the third is a column vector of unknowns <var>yp</var>.
</p>
<p><var>t0</var> is the initial time such that
<code><var>fun</var>(<var>t0</var>, <var>y0_new</var>, <var>yp0_new</var>) = 0</code>, specified as a
scalar.
</p>
<p><var>y0</var> is a vector used as the initial guess for <var>y</var>.
</p>
<p><var>fixed_y0</var> is a vector which specifies the components of <var>y0</var> to
hold fixed. Choose a maximum of <code>length (<var>y0</var>)</code> components between
<var>fixed_y0</var> and <var>fixed_yp0</var> as fixed values.
Set <var>fixed_y0</var>(i) component to 1 if you want to fix the value of
<var>y0</var>(i).
Set <var>fixed_y0</var>(i) component to 0 if you want to allow the value of
<var>y0</var>(i) to change.
</p>
<p><var>yp0</var> is a vector used as the initial guess for <var>yp</var>.
</p>
<p><var>fixed_yp0</var> is a vector which specifies the components of <var>yp0</var> to
hold fixed. Choose a maximum of <code>length (<var>yp0</var>)</code> components
between <var>fixed_y0</var> and <var>fixed_yp0</var> as fixed values.
Set <var>fixed_yp0</var>(i) component to 1 if you want to fix the value of
<var>yp0</var>(i).
Set <var>fixed_yp0</var>(i) component to 0 if you want to allow the value of
<var>yp0</var>(i) to change.
</p>
<p>The optional seventh argument <var>options</var> is a structure array. Use
<code>odeset</code> to generate this structure. The relevant options are
<code>RelTol</code> and <code>AbsTol</code> which specify the error thresholds used to
compute the initial conditions.
</p>
<p>The function typically returns two outputs. Variable <var>y0_new</var> is a
column vector and contains the consistent initial value of <var>y</var>. The
output <var>yp0_new</var> is a column vector and contains the consistent initial
value of <var>yp</var>.
</p>
<p>The optional third output <var>resnorm</var> is the norm of the vector of
residuals. If <var>resnorm</var> is small, <code>decic</code> has successfully
computed the initial conditions. If the value of <var>resnorm</var> is large,
use <code>RelTol</code> and <code>AbsTol</code> to adjust it.
</p>
<p>Example: Compute initial conditions for Robertson’s equations:
</p>
<div class="smallexample">
<pre class="smallexample">function r = robertson_dae (<var>t</var>, <var>y</var>, <var>yp</var>)
r = [ -(<var>yp</var>(1) + 0.04*<var>y</var>(1) - 1e4*<var>y</var>(2)*<var>y</var>(3))
-(<var>yp</var>(2) - 0.04*<var>y</var>(1) + 1e4*<var>y</var>(2)*<var>y</var>(3) + 3e7*<var>y</var>(2)^2)
<var>y</var>(1) + <var>y</var>(2) + <var>y</var>(3) - 1 ];
endfunction
</pre><pre class="smallexample">[<var>y0_new</var>,<var>yp0_new</var>] = decic (@robertson_dae, 0, [1; 0; 0], [1; 1; 0],
[-1e-4; 1; 0], [0; 0; 0]);
</pre></div>
<p><strong>See also:</strong> <a href="#XREFode15i">ode15i</a>, <a href="#XREFodeset">odeset</a>.
</p></dd></dl>
<a name="XREFodeset"></a><dl>
<dt><a name="index-odeset"></a><em><var>odestruct</var> =</em> <strong>odeset</strong> <em>()</em></dt>
<dt><a name="index-odeset-1"></a><em><var>odestruct</var> =</em> <strong>odeset</strong> <em>(<var>"field1"</var>, <var>value1</var>, <var>"field2"</var>, <var>value2</var>, …)</em></dt>
<dt><a name="index-odeset-2"></a><em><var>odestruct</var> =</em> <strong>odeset</strong> <em>(<var>oldstruct</var>, <var>"field1"</var>, <var>value1</var>, <var>"field2"</var>, <var>value2</var>, …)</em></dt>
<dt><a name="index-odeset-3"></a><em><var>odestruct</var> =</em> <strong>odeset</strong> <em>(<var>oldstruct</var>, <var>newstruct</var>)</em></dt>
<dt><a name="index-odeset-4"></a><em></em> <strong>odeset</strong> <em>()</em></dt>
<dd>
<p>Create or modify an ODE options structure.
</p>
<p>When called with no input argument and one output argument, return a new ODE
options structure that contains all possible fields initialized to their
default values. If no output argument is requested, display a list of
the common ODE solver options along with their default value.
</p>
<p>If called with name-value input argument pairs <var>"field1"</var>,
<var>"value1"</var>, <var>"field2"</var>, <var>"value2"</var>, … return a new
ODE options structure with all the most common option fields
initialized, <strong>and</strong> set the values of the fields <var>"field1"</var>,
<var>"field2"</var>, … to the values <var>value1</var>, <var>value2</var>,
….
</p>
<p>If called with an input structure <var>oldstruct</var> then overwrite the
values of the options <var>"field1"</var>, <var>"field2"</var>, … with
new values <var>value1</var>, <var>value2</var>, … and return the
modified structure.
</p>
<p>When called with two input ODE options structures <var>oldstruct</var> and
<var>newstruct</var> overwrite all values from the structure
<var>oldstruct</var> with new values from the structure <var>newstruct</var>.
Empty values in <var>newstruct</var> will not overwrite values in
<var>oldstruct</var>.
</p>
<p>The most commonly used ODE options, which are always assigned a value
by <code>odeset</code>, are the following:
</p>
<dl compact="compact">
<dt><code>AbsTol</code>: positive scalar | vector, def. <code>1e-6</code></dt>
<dd><p>Absolute error tolerance.
</p>
</dd>
<dt><code>BDF</code>: {<code>"off"</code>} | <code>"on"</code></dt>
<dd><p>Use BDF formulas in implicit multistep methods.
<em>Note</em>: This option is not yet implemented.
</p>
</dd>
<dt><code>Events</code>: function_handle</dt>
<dd><p>Event function. An event function must have the form
<code>[value, isterminal, direction] = my_events_f (t, y)</code>
</p>
</dd>
<dt><code>InitialSlope</code>: vector</dt>
<dd><p>Consistent initial slope vector for DAE solvers.
</p>
</dd>
<dt><code>InitialStep</code>: positive scalar</dt>
<dd><p>Initial time step size.
</p>
</dd>
<dt><code>Jacobian</code>: matrix | function_handle</dt>
<dd><p>Jacobian matrix, specified as a constant matrix or a function of
time and state.
</p>
</dd>
<dt><code>JConstant</code>: {<code>"off"</code>} | <code>"on"</code></dt>
<dd><p>Specify whether the Jacobian is a constant matrix or depends on the
state.
</p>
</dd>
<dt><code>JPattern</code>: sparse matrix</dt>
<dd><p>If the Jacobian matrix is sparse and non-constant but maintains a
constant sparsity pattern, specify the sparsity pattern.
</p>
</dd>
<dt><code>Mass</code>: matrix | function_handle</dt>
<dd><p>Mass matrix, specified as a constant matrix or a function of
time and state.
</p>
</dd>
<dt><code>MassSingular</code>: {<code>"maybe"</code>} | <code>"yes"</code> | <code>"on"</code></dt>
<dd><p>Specify whether the mass matrix is singular.
</p>
</dd>
<dt><code>MaxOrder</code>: {<code>5</code>} | <code>4</code> | <code>3</code> | <code>2</code> | <code>1</code></dt>
<dd><p>Maximum order of formula.
</p>
</dd>
<dt><code>MaxStep</code>: positive scalar</dt>
<dd><p>Maximum time step value.
</p>
</dd>
<dt><code>MStateDependence</code>: {<code>"weak"</code>} | <code>"none"</code> | <code>"strong"</code></dt>
<dd><p>Specify whether the mass matrix depends on the state or only on time.
</p>
</dd>
<dt><code>MvPattern</code>: sparse matrix</dt>
<dd><p>If the mass matrix is sparse and non-constant but maintains a
constant sparsity pattern, specify the sparsity pattern.
<em>Note</em>: This option is not yet implemented.
</p>
</dd>
<dt><code>NonNegative</code>: scalar | vector</dt>
<dd><p>Specify elements of the state vector that are expected to remain
non-negative during the simulation.
</p>
</dd>
<dt><code>NormControl</code>: {<code>"off"</code>} | <code>"on"</code></dt>
<dd><p>Control error relative to the 2-norm of the solution, rather than its
absolute value.
</p>
</dd>
<dt><code>OutputFcn</code>: function_handle</dt>
<dd><p>Function to monitor the state during the simulation. For the form of
the function to use see <code>odeplot</code>.
</p>
</dd>
<dt><code>OutputSel</code>: scalar | vector</dt>
<dd><p>Indices of elements of the state vector to be passed to the output
monitoring function.
</p>
</dd>
<dt><code>Refine</code>: positive scalar</dt>
<dd><p>Specify whether output should be returned only at the end of each
time step or also at intermediate time instances. The value should be
a scalar indicating the number of equally spaced time points to use
within each timestep at which to return output.
<em>Note</em>: This option is not yet implemented.
</p>
</dd>
<dt><code>RelTol</code>: positive scalar</dt>
<dd><p>Relative error tolerance.
</p>
</dd>
<dt><code>Stats</code>: {<code>"off"</code>} | <code>"on"</code></dt>
<dd><p>Print solver statistics after simulation.
</p>
</dd>
<dt><code>Vectorized</code>: {<code>"off"</code>} | <code>"on"</code></dt>
<dd><p>Specify whether <code>odefun</code> can be passed multiple values of the
state at once.
</p>
</dd>
</dl>
<p>Field names that are not in the above list are also accepted and
added to the result structure.
</p>
<p><strong>See also:</strong> <a href="#XREFodeget">odeget</a>.
</p></dd></dl>
<a name="XREFodeget"></a><dl>
<dt><a name="index-odeget"></a><em><var>val</var> =</em> <strong>odeget</strong> <em>(<var>ode_opt</var>, <var>field</var>)</em></dt>
<dt><a name="index-odeget-1"></a><em><var>val</var> =</em> <strong>odeget</strong> <em>(<var>ode_opt</var>, <var>field</var>, <var>default</var>)</em></dt>
<dd>
<p>Query the value of the property <var>field</var> in the ODE options structure
<var>ode_opt</var>.
</p>
<p>If called with two input arguments and the first input argument
<var>ode_opt</var> is an ODE option structure and the second input argument
<var>field</var> is a string specifying an option name, then return the option
value <var>val</var> corresponding to <var>field</var> from <var>ode_opt</var>.
</p>
<p>If called with an optional third input argument, and <var>field</var> is
not set in the structure <var>ode_opt</var>, then return the default value
<var>default</var> instead.
</p>
<p><strong>See also:</strong> <a href="#XREFodeset">odeset</a>.
</p></dd></dl>
<a name="XREFodeplot"></a><dl>
<dt><a name="index-odeplot"></a><em><var>stop_solve</var> =</em> <strong>odeplot</strong> <em>(<var>t</var>, <var>y</var>, <var>flag</var>)</em></dt>
<dd>
<p>Open a new figure window and plot the solution of an ode problem at each
time step during the integration.
</p>
<p>The types and values of the input parameters <var>t</var> and <var>y</var> depend on
the input <var>flag</var> that is of type string. Valid values of <var>flag</var>
are:
</p>
<dl compact="compact">
<dt><samp><code>"init"</code></samp></dt>
<dd><p>The input <var>t</var> must be a column vector of length 2 with the first and
last time step (<code>[<var>tfirst</var> <var>tlast</var>]</code>. The input <var>y</var>
contains the initial conditions for the ode problem (<var>y0</var>).
</p>
</dd>
<dt><samp><code>""</code></samp></dt>
<dd><p>The input <var>t</var> must be a scalar double specifying the time for which
the solution in input <var>y</var> was calculated.
</p>
</dd>
<dt><samp><code>"done"</code></samp></dt>
<dd><p>The inputs should be empty, but are ignored if they are present.
</p></dd>
</dl>
<p><code>odeplot</code> always returns false, i.e., don’t stop the ode solver.
</p>
<p>Example: solve an anonymous implementation of the
<code>"Van der Pol"</code> equation and display the results while
solving.
</p>
<div class="example">
<pre class="example">fvdp = @(t,y) [y(2); (1 - y(1)^2) * y(2) - y(1)];
opt = odeset ("OutputFcn", @odeplot, "RelTol", 1e-6);
sol = ode45 (fvdp, [0 20], [2 0], opt);
</pre></div>
<p>Background Information:
This function is called by an ode solver function if it was specified in
the <code>"OutputFcn"</code> property of an options structure created with
<code>odeset</code>. The ode solver will initially call the function with the
syntax <code>odeplot ([<var>tfirst</var>, <var>tlast</var>], <var>y0</var>, "init")</code>. The
function initializes internal variables, creates a new figure window, and
sets the x limits of the plot. Subsequently, at each time step during the
integration the ode solver calls <code>odeplot (<var>t</var>, <var>y</var>, [])</code>.
At the end of the solution the ode solver calls
<code>odeplot ([], [], "done")</code> so that odeplot can perform any clean-up
actions required.
</p>
<p><strong>See also:</strong> <a href="#XREFodeset">odeset</a>, <a href="#XREFodeget">odeget</a>, <a href="#XREFode23">ode23</a>, <a href="#XREFode45">ode45</a>.
</p></dd></dl>
<hr>
<div class="header">
<p>
Up: <a href="Ordinary-Differential-Equations.html#Ordinary-Differential-Equations" accesskey="u" rel="up">Ordinary Differential Equations</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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