<html lang="en"> <head> <title>Finding Roots - GNU Octave</title> <meta http-equiv="Content-Type" content="text/html"> <meta name="description" content="GNU Octave"> <meta name="generator" content="makeinfo 4.13"> <link title="Top" rel="start" href="index.html#Top"> <link rel="up" href="Polynomial-Manipulations.html#Polynomial-Manipulations" title="Polynomial Manipulations"> <link rel="prev" href="Evaluating-Polynomials.html#Evaluating-Polynomials" title="Evaluating Polynomials"> <link rel="next" href="Products-of-Polynomials.html#Products-of-Polynomials" title="Products of Polynomials"> <link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage"> <meta http-equiv="Content-Style-Type" content="text/css"> <style type="text/css"><!-- pre.display { font-family:inherit } pre.format { font-family:inherit } pre.smalldisplay { font-family:inherit; font-size:smaller } pre.smallformat { font-family:inherit; font-size:smaller } pre.smallexample { font-size:smaller } pre.smalllisp { font-size:smaller } span.sc { font-variant:small-caps } span.roman { font-family:serif; font-weight:normal; } span.sansserif { font-family:sans-serif; font-weight:normal; } --></style> </head> <body> <div class="node"> <a name="Finding-Roots"></a> <p> Next: <a rel="next" accesskey="n" href="Products-of-Polynomials.html#Products-of-Polynomials">Products of Polynomials</a>, Previous: <a rel="previous" accesskey="p" href="Evaluating-Polynomials.html#Evaluating-Polynomials">Evaluating Polynomials</a>, Up: <a rel="up" accesskey="u" href="Polynomial-Manipulations.html#Polynomial-Manipulations">Polynomial Manipulations</a> <hr> </div> <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. <!-- Texinfo @sp should work but in practice produces ugly results for HTML. --> <!-- A simple blank line produces the correct behavior. --> <!-- @sp 1 --> <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. <!-- Texinfo @sp should work but in practice produces ugly results for HTML. --> <!-- A simple blank line produces the correct behavior. --> <!-- @sp 1 --> <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> <!-- Texinfo @sp should work but in practice produces ugly results for HTML. --> <!-- A simple blank line produces the correct behavior. --> <!-- @sp 1 --> <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> </body></html>