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<h3 class="section" id="Finding-Roots-1"><span>28.2 Finding Roots<a class="copiable-link" href="#Finding-Roots-1"> ¶</a></span></h3>
<p>Octave can find the roots of a given polynomial. This is done by computing
the companion matrix of the polynomial (see the <code class="code">compan</code> function
for a definition), and then finding its eigenvalues.
</p>
<a class="anchor" id="XREFroots"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-roots"><span><code class="def-type"><var class="var">r</var> =</code> <strong class="def-name">roots</strong> <code class="def-code-arguments">(<var class="var">c</var>)</code><a class="copiable-link" href="#index-roots"> ¶</a></span></dt>
<dd>
<p>Compute the roots of the polynomial <var class="var">c</var>.
</p>
<p>For a vector <var class="var">c</var> with <em class="math">N</em> components, return the roots of the
polynomial
</p>
<div class="example">
<pre class="example-preformatted">c(1) * x^(N-1) + ... + c(N-1) * x + c(N)
</pre></div>
<p>As an example, the following code finds the roots of the quadratic
polynomial
</p>
<div class="example">
<pre class="example-preformatted">p(x) = x^2 - 5.
</pre></div>
<div class="example">
<div class="group"><pre class="example-preformatted">c = [1, 0, -5];
roots (c)
⇒ 2.2361
⇒ -2.2361
</pre></div></div>
<p>Note that the true result is
<em class="math">+/- sqrt(5)</em>
which is roughly
<em class="math">+/- 2.2361</em>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Miscellaneous-Functions.html#XREFpoly">poly</a>, <a class="ref" href="#XREFcompan">compan</a>, <a class="ref" href="Solvers.html#XREFfzero">fzero</a>.
</p></dd></dl>
<a class="anchor" id="XREFpolyeig"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-polyeig"><span><code class="def-type"><var class="var">z</var> =</code> <strong class="def-name">polyeig</strong> <code class="def-code-arguments">(<var class="var">C0</var>, <var class="var">C1</var>, …, <var class="var">Cl</var>)</code><a class="copiable-link" href="#index-polyeig"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-polyeig-1"><span><code class="def-type">[<var class="var">v</var>, <var class="var">z</var>] =</code> <strong class="def-name">polyeig</strong> <code class="def-code-arguments">(<var class="var">C0</var>, <var class="var">C1</var>, …, <var class="var">Cl</var>)</code><a class="copiable-link" href="#index-polyeig-1"> ¶</a></span></dt>
<dd>
<p>Solve the polynomial eigenvalue problem of degree <var class="var">l</var>.
</p>
<p>Given an <var class="var">n</var>x<var class="var">n</var> matrix polynomial
</p>
<p><code class="code"><var class="var">C</var>(<var class="var">s</var>) = <var class="var">C0</var> + <var class="var">C1</var> <var class="var">s</var> + … + <var class="var">Cl</var>
<var class="var">s</var>^<var class="var">l</var></code>
</p>
<p><code class="code">polyeig</code> solves the eigenvalue problem
</p>
<p><code class="code">(<var class="var">C0</var> + <var class="var">C1</var> <var class="var">z</var> + … + <var class="var">Cl</var> <var class="var">z</var>^<var class="var">l</var>)
<var class="var">v</var> = 0</code>.
</p>
<p>Note that the eigenvalues <var class="var">z</var> are the zeros of the matrix polynomial.
<var class="var">z</var> is a row vector with <code class="code"><var class="var">n</var>*<var class="var">l</var></code> elements. <var class="var">v</var> is a
matrix (<var class="var">n</var> x <var class="var">n</var>*<var class="var">l</var>) with columns that correspond to the
eigenvectors.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Basic-Matrix-Functions.html#XREFeig">eig</a>, <a class="ref" href="Sparse-Linear-Algebra.html#XREFeigs">eigs</a>, <a class="ref" href="#XREFcompan">compan</a>.
</p></dd></dl>
<a class="anchor" id="XREFcompan"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-compan"><span><code class="def-type"><var class="var">A</var> =</code> <strong class="def-name">compan</strong> <code class="def-code-arguments">(<var class="var">c</var>)</code><a class="copiable-link" href="#index-compan"> ¶</a></span></dt>
<dd><p>Compute the companion matrix corresponding to polynomial coefficient vector
<var class="var">c</var>.
</p>
<p>The companion matrix is
</p>
<div class="example smallexample">
<div class="group"><pre class="example-preformatted"> _ _
| -c(2)/c(1) -c(3)/c(1) ... -c(N)/c(1) -c(N+1)/c(1) |
| 1 0 ... 0 0 |
| 0 1 ... 0 0 |
A = | . . . . . |
| . . . . . |
| . . . . . |
|_ 0 0 ... 1 0 _|
</pre></div></div>
<p>The eigenvalues of the companion matrix are equal to the roots of the
polynomial.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFroots">roots</a>, <a class="ref" href="Miscellaneous-Functions.html#XREFpoly">poly</a>, <a class="ref" href="Basic-Matrix-Functions.html#XREFeig">eig</a>.
</p></dd></dl>
<a class="anchor" id="XREFmpoles"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-mpoles"><span><code class="def-type">[<var class="var">multp</var>, <var class="var">idxp</var>] =</code> <strong class="def-name">mpoles</strong> <code class="def-code-arguments">(<var class="var">p</var>)</code><a class="copiable-link" href="#index-mpoles"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-mpoles-1"><span><code class="def-type">[<var class="var">multp</var>, <var class="var">idxp</var>] =</code> <strong class="def-name">mpoles</strong> <code class="def-code-arguments">(<var class="var">p</var>, <var class="var">tol</var>)</code><a class="copiable-link" href="#index-mpoles-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-mpoles-2"><span><code class="def-type">[<var class="var">multp</var>, <var class="var">idxp</var>] =</code> <strong class="def-name">mpoles</strong> <code class="def-code-arguments">(<var class="var">p</var>, <var class="var">tol</var>, <var class="var">reorder</var>)</code><a class="copiable-link" href="#index-mpoles-2"> ¶</a></span></dt>
<dd><p>Identify unique poles in <var class="var">p</var> and their associated multiplicity.
</p>
<p>By default, the output is ordered from the pole with the largest magnitude
to the smallest magnitude.
</p>
<p>Two poles are considered to be multiples if the difference between them
is less than the relative tolerance <var class="var">tol</var>.
</p>
<div class="example">
<pre class="example-preformatted">abs (<var class="var">p1</var> - <var class="var">p0</var>) / abs (<var class="var">p0</var>) < <var class="var">tol</var>
</pre></div>
<p>If the pole is 0 then no scaling is done and <var class="var">tol</var> is interpreted as an
absolute tolerance. The default value for <var class="var">tol</var> is 0.001.
</p>
<p>If the optional parameter <var class="var">reorder</var> is false/zero, poles are not
sorted.
</p>
<p>The output <var class="var">multp</var> is a vector specifying the multiplicity of the poles.
<code class="code"><var class="var">multp</var>(n)</code> refers to the multiplicity of the Nth pole
<code class="code"><var class="var">p</var>(<var class="var">idxp</var>(n))</code>.
</p>
<p>For example:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">p = [2 3 1 1 2];
[m, n] = mpoles (p)
⇒ m = [1; 1; 2; 1; 2]
⇒ n = [2; 5; 1; 4; 3]
⇒ p(n) = [3, 2, 2, 1, 1]
</pre></div></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="Products-of-Polynomials.html#XREFresidue">residue</a>, <a class="ref" href="Miscellaneous-Functions.html#XREFpoly">poly</a>, <a class="ref" href="#XREFroots">roots</a>, <a class="ref" href="Products-of-Polynomials.html#XREFconv">conv</a>, <a class="ref" href="Products-of-Polynomials.html#XREFdeconv">deconv</a>.
</p></dd></dl>
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