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<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.
<!-- roots scripts/polynomial/roots.m -->
<p><a name="doc_002droots"></a>
<div class="defun">
— Function File: <b>roots</b> (<var>v</var>)<var><a name="index-roots-2705"></a></var><br>
<blockquote>
<p>For a vector <var>v</var> with N components, return
the roots of the polynomial
<pre class="example"> v(1) * z^(N-1) + ... + v(N-1) * z + v(N)
</pre>
<p>As an example, the following code finds the roots of the quadratic
polynomial
<pre class="example"> p(x) = x^2 - 5.
</pre>
<pre class="example"> c = [1, 0, -5];
roots (c)
⇒ 2.2361
⇒ -2.2361
</pre>
<p>Note that the true result is
+/- sqrt(5)
which is roughly
+/- 2.2361.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dpoly.html#doc_002dpoly">poly</a>, <a href="doc_002dcompan.html#doc_002dcompan">compan</a>, <a href="doc_002dfzero.html#doc_002dfzero">fzero</a>.
</p></blockquote></div>
<!-- compan scripts/polynomial/compan.m -->
<p><a name="doc_002dcompan"></a>
<div class="defun">
— Function File: <b>compan</b> (<var>c</var>)<var><a name="index-compan-2706"></a></var><br>
<blockquote><p>Compute the companion matrix corresponding to polynomial coefficient
vector <var>c</var>.
<p>The companion matrix is
<!-- Set example in small font to prevent overfull line -->
<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>
<p>The eigenvalues of the companion matrix are equal to the roots of the
polynomial.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002droots.html#doc_002droots">roots</a>, <a href="doc_002dpoly.html#doc_002dpoly">poly</a>, <a href="doc_002deig.html#doc_002deig">eig</a>.
</p></blockquote></div>
<!-- mpoles scripts/polynomial/mpoles.m -->
<p><a name="doc_002dmpoles"></a>
<div class="defun">
— Function File: [<var>multp</var>, <var>idxp</var>] = <b>mpoles</b> (<var>p</var>)<var><a name="index-mpoles-2707"></a></var><br>
— Function File: [<var>multp</var>, <var>idxp</var>] = <b>mpoles</b> (<var>p, tol</var>)<var><a name="index-mpoles-2708"></a></var><br>
— Function File: [<var>multp</var>, <var>idxp</var>] = <b>mpoles</b> (<var>p, tol, reorder</var>)<var><a name="index-mpoles-2709"></a></var><br>
<blockquote><p>Identify unique poles in <var>p</var> and their associated multiplicity. The
output is ordered from largest pole to smallest pole.
<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>If the optional parameter <var>reorder</var> is zero, poles are not sorted.
<p>The output <var>multp</var> is a vector specifying the multiplicity of the
poles. <var>multp</var><code>(n)</code> refers to the multiplicity of the Nth pole
<var>p</var><code>(</code><var>idxp</var><code>(n))</code>.
<p>For 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>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dresidue.html#doc_002dresidue">residue</a>, <a href="doc_002dpoly.html#doc_002dpoly">poly</a>, <a href="doc_002droots.html#doc_002droots">roots</a>, <a href="doc_002dconv.html#doc_002dconv">conv</a>, <a href="doc_002ddeconv.html#doc_002ddeconv">deconv</a>.
</p></blockquote></div>
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