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<h2 class="chapter">19 Nonlinear Equations</h2>
<p><a name="index-nonlinear-equations-1631"></a><a name="index-equations_002c-nonlinear-1632"></a>
Octave can solve sets of nonlinear equations of the form
<pre class="example"> F (x) = 0
</pre>
<p class="noindent">using the function <code>fsolve</code>, which is based on the <span class="sc">Minpack</span>
subroutine <code>hybrd</code>. This is an iterative technique so a starting
point will have to be provided. This also has the consequence that
convergence is not guaranteed even if a solution exists.
<!-- ./optimization/fsolve.m -->
<p><a name="doc_002dfsolve"></a>
<div class="defun">
— Function File: <b>fsolve</b> (<var>fcn, x0, options</var>)<var><a name="index-fsolve-1633"></a></var><br>
— Function File: [<var>x</var>, <var>fvec</var>, <var>info</var>, <var>output</var>, <var>fjac</var>] <b>=</b><var> fsolve </var>(<var>fcn, <small class="dots">...</small></var>)<var><a name="index-g_t_003d-1634"></a></var><br>
<blockquote><p>Solve a system of nonlinear equations defined by the function <var>fcn</var>.
<var>fcn</var> should accepts a vector (array) defining the unknown variables,
and return a vector of left-hand sides of the equations. Right-hand sides
are defined to be zeros.
In other words, this function attempts to determine a vector <var>x</var> such
that <var>fcn</var><code> (</code><var>x</var><code>)</code> gives (approximately) all zeros.
<var>x0</var> determines a starting guess. The shape of <var>x0</var> is preserved
in all calls to <var>fcn</var>, but otherwise it is treated as a column vector.
<var>options</var> is a structure specifying additional options.
Currently, <code>fsolve</code> recognizes these options:
<code>"FunValCheck"</code>, <code>"OutputFcn"</code>, <code>"TolX"</code>,
<code>"TolFun"</code>, <code>"MaxIter"</code>, <code>"MaxFunEvals"</code>,
<code>"Jacobian"</code>, <code>"Updating"</code> and <code>"ComplexEqn"</code>.
<p>If <code>"Jacobian"</code> is <code>"on"</code>, it specifies that <var>fcn</var>,
called with 2 output arguments, also returns the Jacobian matrix
of right-hand sides at the requested point. <code>"TolX"</code> specifies
the termination tolerance in the unknown variables, while
<code>"TolFun"</code> is a tolerance for equations. Default is <code>1e-7</code>
for both <code>"TolX"</code> and <code>"TolFun"</code>.
If <code>"Updating"</code> is "on", the function will attempt to use Broyden
updates to update the Jacobian, in order to reduce the amount of jacobian
calculations.
If your user function always calculates the Jacobian (regardless of number
of output arguments), this option provides no advantage and should be set to
false.
<p><code>"ComplexEqn"</code> is <code>"on"</code>, <code>fsolve</code> will attempt to solve
complex equations in complex variables, assuming that the equations possess a
complex derivative (i.e., are holomorphic). If this is not what you want,
should unpack the real and imaginary parts of the system to get a real
system.
<p>For description of the other options, see <code>optimset</code>.
<p>On return, <var>fval</var> contains the value of the function <var>fcn</var>
evaluated at <var>x</var>, and <var>info</var> may be one of the following values:
<dl>
<dt>1<dd>Converged to a solution point. Relative residual error is less than specified
by TolFun.
<br><dt>2<dd>Last relative step size was less that TolX.
<br><dt>3<dd>Last relative decrease in residual was less than TolF.
<br><dt>0<dd>Iteration limit exceeded.
<br><dt>-3<dd>The trust region radius became excessively small.
</dl>
<p>Note: If you only have a single nonlinear equation of one variable, using
<code>fzero</code> is usually a much better idea.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dfzero.html#doc_002dfzero">fzero</a>, <a href="doc_002doptimset.html#doc_002doptimset">optimset</a>.
<p>Note about user-supplied jacobians:
As an inherent property of the algorithm, jacobian is always requested for a
solution vector whose residual vector is already known, and it is the last
accepted successful step. Often this will be one of the last two calls, but
not always. If the savings by reusing intermediate results from residual
calculation in jacobian calculation are significant, the best strategy is to
employ OutputFcn: After a vector is evaluated for residuals, if OutputFcn is
called with that vector, then the intermediate results should be saved for
future jacobian evaluation, and should be kept until a jacobian evaluation
is requested or until outputfcn is called with a different vector, in which
case they should be dropped in favor of this most recent vector. A short
example how this can be achieved follows:
<pre class="example"> function [fvec, fjac] = user_func (x, optimvalues, state)
persistent sav = [], sav0 = [];
if (nargin == 1)
## evaluation call
if (nargout == 1)
sav0.x = x; # mark saved vector
## calculate fvec, save results to sav0.
elseif (nargout == 2)
## calculate fjac using sav.
endif
else
## outputfcn call.
if (all (x == sav0.x))
sav = sav0;
endif
## maybe output iteration status, etc.
endif
endfunction
....
fsolve (@user_func, x0, optimset ("OutputFcn", @user_func, ...))
</pre>
</blockquote></div>
<p>Here is a complete example. To solve the set of equations
<pre class="example"> -2x^2 + 3xy + 4 sin(y) = 6
3x^2 - 2xy^2 + 3 cos(x) = -4
</pre>
<p class="noindent">you first need to write a function to compute the value of the given
function. For example:
<pre class="example"> function y = f (x)
y(1) = -2*x(1)^2 + 3*x(1)*x(2) + 4*sin(x(2)) - 6;
y(2) = 3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4;
endfunction
</pre>
<p>Then, call <code>fsolve</code> with a specified initial condition to find the
roots of the system of equations. For example, given the function
<code>f</code> defined above,
<pre class="example"> [x, fval, info] = fsolve (@f, [1; 2])
</pre>
<p class="noindent">results in the solution
<pre class="example"> x =
0.57983
2.54621
fval =
-5.7184e-10
5.5460e-10
info = 1
</pre>
<p class="noindent">A value of <code>info = 1</code> indicates that the solution has converged.
<p>The function <code>perror</code> may be used to print English messages
corresponding to the numeric error codes. For example,
<pre class="example"> perror ("fsolve", 1)
-| solution converged to requested tolerance
</pre>
<p>When no Jacobian is supplied (as in the example above) it is approximated
numerically. This requires more function evaluations, and hence is
less efficient. In the example above we could compute the Jacobian
analytically as
<pre class="example"> function J = jacobian(x)
J(1,1) = 3*x(2) - 4*x(1);
J(1,2) = 4*cos(x(2)) + 3*x(1);
J(2,1) = -2*x(2)^2 - 3*sin(x(1)) + 6*x(1);
J(2,2) = -4*x(1)*x(2);
endfunction
</pre>
<p class="noindent">The Jacobian can then be used with the following call to <code>fsolve</code>:
<pre class="example"> [x, fval, info] = fsolve ({@f, @jacobian}, [1; 2]);
</pre>
<p class="noindent">which gives the same solution as before.
<!-- ./optimization/fzero.m -->
<p><a name="doc_002dfzero"></a>
<div class="defun">
— Function File: [<var>x</var>, <var>fval</var>, <var>info</var>, <var>output</var>] = <b>fzero</b> (<var>fun, x0, options</var>)<var><a name="index-fzero-1635"></a></var><br>
<blockquote><p>Find a zero point of a univariate function. <var>fun</var> should be a function
handle or name. <var>x0</var> specifies a starting point. <var>options</var> is a
structure specifying additional options. Currently, <code>fzero</code>
recognizes these options: <code>"FunValCheck"</code>, <code>"OutputFcn"</code>,
<code>"TolX"</code>, <code>"MaxIter"</code>, <code>"MaxFunEvals"</code>.
For description of these options, see <a href="doc_002doptimset.html#doc_002doptimset">optimset</a>.
<p>On exit, the function returns <var>x</var>, the approximate zero point
and <var>fval</var>, the function value thereof.
<var>info</var> is an exit flag that can have these values:
<ul>
<li>1
The algorithm converged to a solution.
<li>0
Maximum number of iterations or function evaluations has been exhausted.
<li>-1
The algorithm has been terminated from user output function.
<li>-2
A general unexpected error.
<li>-3
A non-real value encountered.
<li>-4
A NaN value encountered.
</ul>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002doptimset.html#doc_002doptimset">optimset</a>, <a href="doc_002dfsolve.html#doc_002dfsolve">fsolve</a>.
</p></blockquote></div>
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