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<p>
Up: <a href="Ordinary-Differential-Equations.html" 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" title="Index" rel="index">Index</a>]</p>
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<hr>
<span id="Matlab_002dcompatible-solvers-1"></span><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 (ODEs) that have a <small>MATLAB</small>-compatible interface.
The options for this class of methods are set using the functions.
</p>
<ul>
<li> <a href="#XREFodeset">odeset</a>
</li><li> <a href="#XREFodeget">odeget</a>
</li></ul>
<p>Currently implemented solvers are:
</p>
<ul>
<li> Runge-Kutta methods
<ul>
<li> <a href="#XREFode45">ode45</a> integrates a system of non-stiff ODEs or
index-1 differential-algebraic equations (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><li> <a href="#XREFode23">ode23</a> integrates a system of non-stiff ODEs or (or
index-1 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> <a href="#XREFode23s">ode23s</a> integrates a system of stiff ODEs (or
index-1 DAEs) using a modified second-order Rosenbrock method.
</li></ul>
</li><li> Linear multistep methods
<ul>
<li> <a href="#XREFode15s">ode15s</a> 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> <a href="#XREFode15i">ode15i</a> integrates a system of fully-implicit ODEs
(or index-1 DAEs) using the same variable step, variable order method as
<code>ode15s</code>. <a href="#XREFdecic">decic</a> can be used to compute consistent
initial conditions for <code>ode15i</code>.
</li></ul>
</li></ul>
<p>Detailed information on the solvers are given in L. F. Shampine and
M. W. Reichelt, <cite>The MATLAB ODE Suite</cite>, SIAM Journal on
Scientific Computing, Vol. 18, 1997, pp. 1–22.
</p>
<span id="XREFode45"></span><dl>
<dt id="index-ode45">: <em>[<var>t</var>, <var>y</var>] =</em> <strong>ode45</strong> <em>(<var>fun</var>, <var>trange</var>, <var>init</var>)</em></dt>
<dt id="index-ode45-1">: <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 id="index-ode45-2">: <em>[<var>t</var>, <var>y</var>, <var>te</var>, <var>ye</var>, <var>ie</var>] =</em> <strong>ode45</strong> <em>(…)</em></dt>
<dt id="index-ode45-3">: <em><var>solution</var> =</em> <strong>ode45</strong> <em>(…)</em></dt>
<dt id="index-ode45-4">: <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>
<span id="XREFode23"></span><dl>
<dt id="index-ode23">: <em>[<var>t</var>, <var>y</var>] =</em> <strong>ode23</strong> <em>(<var>fun</var>, <var>trange</var>, <var>init</var>)</em></dt>
<dt id="index-ode23-1">: <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 id="index-ode23-2">: <em>[<var>t</var>, <var>y</var>, <var>te</var>, <var>ye</var>, <var>ie</var>] =</em> <strong>ode23</strong> <em>(…)</em></dt>
<dt id="index-ode23-3">: <em><var>solution</var> =</em> <strong>ode23</strong> <em>(…)</em></dt>
<dt id="index-ode23-4">: <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>
<span id="XREFode23s"></span><dl>
<dt id="index-ode23s">: <em>[<var>t</var>, <var>y</var>] =</em> <strong>ode23s</strong> <em>(<var>fun</var>, <var>trange</var>, <var>init</var>)</em></dt>
<dt id="index-ode23s-1">: <em>[<var>t</var>, <var>y</var>] =</em> <strong>ode23s</strong> <em>(<var>fun</var>, <var>trange</var>, <var>init</var>, <var>ode_opt</var>)</em></dt>
<dt id="index-ode23s-2">: <em>[<var>t</var>, <var>y</var>] =</em> <strong>ode23s</strong> <em>(…, <var>par1</var>, <var>par2</var>, …)</em></dt>
<dt id="index-ode23s-3">: <em>[<var>t</var>, <var>y</var>, <var>te</var>, <var>ye</var>, <var>ie</var>] =</em> <strong>ode23s</strong> <em>(…)</em></dt>
<dt id="index-ode23s-4">: <em><var>solution</var> =</em> <strong>ode23s</strong> <em>(…)</em></dt>
<dd>
<p>Solve a set of stiff Ordinary Differential Equations (stiff ODEs) with a
Rosenbrock method of order (2,3).
</p>
<p><var>fun</var> is a function handle, inline function, or string containing the
name of the function that defines the ODE: <code>M 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>. <var>M</var> is a constant mass
matrix, non-singular and possibly sparse. Set the field <code>"Mass"</code> in
<var>odeopts</var> using <var>odeset</var> to specify a mass matrix.
</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 using an interpolation procedure of the same order as the
one of the solver.
</p>
<p>By default, <code>ode23s</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>.
<code>ode23s</code> will ignore the following options: <code>"BDF"</code>,
<code>"InitialSlope"</code>, <code>"MassSingular"</code>, <code>"MStateDependence"</code>,
<code>"MvPattern"</code>, <code>"MaxOrder"</code>, <code>"Non-negative"</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>. If
<var>trange</var> specifies intermediate time steps, only those will be returned.
</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 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 stiff Van der Pol equation
</p>
<div class="example">
<pre class="example">f = @(<var>t</var>,<var>y</var>) [<var>y</var>(2); 1000*(1 - <var>y</var>(1)^2) * <var>y</var>(2) - <var>y</var>(1)];
opt = odeset ('Mass', [1 0; 0 1], 'MaxStep', 1e-1);
[vt, vy] = ode23s (f, [0 2000], [2 0], opt);
</pre></div>
<p><strong>See also:</strong> <a href="#XREFodeset">odeset</a>, <a href="Differential_002dAlgebraic-Equations.html#XREFdaspk">daspk</a>, <a href="Differential_002dAlgebraic-Equations.html#XREFdassl">dassl</a>.
</p></dd></dl>
<span id="XREFode15s"></span><dl>
<dt id="index-ode15s">: <em>[<var>t</var>, <var>y</var>] =</em> <strong>ode15s</strong> <em>(<var>fun</var>, <var>trange</var>, <var>y0</var>)</em></dt>
<dt id="index-ode15s-1">: <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 id="index-ode15s-2">: <em>[<var>t</var>, <var>y</var>, <var>te</var>, <var>ye</var>, <var>ie</var>] =</em> <strong>ode15s</strong> <em>(…)</em></dt>
<dt id="index-ode15s-3">: <em><var>solution</var> =</em> <strong>ode15s</strong> <em>(…)</em></dt>
<dt id="index-ode15s-4">: <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="example">
<pre class="example">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>
<span id="XREFode15i"></span><dl>
<dt id="index-ode15i">: <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 id="index-ode15i-1">: <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 id="index-ode15i-2">: <em>[<var>t</var>, <var>y</var>, <var>te</var>, <var>ye</var>, <var>ie</var>] =</em> <strong>ode15i</strong> <em>(…)</em></dt>
<dt id="index-ode15i-3">: <em><var>solution</var> =</em> <strong>ode15i</strong> <em>(…)</em></dt>
<dt id="index-ode15i-4">: <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="example">
<pre class="example">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>
<span id="XREFdecic"></span><dl>
<dt id="index-decic">: <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 id="index-decic-1">: <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 id="index-decic-2">: <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="example">
<pre class="example">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="example">[<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>
<span id="XREFodeset"></span><dl>
<dt id="index-odeset">: <em><var>odestruct</var> =</em> <strong>odeset</strong> <em>()</em></dt>
<dt id="index-odeset-1">: <em><var>odestruct</var> =</em> <strong>odeset</strong> <em>(<var>"field1"</var>, <var>value1</var>, <var>"field2"</var>, <var>value2</var>, …)</em></dt>
<dt id="index-odeset-2">: <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 id="index-odeset-3">: <em><var>odestruct</var> =</em> <strong>odeset</strong> <em>(<var>oldstruct</var>, <var>newstruct</var>)</em></dt>
<dt id="index-odeset-4">: <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>
<span id="XREFodeget"></span><dl>
<dt id="index-odeget">: <em><var>val</var> =</em> <strong>odeget</strong> <em>(<var>ode_opt</var>, <var>field</var>)</em></dt>
<dt id="index-odeget-1">: <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>
<span id="XREFodeplot"></span><dl>
<dt id="index-odeplot">: <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>
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