| <!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="Source of the Rust file `/root/.cargo/git/checkouts/rulinalg-309246e5a94bf5cf/1ed8b93/src/matrix/decomposition/mod.rs`."><meta name="keywords" content="rust, rustlang, rust-lang"><title>mod.rs - source</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="../../../../source-script.js"></script><script defer src="../../../../source-files.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 source"><!--[if lte IE 11]><div class="warning">This old browser is unsupported and will most likely display funky things.</div><![endif]--><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></nav><main><div class="width-limiter"><nav class="sub"><a class="sub-logo-container" href="../../../../rulinalg/index.html"><img class="rust-logo" src="../../../../rust-logo.svg" alt="logo"></a><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="example-wrap"><pre class="src-line-numbers"><span id="1">1</span> |
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| </pre><pre class="rust"><code><span class="doccomment">//! Decompositions for matrices. |
| //! |
| //! This module houses the decomposition API of `rulinalg`. |
| //! A decomposition - or factorization - of a matrix is an |
| //! ordered set of *factors* such that when multiplied reconstructs |
| //! the original matrix. The [Decomposition](trait.Decomposition.html) |
| //! trait encodes this property. |
| //! |
| //! # The decomposition API |
| //! |
| //! Decompositions in `rulinalg` are in general modeled after |
| //! the following: |
| //! |
| //! 1. Given an appropriate matrix, an opaque decomposition object |
| //! may be computed which internally stores the factors |
| //! in an efficient and appropriate format. |
| //! 2. 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 `unpack` the decomposition. |
| //! 3. 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. |
| //! |
| //! 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 |
| //! `A`, one may find matrices `P`, `L` and `U` such that |
| //! `PA = LU`. Here `P` is a permutation matrix, `L` is a lower |
| //! triangular matrix and `U` is an upper triangular matrix. |
| //! |
| //! 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 `Ax = b` 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 `b` has no bearing on the LU decomposition, |
| //! it follows that one can efficiently solve this system for any `b`. |
| //! |
| //! It turns out that the matrices `L` and `U` can be stored compactly |
| //! in the space of a single matrix. Indeed, this is how `PartialPivLu` |
| //! stores the LU decomposition internally. This allows `rulinalg` 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. |
| //! |
| //! # Available decompositions |
| //! |
| //! **The decompositions API is a work in progress.** |
| //! |
| //! Currently, only a portion of the available decompositions in `rulinalg` |
| //! are available through the decomposition API. Please see the |
| //! [Matrix](../struct.Matrix.html) API for the old decomposition |
| //! implementations that have yet not been implemented within |
| //! this framework. |
| //! |
| //! <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> |
| |
| </span><span class="comment">// References: |
| // |
| // 1. [On Matrix Balancing and EigenVector computation] |
| // (http://arxiv.org/pdf/1401.5766v1.pdf), James, Langou and Lowery |
| // |
| // 2. [The QR algorithm for eigen decomposition] |
| // (http://people.inf.ethz.ch/arbenz/ewp/Lnotes/chapter4.pdf) |
| // |
| // 3. [Computation of the SVD] |
| // (http://www.cs.utexas.edu/users/inderjit/public_papers/HLA_SVD.pdf) |
| |
| </span><span class="kw">mod </span>qr; |
| <span class="kw">mod </span>cholesky; |
| <span class="kw">mod </span>bidiagonal; |
| <span class="kw">mod </span>svd; |
| <span class="kw">mod </span>hessenberg; |
| <span class="kw">mod </span>lu; |
| <span class="kw">mod </span>eigen; |
| <span class="kw">mod </span>householder; |
| |
| <span class="kw">use </span>std::any::Any; |
| |
| <span class="kw">use </span>matrix::{Matrix, BaseMatrix}; |
| <span class="kw">use </span>norm::Euclidean; |
| <span class="kw">use </span>vector::Vector; |
| <span class="kw">use </span>utils; |
| <span class="kw">use </span>error::{Error, ErrorKind}; |
| |
| <span class="kw">use </span><span class="self">self</span>::householder::HouseholderReflection; |
| |
| <span class="kw">pub use </span><span class="self">self</span>::householder::HouseholderComposition; |
| <span class="kw">pub use </span><span class="self">self</span>::lu::{PartialPivLu, LUP, FullPivLu, LUPQ}; |
| <span class="kw">pub use </span><span class="self">self</span>::cholesky::Cholesky; |
| <span class="kw">pub use </span><span class="self">self</span>::qr::{HouseholderQr, QR, ThinQR}; |
| |
| <span class="kw">use </span>libnum::{Float}; |
| |
| <span class="doccomment">/// Base trait for decompositions. |
| /// |
| /// A matrix decomposition, or factorization, |
| /// is a procedure which takes a matrix `X` and returns |
| /// a set of `k` factors `X_1, X_2, ..., X_k` such that |
| /// `X = X_1 * X_2 * ... * X_k`. |
| </span><span class="kw">pub trait </span>Decomposition { |
| <span class="doccomment">/// The type representing the ordered set of factors |
| /// that when multiplied yields the decomposed matrix. |
| </span><span class="kw">type </span>Factors; |
| |
| <span class="doccomment">/// Extract the individual factors from this decomposition. |
| </span><span class="kw">fn </span>unpack(<span class="self">self</span>) -> <span class="self">Self</span>::Factors; |
| } |
| |
| <span class="kw">impl</span><T> Matrix<T> |
| <span class="kw">where </span>T: Any + Float |
| { |
| <span class="doccomment">/// Compute the cos and sin values for the givens rotation. |
| /// |
| /// Returns a tuple (c, s). |
| </span><span class="kw">fn </span>givens_rot(a: T, b: T) -> (T, T) { |
| <span class="kw">let </span>r = a.hypot(b); |
| |
| (a / r, -b / r) |
| } |
| |
| <span class="kw">fn </span>make_householder(column: <span class="kw-2">&</span>[T]) -> <span class="prelude-ty">Result</span><Matrix<T>, Error> { |
| <span class="kw">let </span>size = column.len(); |
| |
| <span class="kw">if </span>size == <span class="number">0 </span>{ |
| <span class="kw">return </span><span class="prelude-val">Err</span>(Error::new(ErrorKind::InvalidArg, |
| <span class="string">"Column for householder transform cannot be empty."</span>)); |
| } |
| |
| <span class="kw">let </span>denom = column[<span class="number">0</span>] + column[<span class="number">0</span>].signum() * utils::dot(column, column).sqrt(); |
| |
| <span class="kw">if </span>denom == T::zero() { |
| <span class="kw">return </span><span class="prelude-val">Err</span>(Error::new(ErrorKind::DecompFailure, |
| <span class="string">"Cannot produce househoulder transform from column as first \ |
| entry is 0."</span>)); |
| } |
| |
| <span class="kw">let </span><span class="kw-2">mut </span>v = column.into_iter().map(|<span class="kw-2">&</span>x| x / denom).collect::<Vec<T>>(); |
| <span class="comment">// Ensure first element is fixed to 1. |
| </span>v[<span class="number">0</span>] = T::one(); |
| <span class="kw">let </span>v = Vector::new(v); |
| <span class="kw">let </span>v_norm_sq = v.dot(<span class="kw-2">&</span>v); |
| |
| <span class="kw">let </span>v_vert = Matrix::new(size, <span class="number">1</span>, v.data().clone()); |
| <span class="kw">let </span>v_hor = Matrix::new(<span class="number">1</span>, size, v.into_vec()); |
| <span class="prelude-val">Ok</span>(Matrix::<T>::identity(size) - (v_vert * v_hor) * ((T::one() + T::one()) / v_norm_sq)) |
| } |
| |
| <span class="kw">fn </span>make_householder_vec(column: <span class="kw-2">&</span>[T]) -> <span class="prelude-ty">Result</span><Matrix<T>, Error> { |
| <span class="kw">let </span>size = column.len(); |
| |
| <span class="kw">if </span>size == <span class="number">0 </span>{ |
| <span class="kw">return </span><span class="prelude-val">Err</span>(Error::new(ErrorKind::InvalidArg, |
| <span class="string">"Column for householder transform cannot be empty."</span>)); |
| } |
| |
| <span class="kw">let </span>denom = column[<span class="number">0</span>] + column[<span class="number">0</span>].signum() * utils::dot(column, column).sqrt(); |
| |
| <span class="kw">if </span>denom == T::zero() { |
| <span class="kw">return </span><span class="prelude-val">Err</span>(Error::new(ErrorKind::DecompFailure, |
| <span class="string">"Cannot produce househoulder transform from column as first \ |
| entry is 0."</span>)); |
| } |
| |
| <span class="kw">let </span><span class="kw-2">mut </span>v = column.into_iter().map(|<span class="kw-2">&</span>x| x / denom).collect::<Vec<T>>(); |
| <span class="comment">// Ensure first element is fixed to 1. |
| </span>v[<span class="number">0</span>] = T::one(); |
| <span class="kw">let </span>v = Matrix::new(size, <span class="number">1</span>, v); |
| |
| <span class="prelude-val">Ok</span>(<span class="kw-2">&</span>v / v.norm(Euclidean)) |
| } |
| } |
| </code></pre></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> |
