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<a name="Finding-Roots"></a>
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<p>
Next: <a href="Products-of-Polynomials.html#Products-of-Polynomials" accesskey="n" rel="next">Products of Polynomials</a>, Previous: <a href="Evaluating-Polynomials.html#Evaluating-Polynomials" accesskey="p" rel="prev">Evaluating Polynomials</a>, Up: <a href="Polynomial-Manipulations.html#Polynomial-Manipulations" accesskey="u" rel="up">Polynomial Manipulations</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="Finding-Roots-1"></a>
<h3 class="section">28.2 Finding Roots</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>compan</code> function
for a definition), and then finding its eigenvalues.
</p>
<a name="XREFroots"></a><dl>
<dt><a name="index-roots"></a>Function File: <em></em> <strong>roots</strong> <em>(<var>v</var>)</em></dt>
<dd>
<p>For a vector <var>v</var> with <em>N</em> components, return
the roots of the polynomial
</p>
<div class="example">
<pre class="example">v(1) * z^(N-1) + … + v(N-1) * z + v(N)
</pre></div>
<p>As an example, the following code finds the roots of the quadratic
polynomial
</p>
<div class="example">
<pre class="example">p(x) = x^2 - 5.
</pre></div>
<div class="example">
<pre class="example">c = [1, 0, -5];
roots (c)
⇒ 2.2361
⇒ -2.2361
</pre></div>
<p>Note that the true result is
<em>+/- sqrt(5)</em>
which is roughly
<em>+/- 2.2361</em>.
</p>
<p><strong>See also:</strong> <a href="Miscellaneous-Functions.html#XREFpoly">poly</a>, <a href="#XREFcompan">compan</a>, <a href="Solvers.html#XREFfzero">fzero</a>.
</p></dd></dl>
<a name="XREFpolyeig"></a><dl>
<dt><a name="index-polyeig"></a>Function File: <em><var>z</var> =</em> <strong>polyeig</strong> <em>(<var>C0</var>, <var>C1</var>, …, <var>Cl</var>)</em></dt>
<dt><a name="index-polyeig-1"></a>Function File: <em>[<var>v</var>, <var>z</var>] =</em> <strong>polyeig</strong> <em>(<var>C0</var>, <var>C1</var>, …, <var>Cl</var>)</em></dt>
<dd>
<p>Solve the polynomial eigenvalue problem of degree <var>l</var>.
</p>
<p>Given an <var>n*n</var> matrix polynomial
<code><var>C</var>(s) = <var>C0</var> + <var>C1</var> s + … + <var>Cl</var> s^l</code>
polyeig solves the eigenvalue problem
<code>(<var>C0</var> + <var>C1</var> + … + <var>Cl</var>)v = 0</code>.
Note that the eigenvalues <var>z</var> are the zeros of the matrix polynomial.
<var>z</var> is an <var>lxn</var> vector and <var>v</var> is an (<var>n</var> x <var>n</var>)l matrix
with columns that correspond to the eigenvectors.
</p>
<p><strong>See also:</strong> <a href="Basic-Matrix-Functions.html#XREFeig">eig</a>, <a href="Sparse-Linear-Algebra.html#XREFeigs">eigs</a>, <a href="#XREFcompan">compan</a>.
</p></dd></dl>
<a name="XREFcompan"></a><dl>
<dt><a name="index-compan"></a>Function File: <em></em> <strong>compan</strong> <em>(<var>c</var>)</em></dt>
<dd><p>Compute the companion matrix corresponding to polynomial coefficient
vector <var>c</var>.
</p>
<p>The companion matrix is
</p>
<div class="smallexample">
<pre class="smallexample"> _ _
| -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>
<p>The eigenvalues of the companion matrix are equal to the roots of the
polynomial.
</p>
<p><strong>See also:</strong> <a href="#XREFroots">roots</a>, <a href="Miscellaneous-Functions.html#XREFpoly">poly</a>, <a href="Basic-Matrix-Functions.html#XREFeig">eig</a>.
</p></dd></dl>
<a name="XREFmpoles"></a><dl>
<dt><a name="index-mpoles"></a>Function File: <em>[<var>multp</var>, <var>idxp</var>] =</em> <strong>mpoles</strong> <em>(<var>p</var>)</em></dt>
<dt><a name="index-mpoles-1"></a>Function File: <em>[<var>multp</var>, <var>idxp</var>] =</em> <strong>mpoles</strong> <em>(<var>p</var>, <var>tol</var>)</em></dt>
<dt><a name="index-mpoles-2"></a>Function File: <em>[<var>multp</var>, <var>idxp</var>] =</em> <strong>mpoles</strong> <em>(<var>p</var>, <var>tol</var>, <var>reorder</var>)</em></dt>
<dd><p>Identify unique poles in <var>p</var> and their associated multiplicity. The
output is ordered from largest pole to smallest pole.
</p>
<p>If the relative difference of two poles is less than <var>tol</var> then
they are considered to be multiples. The default value for <var>tol</var>
is 0.001.
</p>
<p>If the optional parameter <var>reorder</var> is zero, poles are not sorted.
</p>
<p>The output <var>multp</var> is a vector specifying the multiplicity of the
poles. <code><var>multp</var>(n)</code> refers to the multiplicity of the Nth pole
<code><var>p</var>(<var>idxp</var>(n))</code>.
</p>
<p>For example:
</p>
<div class="example">
<pre class="example">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>
<p><strong>See also:</strong> <a href="Products-of-Polynomials.html#XREFresidue">residue</a>, <a href="Miscellaneous-Functions.html#XREFpoly">poly</a>, <a href="#XREFroots">roots</a>, <a href="Products-of-Polynomials.html#XREFconv">conv</a>, <a href="Products-of-Polynomials.html#XREFdeconv">deconv</a>.
</p></dd></dl>
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Next: <a href="Products-of-Polynomials.html#Products-of-Polynomials" accesskey="n" rel="next">Products of Polynomials</a>, Previous: <a href="Evaluating-Polynomials.html#Evaluating-Polynomials" accesskey="p" rel="prev">Evaluating Polynomials</a>, Up: <a href="Polynomial-Manipulations.html#Polynomial-Manipulations" accesskey="u" rel="up">Polynomial Manipulations</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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