| <!DOCTYPE html><html lang="en"><head><meta charset="utf-8"><meta name="viewport" content="width=device-width, initial-scale=1.0"><meta name="generator" content="rustdoc"><meta name="description" content="Decompositions for matrices."><meta name="keywords" content="rust, rustlang, rust-lang, decomposition"><title>rulinalg::matrix::decomposition - Rust</title><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../SourceSerif4-Regular.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../FiraSans-Regular.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../FiraSans-Medium.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../SourceCodePro-Regular.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../SourceSerif4-Bold.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../SourceCodePro-Semibold.ttf.woff2"><link rel="stylesheet" href="../../../normalize.css"><link rel="stylesheet" href="../../../rustdoc.css" id="mainThemeStyle"><link rel="stylesheet" href="../../../ayu.css" disabled><link rel="stylesheet" href="../../../dark.css" disabled><link rel="stylesheet" href="../../../light.css" id="themeStyle"><script id="default-settings" ></script><script src="../../../storage.js"></script><script defer src="../../../main.js"></script><noscript><link rel="stylesheet" href="../../../noscript.css"></noscript><link rel="alternate icon" type="image/png" href="../../../favicon-16x16.png"><link rel="alternate icon" type="image/png" href="../../../favicon-32x32.png"><link rel="icon" type="image/svg+xml" href="../../../favicon.svg"></head><body class="rustdoc mod"><!--[if lte IE 11]><div class="warning">This old browser is unsupported and will most likely display funky things.</div><![endif]--><nav class="mobile-topbar"><button class="sidebar-menu-toggle">☰</button><a class="sidebar-logo" href="../../../rulinalg/index.html"><div class="logo-container"><img class="rust-logo" src="../../../rust-logo.svg" alt="logo"></div></a><h2></h2></nav><nav class="sidebar"><a class="sidebar-logo" href="../../../rulinalg/index.html"><div class="logo-container"><img class="rust-logo" src="../../../rust-logo.svg" alt="logo"></div></a><h2 class="location"><a href="#">Module decomposition</a></h2><div class="sidebar-elems"><section><ul class="block"><li><a href="#structs">Structs</a></li><li><a href="#traits">Traits</a></li></ul></section></div></nav><main><div class="width-limiter"><nav class="sub"><form class="search-form"><div class="search-container"><span></span><input class="search-input" name="search" autocomplete="off" spellcheck="false" placeholder="Click or press ‘S’ to search, ‘?’ for more options…" type="search"><div id="help-button" title="help" tabindex="-1"><a href="../../../help.html">?</a></div><div id="settings-menu" tabindex="-1"><a href="../../../settings.html" title="settings"><img width="22" height="22" alt="Change settings" src="../../../wheel.svg"></a></div></div></form></nav><section id="main-content" class="content"><div class="main-heading"><h1 class="fqn">Module <a href="../../index.html">rulinalg</a>::<wbr><a href="../index.html">matrix</a>::<wbr><a class="mod" href="#">decomposition</a><button id="copy-path" onclick="copy_path(this)" title="Copy item path to clipboard"><img src="../../../clipboard.svg" width="19" height="18" alt="Copy item path"></button></h1><span class="out-of-band"><a class="srclink" href="../../../src/rulinalg/matrix/decomposition/mod.rs.html#1-230">source</a> · <a id="toggle-all-docs" href="javascript:void(0)" title="collapse all docs">[<span class="inner">−</span>]</a></span></div><details class="rustdoc-toggle top-doc" open><summary class="hideme"><span>Expand description</span></summary><div class="docblock"><p>Decompositions for matrices.</p> |
| <p>This module houses the decomposition API of <code>rulinalg</code>. |
| A decomposition - or factorization - of a matrix is an |
| ordered set of <em>factors</em> such that when multiplied reconstructs |
| the original matrix. The <a href="trait.Decomposition.html">Decomposition</a> |
| trait encodes this property.</p> |
| <h2 id="the-decomposition-api"><a href="#the-decomposition-api">The decomposition API</a></h2> |
| <p>Decompositions in <code>rulinalg</code> are in general modeled after |
| the following:</p> |
| <ol> |
| <li>Given an appropriate matrix, an opaque decomposition object |
| may be computed which internally stores the factors |
| in an efficient and appropriate format.</li> |
| <li>In general, the factors may not be immediately available |
| as distinct matrices after decomposition. If the user |
| desires the explicit matrix factors involved in the |
| decomposition, the user must <code>unpack</code> the decomposition.</li> |
| <li>Before unpacking the decomposition, the decomposition |
| data structure in question may offer an API that provides |
| efficient implementations for some of the most common |
| applications of the decomposition. The user is encouraged |
| to use the decomposition-specific API rather than unpacking |
| the decompositions whenever possible.</li> |
| </ol> |
| <p>For a motivating example that explains the rationale behind |
| this design, let us consider the typical LU decomposition with |
| partial pivoting. In this case, given a square invertible matrix |
| <code>A</code>, one may find matrices <code>P</code>, <code>L</code> and <code>U</code> such that |
| <code>PA = LU</code>. Here <code>P</code> is a permutation matrix, <code>L</code> is a lower |
| triangular matrix and <code>U</code> is an upper triangular matrix.</p> |
| <p>Once the decomposition has been obtained, one of its applications |
| is the efficient solution of multiple similar linear systems. |
| Consider that while computing the LU decomposition requires |
| O(n<sup>3</sup>) floating point operations, the solution to |
| the system <code>Ax = b</code> can be computed in O(n<sup>2</sup>) floating |
| point operations if the LU decomposition has already been obtained. |
| Since the right-hand side <code>b</code> has no bearing on the LU decomposition, |
| it follows that one can efficiently solve this system for any <code>b</code>.</p> |
| <p>It turns out that the matrices <code>L</code> and <code>U</code> can be stored compactly |
| in the space of a single matrix. Indeed, this is how <code>PartialPivLu</code> |
| stores the LU decomposition internally. This allows <code>rulinalg</code> to |
| provide the user with efficient implementations of common applications |
| for the LU decomposition. However, the full matrix factors are easily |
| available to the user by unpacking the decomposition.</p> |
| <h2 id="available-decompositions"><a href="#available-decompositions">Available decompositions</a></h2> |
| <p><strong>The decompositions API is a work in progress.</strong></p> |
| <p>Currently, only a portion of the available decompositions in <code>rulinalg</code> |
| are available through the decomposition API. Please see the |
| <a href="../struct.Matrix.html">Matrix</a> API for the old decomposition |
| implementations that have yet not been implemented within |
| this framework.</p> |
| <table> |
| <thead> |
| <tr> |
| <th>Decomposition</th> |
| <th>Matrix requirements</th> |
| <th>Supported features</th> |
| </tr> |
| <tbody> |
| <tr> |
| <td><a href="struct.PartialPivLu.html">PartialPivLu</a></td> |
| <td>Square, invertible</td> |
| <td> |
| <ul> |
| <li>Linear system solving</li> |
| <li>Matrix inverse</li> |
| <li>Determinant computation</li> |
| </ul> |
| </td> |
| </tr> |
| <tr> |
| <td><a href="struct.FullPivLu.html">FullPivLu</a></td> |
| <td>Square matrices</td> |
| <td> |
| <ul> |
| <li>Linear system solving</li> |
| <li>Matrix inverse</li> |
| <li>Determinant computation</li> |
| <li>Rank computation</li> |
| </ul> |
| </td> |
| </tr> |
| <tr> |
| <td><a href="struct.Cholesky.html">Cholesky</a></td> |
| <td>Square, symmetric positive definite</td> |
| <td> |
| <ul> |
| <li>Linear system solving</li> |
| <li>Matrix inverse</li> |
| <li>Determinant computation</li> |
| </ul> |
| </td> |
| </tr> |
| <tr> |
| <td><a href="struct.HouseholderQr.html">HouseholderQr</a></td> |
| <td>Any matrix</td> |
| <td></td> |
| </tr> |
| </tbody> |
| </table></div></details><h2 id="structs" class="small-section-header"><a href="#structs">Structs</a></h2><div class="item-table"><div class="item-row"><div class="item-left module-item"><a class="struct" href="struct.Cholesky.html" title="rulinalg::matrix::decomposition::Cholesky struct">Cholesky</a></div><div class="item-right docblock-short">Cholesky decomposition.</div></div><div class="item-row"><div class="item-left module-item"><a class="struct" href="struct.FullPivLu.html" title="rulinalg::matrix::decomposition::FullPivLu struct">FullPivLu</a></div><div class="item-right docblock-short">LU decomposition with complete pivoting.</div></div><div class="item-row"><div class="item-left module-item"><a class="struct" href="struct.HouseholderComposition.html" title="rulinalg::matrix::decomposition::HouseholderComposition struct">HouseholderComposition</a></div><div class="item-right docblock-short">An efficient representation for a composition of |
| Householder transformations.</div></div><div class="item-row"><div class="item-left module-item"><a class="struct" href="struct.HouseholderQr.html" title="rulinalg::matrix::decomposition::HouseholderQr struct">HouseholderQr</a></div><div class="item-right docblock-short">QR decomposition based on Householder reflections.</div></div><div class="item-row"><div class="item-left module-item"><a class="struct" href="struct.LUP.html" title="rulinalg::matrix::decomposition::LUP struct">LUP</a></div><div class="item-right docblock-short">Result of unpacking an instance of |
| <a href="struct.PartialPivLu.html">PartialPivLu</a>.</div></div><div class="item-row"><div class="item-left module-item"><a class="struct" href="struct.LUPQ.html" title="rulinalg::matrix::decomposition::LUPQ struct">LUPQ</a></div><div class="item-right docblock-short">Result of unpacking an instance of |
| <a href="struct.FullPivLu.html">FullPivLu</a>.</div></div><div class="item-row"><div class="item-left module-item"><a class="struct" href="struct.PartialPivLu.html" title="rulinalg::matrix::decomposition::PartialPivLu struct">PartialPivLu</a></div><div class="item-right docblock-short">LU decomposition with partial pivoting.</div></div><div class="item-row"><div class="item-left module-item"><a class="struct" href="struct.QR.html" title="rulinalg::matrix::decomposition::QR struct">QR</a></div><div class="item-right docblock-short">The result of unpacking a QR decomposition.</div></div><div class="item-row"><div class="item-left module-item"><a class="struct" href="struct.ThinQR.html" title="rulinalg::matrix::decomposition::ThinQR struct">ThinQR</a></div><div class="item-right docblock-short">The result of computing a <em>thin</em> (or <em>reduced</em>) QR decomposition.</div></div></div><h2 id="traits" class="small-section-header"><a href="#traits">Traits</a></h2><div class="item-table"><div class="item-row"><div class="item-left module-item"><a class="trait" href="trait.Decomposition.html" title="rulinalg::matrix::decomposition::Decomposition trait">Decomposition</a></div><div class="item-right docblock-short">Base trait for decompositions.</div></div></div></section></div></main><div id="rustdoc-vars" data-root-path="../../../" data-current-crate="rulinalg" data-themes="ayu,dark,light" data-resource-suffix="" data-rustdoc-version="1.66.0-nightly (5c8bff74b 2022-10-21)" ></div></body></html> |
