<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> <html> <!-- Created by GNU Texinfo 6.5, http://www.gnu.org/software/texinfo/ --> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> <title>Example Code (GNU Octave (version 5.1.0))</title> <meta name="description" content="Example Code (GNU Octave (version 5.1.0))"> <meta name="keywords" content="Example Code (GNU Octave (version 5.1.0))"> <meta name="resource-type" content="document"> <meta name="distribution" content="global"> <meta name="Generator" content="makeinfo"> <link href="index.html#Top" rel="start" title="Top"> <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index"> <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents"> <link href="Diagonal-and-Permutation-Matrices.html#Diagonal-and-Permutation-Matrices" rel="up" title="Diagonal and Permutation Matrices"> <link href="Zeros-Treatment.html#Zeros-Treatment" rel="next" title="Zeros Treatment"> <link href="Permutation-Matrix-Functions.html#Permutation-Matrix-Functions" rel="prev" title="Permutation Matrix Functions"> <style type="text/css"> <!-- a.summary-letter {text-decoration: none} blockquote.indentedblock {margin-right: 0em} blockquote.smallindentedblock {margin-right: 0em; font-size: smaller} blockquote.smallquotation {font-size: smaller} div.display {margin-left: 3.2em} div.example {margin-left: 3.2em} div.lisp {margin-left: 3.2em} div.smalldisplay {margin-left: 3.2em} div.smallexample {margin-left: 3.2em} div.smalllisp {margin-left: 3.2em} kbd {font-style: oblique} pre.display {font-family: inherit} pre.format {font-family: inherit} pre.menu-comment {font-family: serif} pre.menu-preformatted {font-family: serif} pre.smalldisplay {font-family: inherit; font-size: smaller} pre.smallexample {font-size: smaller} pre.smallformat {font-family: inherit; font-size: smaller} pre.smalllisp {font-size: smaller} span.nolinebreak {white-space: nowrap} span.roman {font-family: initial; font-weight: normal} span.sansserif {font-family: sans-serif; font-weight: normal} ul.no-bullet {list-style: none} --> </style> <link rel="stylesheet" type="text/css" href="octave.css"> </head> <body lang="en"> <a name="Example-Code"></a> <div class="header"> <p> Next: <a href="Zeros-Treatment.html#Zeros-Treatment" accesskey="n" rel="next">Zeros Treatment</a>, Previous: <a href="Function-Support.html#Function-Support" accesskey="p" rel="prev">Function Support</a>, Up: <a href="Diagonal-and-Permutation-Matrices.html#Diagonal-and-Permutation-Matrices" accesskey="u" rel="up">Diagonal and Permutation Matrices</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="Examples-of-Usage"></a> <h3 class="section">21.4 Examples of Usage</h3> <p>The following can be used to solve a linear system <code>A*x = b</code> using the pivoted LU factorization: </p> <div class="example"> <pre class="example"> [L, U, P] = lu (A); ## now L*U = P*A x = U \ (L \ P) * b; </pre></div> <p>This is one way to normalize columns of a matrix <var>X</var> to unit norm: </p> <div class="example"> <pre class="example"> s = norm (X, "columns"); X /= diag (s); </pre></div> <p>The same can also be accomplished with broadcasting (see <a href="Broadcasting.html#Broadcasting">Broadcasting</a>): </p> <div class="example"> <pre class="example"> s = norm (X, "columns"); X ./= s; </pre></div> <p>The following expression is a way to efficiently calculate the sign of a permutation, given by a permutation vector <var>p</var>. It will also work in earlier versions of Octave, but slowly. </p> <div class="example"> <pre class="example"> det (eye (length (p))(p, :)) </pre></div> <p>Finally, here’s how to solve a linear system <code>A*x = b</code> with Tikhonov regularization (ridge regression) using SVD (a skeleton only): </p> <div class="example"> <pre class="example"> m = rows (A); n = columns (A); [U, S, V] = svd (A); ## determine the regularization factor alpha ## alpha = … ## transform to orthogonal basis b = U'*b; ## Use the standard formula, replacing A with S. ## S is diagonal, so the following will be very fast and accurate. x = (S'*S + alpha^2 * eye (n)) \ (S' * b); ## transform to solution basis x = V*x; </pre></div> </body> </html>