<!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 to the Rust file `/home/mrh/.cargo/registry/src/github.com-1ecc6299db9ec823/nalgebra-0.18.0/src/linalg/bidiagonal.rs`."><meta name="keywords" content="rust, rustlang, rust-lang"><title>bidiagonal.rs.html -- source</title><link rel="stylesheet" type="text/css" href="../../../normalize.css"><link rel="stylesheet" type="text/css" href="../../../rustdoc.css" id="mainThemeStyle"><link rel="stylesheet" type="text/css" href="../../../dark.css"><link rel="stylesheet" type="text/css" href="../../../light.css" id="themeStyle"><script src="../../../storage.js"></script><noscript><link rel="stylesheet" href="../../../noscript.css"></noscript><link rel="shortcut icon" href="http://nalgebra.org/img/favicon.ico"><style type="text/css">#crate-search{background-image:url("../../../down-arrow.svg");}</style></head><body class="rustdoc source"><!--[if lte IE 8]><div class="warning">This old browser is unsupported and will most likely display funky things.</div><![endif]--><nav class="sidebar"><div class="sidebar-menu">☰</div><a href='../../../nalgebra/index.html'><div class='logo-container'><img src='../../../rust-logo.png' alt='logo'></div></a></nav><div class="theme-picker"><button id="theme-picker" aria-label="Pick another theme!"><img src="../../../brush.svg" width="18" alt="Pick another theme!"></button><div id="theme-choices"></div></div><script src="../../../theme.js"></script><nav class="sub"><form class="search-form js-only"><div class="search-container"><div><select id="crate-search"><option value="All crates">All crates</option></select><input class="search-input" name="search" autocomplete="off" spellcheck="false" placeholder="Click or press ‘S’ to search, ‘?’ for more options…" type="search"></div><a id="settings-menu" href="../../../settings.html"><img src="../../../wheel.svg" width="18" alt="Change settings"></a></div></form></nav><section id="main" class="content"><pre class="line-numbers"><span id="1"> 1</span>
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</pre><div class="example-wrap"><pre class="rust ">
<span class="attribute">#[<span class="ident">cfg</span>(<span class="ident">feature</span> <span class="op">=</span> <span class="string">"serde-serialize"</span>)]</span>
<span class="kw">use</span> <span class="ident">serde</span>::{<span class="ident">Deserialize</span>, <span class="ident">Serialize</span>};
<span class="kw">use</span> <span class="ident">alga</span>::<span class="ident">general</span>::<span class="ident">ComplexField</span>;
<span class="kw">use</span> <span class="kw">crate</span>::<span class="ident">allocator</span>::<span class="ident">Allocator</span>;
<span class="kw">use</span> <span class="kw">crate</span>::<span class="ident">base</span>::{<span class="ident">DefaultAllocator</span>, <span class="ident">Matrix</span>, <span class="ident">MatrixMN</span>, <span class="ident">MatrixN</span>, <span class="ident">Unit</span>, <span class="ident">VectorN</span>};
<span class="kw">use</span> <span class="kw">crate</span>::<span class="ident">dimension</span>::{<span class="ident">Dim</span>, <span class="ident">DimDiff</span>, <span class="ident">DimMin</span>, <span class="ident">DimMinimum</span>, <span class="ident">DimSub</span>, <span class="ident">U1</span>};
<span class="kw">use</span> <span class="kw">crate</span>::<span class="ident">storage</span>::<span class="ident">Storage</span>;
<span class="kw">use</span> <span class="kw">crate</span>::<span class="ident">geometry</span>::<span class="ident">Reflection</span>;
<span class="kw">use</span> <span class="kw">crate</span>::<span class="ident">linalg</span>::<span class="ident">householder</span>;
<span class="doccomment">/// The bidiagonalization of a general matrix.</span>
<span class="attribute">#[<span class="ident">cfg_attr</span>(<span class="ident">feature</span> <span class="op">=</span> <span class="string">"serde-serialize"</span>, <span class="ident">derive</span>(<span class="ident">Serialize</span>, <span class="ident">Deserialize</span>))]</span>
<span class="attribute">#[<span class="ident">cfg_attr</span>(
<span class="ident">feature</span> <span class="op">=</span> <span class="string">"serde-serialize"</span>,
<span class="ident">serde</span>(<span class="ident">bound</span>(
<span class="ident">serialize</span> <span class="op">=</span> <span class="string">"DimMinimum<R, C>: DimSub<U1>,
DefaultAllocator: Allocator<N, R, C> +
Allocator<N, DimMinimum<R, C>> +
Allocator<N, DimDiff<DimMinimum<R, C>, U1>>,
MatrixMN<N, R, C>: Serialize,
VectorN<N, DimMinimum<R, C>>: Serialize,
VectorN<N, DimDiff<DimMinimum<R, C>, U1>>: Serialize"</span>
))
)]</span>
<span class="attribute">#[<span class="ident">cfg_attr</span>(
<span class="ident">feature</span> <span class="op">=</span> <span class="string">"serde-serialize"</span>,
<span class="ident">serde</span>(<span class="ident">bound</span>(
<span class="ident">deserialize</span> <span class="op">=</span> <span class="string">"DimMinimum<R, C>: DimSub<U1>,
DefaultAllocator: Allocator<N, R, C> +
Allocator<N, DimMinimum<R, C>> +
Allocator<N, DimDiff<DimMinimum<R, C>, U1>>,
MatrixMN<N, R, C>: Deserialize<'de>,
VectorN<N, DimMinimum<R, C>>: Deserialize<'de>,
VectorN<N, DimDiff<DimMinimum<R, C>, U1>>: Deserialize<'de>"</span>
))
)]</span>
<span class="attribute">#[<span class="ident">derive</span>(<span class="ident">Clone</span>, <span class="ident">Debug</span>)]</span>
<span class="kw">pub</span> <span class="kw">struct</span> <span class="ident">Bidiagonal</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">ComplexField</span>, <span class="ident">R</span>: <span class="ident">DimMin</span><span class="op"><</span><span class="ident">C</span><span class="op">></span>, <span class="ident">C</span>: <span class="ident">Dim</span><span class="op">></span>
<span class="kw">where</span>
<span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>: <span class="ident">DimSub</span><span class="op"><</span><span class="ident">U1</span><span class="op">></span>,
<span class="ident">DefaultAllocator</span>: <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimDiff</span><span class="op"><</span><span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">U1</span><span class="op">></span><span class="op">></span>,
{
<span class="comment">// FIXME: perhaps we should pack the axises into different vectors so that axises for `v_t` are</span>
<span class="comment">// contiguous. This prevents some useless copies.</span>
<span class="ident">uv</span>: <span class="ident">MatrixMN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>,
<span class="doccomment">/// The diagonal elements of the decomposed matrix.</span>
<span class="ident">diagonal</span>: <span class="ident">VectorN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>,
<span class="doccomment">/// The off-diagonal elements of the decomposed matrix.</span>
<span class="ident">off_diagonal</span>: <span class="ident">VectorN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimDiff</span><span class="op"><</span><span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">U1</span><span class="op">></span><span class="op">></span>,
<span class="ident">upper_diagonal</span>: <span class="ident">bool</span>,
}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">ComplexField</span>, <span class="ident">R</span>: <span class="ident">DimMin</span><span class="op"><</span><span class="ident">C</span><span class="op">></span>, <span class="ident">C</span>: <span class="ident">Dim</span><span class="op">></span> <span class="ident">Copy</span> <span class="kw">for</span> <span class="ident">Bidiagonal</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>
<span class="kw">where</span>
<span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>: <span class="ident">DimSub</span><span class="op"><</span><span class="ident">U1</span><span class="op">></span>,
<span class="ident">DefaultAllocator</span>: <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimDiff</span><span class="op"><</span><span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">U1</span><span class="op">></span><span class="op">></span>,
<span class="ident">MatrixMN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>: <span class="ident">Copy</span>,
<span class="ident">VectorN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>: <span class="ident">Copy</span>,
<span class="ident">VectorN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimDiff</span><span class="op"><</span><span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">U1</span><span class="op">></span><span class="op">></span>: <span class="ident">Copy</span>,
{}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">ComplexField</span>, <span class="ident">R</span>: <span class="ident">DimMin</span><span class="op"><</span><span class="ident">C</span><span class="op">></span>, <span class="ident">C</span>: <span class="ident">Dim</span><span class="op">></span> <span class="ident">Bidiagonal</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>
<span class="kw">where</span>
<span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>: <span class="ident">DimSub</span><span class="op"><</span><span class="ident">U1</span><span class="op">></span>,
<span class="ident">DefaultAllocator</span>: <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">C</span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimDiff</span><span class="op"><</span><span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">U1</span><span class="op">></span><span class="op">></span>,
{
<span class="doccomment">/// Computes the Bidiagonal decomposition using householder reflections.</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">new</span>(<span class="kw-2">mut</span> <span class="ident">matrix</span>: <span class="ident">MatrixMN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw">let</span> (<span class="ident">nrows</span>, <span class="ident">ncols</span>) <span class="op">=</span> <span class="ident">matrix</span>.<span class="ident">data</span>.<span class="ident">shape</span>();
<span class="kw">let</span> <span class="ident">min_nrows_ncols</span> <span class="op">=</span> <span class="ident">nrows</span>.<span class="ident">min</span>(<span class="ident">ncols</span>);
<span class="kw">let</span> <span class="ident">dim</span> <span class="op">=</span> <span class="ident">min_nrows_ncols</span>.<span class="ident">value</span>();
<span class="macro">assert</span><span class="macro">!</span>(
<span class="ident">dim</span> <span class="op">!</span><span class="op">=</span> <span class="number">0</span>,
<span class="string">"Cannot compute the bidiagonalization of an empty matrix."</span>
);
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">diagonal</span> <span class="op">=</span> <span class="kw">unsafe</span> { <span class="ident">MatrixMN</span>::<span class="ident">new_uninitialized_generic</span>(<span class="ident">min_nrows_ncols</span>, <span class="ident">U1</span>) };
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">off_diagonal</span> <span class="op">=</span>
<span class="kw">unsafe</span> { <span class="ident">MatrixMN</span>::<span class="ident">new_uninitialized_generic</span>(<span class="ident">min_nrows_ncols</span>.<span class="ident">sub</span>(<span class="ident">U1</span>), <span class="ident">U1</span>) };
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">axis_packed</span> <span class="op">=</span> <span class="kw">unsafe</span> { <span class="ident">MatrixMN</span>::<span class="ident">new_uninitialized_generic</span>(<span class="ident">ncols</span>, <span class="ident">U1</span>) };
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">work</span> <span class="op">=</span> <span class="kw">unsafe</span> { <span class="ident">MatrixMN</span>::<span class="ident">new_uninitialized_generic</span>(<span class="ident">nrows</span>, <span class="ident">U1</span>) };
<span class="kw">let</span> <span class="ident">upper_diagonal</span> <span class="op">=</span> <span class="ident">nrows</span>.<span class="ident">value</span>() <span class="op">></span><span class="op">=</span> <span class="ident">ncols</span>.<span class="ident">value</span>();
<span class="kw">if</span> <span class="ident">upper_diagonal</span> {
<span class="kw">for</span> <span class="ident">ite</span> <span class="kw">in</span> <span class="number">0</span>..<span class="ident">dim</span> <span class="op">-</span> <span class="number">1</span> {
<span class="ident">householder</span>::<span class="ident">clear_column_unchecked</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">matrix</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">diagonal</span>[<span class="ident">ite</span>], <span class="ident">ite</span>, <span class="number">0</span>, <span class="prelude-val">None</span>);
<span class="ident">householder</span>::<span class="ident">clear_row_unchecked</span>(
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">matrix</span>,
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">off_diagonal</span>[<span class="ident">ite</span>],
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">axis_packed</span>,
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">work</span>,
<span class="ident">ite</span>,
<span class="number">1</span>,
);
}
<span class="ident">householder</span>::<span class="ident">clear_column_unchecked</span>(
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">matrix</span>,
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">diagonal</span>[<span class="ident">dim</span> <span class="op">-</span> <span class="number">1</span>],
<span class="ident">dim</span> <span class="op">-</span> <span class="number">1</span>,
<span class="number">0</span>,
<span class="prelude-val">None</span>,
);
} <span class="kw">else</span> {
<span class="kw">for</span> <span class="ident">ite</span> <span class="kw">in</span> <span class="number">0</span>..<span class="ident">dim</span> <span class="op">-</span> <span class="number">1</span> {
<span class="ident">householder</span>::<span class="ident">clear_row_unchecked</span>(
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">matrix</span>,
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">diagonal</span>[<span class="ident">ite</span>],
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">axis_packed</span>,
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">work</span>,
<span class="ident">ite</span>,
<span class="number">0</span>,
);
<span class="ident">householder</span>::<span class="ident">clear_column_unchecked</span>(
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">matrix</span>,
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">off_diagonal</span>[<span class="ident">ite</span>],
<span class="ident">ite</span>,
<span class="number">1</span>,
<span class="prelude-val">None</span>,
);
}
<span class="ident">householder</span>::<span class="ident">clear_row_unchecked</span>(
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">matrix</span>,
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">diagonal</span>[<span class="ident">dim</span> <span class="op">-</span> <span class="number">1</span>],
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">axis_packed</span>,
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">work</span>,
<span class="ident">dim</span> <span class="op">-</span> <span class="number">1</span>,
<span class="number">0</span>,
);
}
<span class="ident">Bidiagonal</span> {
<span class="ident">uv</span>: <span class="ident">matrix</span>,
<span class="ident">diagonal</span>,
<span class="ident">off_diagonal</span>,
<span class="ident">upper_diagonal</span>,
}
}
<span class="doccomment">/// Indicates whether this decomposition contains an upper-diagonal matrix.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">is_upper_diagonal</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">bool</span> {
<span class="self">self</span>.<span class="ident">upper_diagonal</span>
}
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">axis_shift</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> (<span class="ident">usize</span>, <span class="ident">usize</span>) {
<span class="kw">if</span> <span class="self">self</span>.<span class="ident">upper_diagonal</span> {
(<span class="number">0</span>, <span class="number">1</span>)
} <span class="kw">else</span> {
(<span class="number">1</span>, <span class="number">0</span>)
}
}
<span class="doccomment">/// Unpacks this decomposition into its three matrix factors `(U, D, V^t)`.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// The decomposed matrix `M` is equal to `U * D * V^t`.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">unpack</span>(
<span class="self">self</span>,
) <span class="op">-</span><span class="op">></span> (
<span class="ident">MatrixMN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>,
<span class="ident">MatrixN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>,
<span class="ident">MatrixMN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">C</span><span class="op">></span>,
)
<span class="kw">where</span>
<span class="ident">DefaultAllocator</span>: <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">C</span><span class="op">></span>,
{
<span class="comment">// FIXME: optimize by calling a reallocator.</span>
(<span class="self">self</span>.<span class="ident">u</span>(), <span class="self">self</span>.<span class="ident">d</span>(), <span class="self">self</span>.<span class="ident">v_t</span>())
}
<span class="doccomment">/// Retrieves the upper trapezoidal submatrix `R` of this decomposition.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">d</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">MatrixN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>
<span class="kw">where</span>
<span class="ident">DefaultAllocator</span>: <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>,
{
<span class="kw">let</span> (<span class="ident">nrows</span>, <span class="ident">ncols</span>) <span class="op">=</span> <span class="self">self</span>.<span class="ident">uv</span>.<span class="ident">data</span>.<span class="ident">shape</span>();
<span class="kw">let</span> <span class="ident">d</span> <span class="op">=</span> <span class="ident">nrows</span>.<span class="ident">min</span>(<span class="ident">ncols</span>);
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">res</span> <span class="op">=</span> <span class="ident">MatrixN</span>::<span class="ident">identity_generic</span>(<span class="ident">d</span>, <span class="ident">d</span>);
<span class="ident">res</span>.<span class="ident">set_partial_diagonal</span>(<span class="self">self</span>.<span class="ident">diagonal</span>.<span class="ident">iter</span>().<span class="ident">map</span>(<span class="op">|</span><span class="ident">e</span><span class="op">|</span> <span class="ident">N</span>::<span class="ident">from_real</span>(<span class="ident">e</span>.<span class="ident">modulus</span>())));
<span class="kw">let</span> <span class="ident">start</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">axis_shift</span>();
<span class="ident">res</span>.<span class="ident">slice_mut</span>(<span class="ident">start</span>, (<span class="ident">d</span>.<span class="ident">value</span>() <span class="op">-</span> <span class="number">1</span>, <span class="ident">d</span>.<span class="ident">value</span>() <span class="op">-</span> <span class="number">1</span>))
.<span class="ident">set_partial_diagonal</span>(<span class="self">self</span>.<span class="ident">off_diagonal</span>.<span class="ident">iter</span>().<span class="ident">map</span>(<span class="op">|</span><span class="ident">e</span><span class="op">|</span> <span class="ident">N</span>::<span class="ident">from_real</span>(<span class="ident">e</span>.<span class="ident">modulus</span>())));
<span class="ident">res</span>
}
<span class="doccomment">/// Computes the orthogonal matrix `U` of this `U * D * V` decomposition.</span>
<span class="comment">// FIXME: code duplication with householder::assemble_q.</span>
<span class="comment">// Except that we are returning a rectangular matrix here.</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">u</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">MatrixMN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>
<span class="kw">where</span> <span class="ident">DefaultAllocator</span>: <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span> {
<span class="kw">let</span> (<span class="ident">nrows</span>, <span class="ident">ncols</span>) <span class="op">=</span> <span class="self">self</span>.<span class="ident">uv</span>.<span class="ident">data</span>.<span class="ident">shape</span>();
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">res</span> <span class="op">=</span> <span class="ident">Matrix</span>::<span class="ident">identity_generic</span>(<span class="ident">nrows</span>, <span class="ident">nrows</span>.<span class="ident">min</span>(<span class="ident">ncols</span>));
<span class="kw">let</span> <span class="ident">dim</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">diagonal</span>.<span class="ident">len</span>();
<span class="kw">let</span> <span class="ident">shift</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">axis_shift</span>().<span class="number">0</span>;
<span class="kw">for</span> <span class="ident">i</span> <span class="kw">in</span> (<span class="number">0</span>..<span class="ident">dim</span> <span class="op">-</span> <span class="ident">shift</span>).<span class="ident">rev</span>() {
<span class="kw">let</span> <span class="ident">axis</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">uv</span>.<span class="ident">slice_range</span>(<span class="ident">i</span> <span class="op">+</span> <span class="ident">shift</span>.., <span class="ident">i</span>);
<span class="comment">// FIXME: sometimes, the axis might have a zero magnitude.</span>
<span class="kw">let</span> <span class="ident">refl</span> <span class="op">=</span> <span class="ident">Reflection</span>::<span class="ident">new</span>(<span class="ident">Unit</span>::<span class="ident">new_unchecked</span>(<span class="ident">axis</span>), <span class="ident">N</span>::<span class="ident">zero</span>());
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">res_rows</span> <span class="op">=</span> <span class="ident">res</span>.<span class="ident">slice_range_mut</span>(<span class="ident">i</span> <span class="op">+</span> <span class="ident">shift</span>.., <span class="ident">i</span>..);
<span class="kw">let</span> <span class="ident">sign</span> <span class="op">=</span> <span class="kw">if</span> <span class="self">self</span>.<span class="ident">upper_diagonal</span> {
<span class="self">self</span>.<span class="ident">diagonal</span>[<span class="ident">i</span>].<span class="ident">signum</span>()
} <span class="kw">else</span> {
<span class="self">self</span>.<span class="ident">off_diagonal</span>[<span class="ident">i</span>].<span class="ident">signum</span>()
};
<span class="ident">refl</span>.<span class="ident">reflect_with_sign</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">res_rows</span>, <span class="ident">sign</span>);
}
<span class="ident">res</span>
}
<span class="doccomment">/// Computes the orthogonal matrix `V_t` of this `U * D * V_t` decomposition.</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">v_t</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">MatrixMN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">C</span><span class="op">></span>
<span class="kw">where</span> <span class="ident">DefaultAllocator</span>: <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">C</span><span class="op">></span> {
<span class="kw">let</span> (<span class="ident">nrows</span>, <span class="ident">ncols</span>) <span class="op">=</span> <span class="self">self</span>.<span class="ident">uv</span>.<span class="ident">data</span>.<span class="ident">shape</span>();
<span class="kw">let</span> <span class="ident">min_nrows_ncols</span> <span class="op">=</span> <span class="ident">nrows</span>.<span class="ident">min</span>(<span class="ident">ncols</span>);
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">res</span> <span class="op">=</span> <span class="ident">Matrix</span>::<span class="ident">identity_generic</span>(<span class="ident">min_nrows_ncols</span>, <span class="ident">ncols</span>);
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">work</span> <span class="op">=</span> <span class="kw">unsafe</span> { <span class="ident">MatrixMN</span>::<span class="ident">new_uninitialized_generic</span>(<span class="ident">min_nrows_ncols</span>, <span class="ident">U1</span>) };
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">axis_packed</span> <span class="op">=</span> <span class="kw">unsafe</span> { <span class="ident">MatrixMN</span>::<span class="ident">new_uninitialized_generic</span>(<span class="ident">ncols</span>, <span class="ident">U1</span>) };
<span class="kw">let</span> <span class="ident">shift</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">axis_shift</span>().<span class="number">1</span>;
<span class="kw">for</span> <span class="ident">i</span> <span class="kw">in</span> (<span class="number">0</span>..<span class="ident">min_nrows_ncols</span>.<span class="ident">value</span>() <span class="op">-</span> <span class="ident">shift</span>).<span class="ident">rev</span>() {
<span class="kw">let</span> <span class="ident">axis</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">uv</span>.<span class="ident">slice_range</span>(<span class="ident">i</span>, <span class="ident">i</span> <span class="op">+</span> <span class="ident">shift</span>..);
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">axis_packed</span> <span class="op">=</span> <span class="ident">axis_packed</span>.<span class="ident">rows_range_mut</span>(<span class="ident">i</span> <span class="op">+</span> <span class="ident">shift</span>..);
<span class="ident">axis_packed</span>.<span class="ident">tr_copy_from</span>(<span class="kw-2">&</span><span class="ident">axis</span>);
<span class="comment">// FIXME: sometimes, the axis might have a zero magnitude.</span>
<span class="kw">let</span> <span class="ident">refl</span> <span class="op">=</span> <span class="ident">Reflection</span>::<span class="ident">new</span>(<span class="ident">Unit</span>::<span class="ident">new_unchecked</span>(<span class="ident">axis_packed</span>), <span class="ident">N</span>::<span class="ident">zero</span>());
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">res_rows</span> <span class="op">=</span> <span class="ident">res</span>.<span class="ident">slice_range_mut</span>(<span class="ident">i</span>.., <span class="ident">i</span> <span class="op">+</span> <span class="ident">shift</span>..);
<span class="kw">let</span> <span class="ident">sign</span> <span class="op">=</span> <span class="kw">if</span> <span class="self">self</span>.<span class="ident">upper_diagonal</span> {
<span class="self">self</span>.<span class="ident">off_diagonal</span>[<span class="ident">i</span>].<span class="ident">signum</span>()
} <span class="kw">else</span> {
<span class="self">self</span>.<span class="ident">diagonal</span>[<span class="ident">i</span>].<span class="ident">signum</span>()
};
<span class="ident">refl</span>.<span class="ident">reflect_rows_with_sign</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">res_rows</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">work</span>.<span class="ident">rows_range_mut</span>(<span class="ident">i</span>..), <span class="ident">sign</span>);
}
<span class="ident">res</span>
}
<span class="doccomment">/// The diagonal part of this decomposed matrix.</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">diagonal</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">VectorN</span><span class="op"><</span><span class="ident">N</span>::<span class="ident">RealField</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>
<span class="kw">where</span> <span class="ident">DefaultAllocator</span>: <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>::<span class="ident">RealField</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span> {
<span class="self">self</span>.<span class="ident">diagonal</span>.<span class="ident">map</span>(<span class="op">|</span><span class="ident">e</span><span class="op">|</span> <span class="ident">e</span>.<span class="ident">modulus</span>())
}
<span class="doccomment">/// The off-diagonal part of this decomposed matrix.</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">off_diagonal</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">VectorN</span><span class="op"><</span><span class="ident">N</span>::<span class="ident">RealField</span>, <span class="ident">DimDiff</span><span class="op"><</span><span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">U1</span><span class="op">></span><span class="op">></span>
<span class="kw">where</span> <span class="ident">DefaultAllocator</span>: <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>::<span class="ident">RealField</span>, <span class="ident">DimDiff</span><span class="op"><</span><span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">U1</span><span class="op">></span><span class="op">></span> {
<span class="self">self</span>.<span class="ident">off_diagonal</span>.<span class="ident">map</span>(<span class="op">|</span><span class="ident">e</span><span class="op">|</span> <span class="ident">e</span>.<span class="ident">modulus</span>())
}
<span class="attribute">#[<span class="ident">doc</span>(<span class="ident">hidden</span>)]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">uv_internal</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="kw-2">&</span><span class="ident">MatrixMN</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">C</span><span class="op">></span> {
<span class="kw-2">&</span><span class="self">self</span>.<span class="ident">uv</span>
}
}
<span class="comment">// impl<N: ComplexField, D: DimMin<D, Output = D> + DimSub<Dynamic>> Bidiagonal<N, D, D></span>
<span class="comment">// where DefaultAllocator: Allocator<N, D, D> +</span>
<span class="comment">// Allocator<N, D> {</span>
<span class="comment">// /// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.</span>
<span class="comment">// pub fn solve<R2: Dim, C2: Dim, S2>(&self, b: &Matrix<N, R2, C2, S2>) -> MatrixMN<N, R2, C2></span>
<span class="comment">// where S2: StorageMut<N, R2, C2>,</span>
<span class="comment">// ShapeConstraint: SameNumberOfRows<R2, D>,</span>
<span class="comment">// DefaultAllocator: Allocator<N, R2, C2> {</span>
<span class="comment">// let mut res = b.clone_owned();</span>
<span class="comment">// self.solve_mut(&mut res);</span>
<span class="comment">// res</span>
<span class="comment">// }</span>
<span class="comment">//</span>
<span class="comment">// /// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.</span>
<span class="comment">// pub fn solve_mut<R2: Dim, C2: Dim, S2>(&self, b: &mut Matrix<N, R2, C2, S2>)</span>
<span class="comment">// where S2: StorageMut<N, R2, C2>,</span>
<span class="comment">// ShapeConstraint: SameNumberOfRows<R2, D> {</span>
<span class="comment">//</span>
<span class="comment">// assert_eq!(self.uv.nrows(), b.nrows(), "Bidiagonal solve matrix dimension mismatch.");</span>
<span class="comment">// assert!(self.uv.is_square(), "Bidiagonal solve: unable to solve a non-square system.");</span>
<span class="comment">//</span>
<span class="comment">// self.q_tr_mul(b);</span>
<span class="comment">// self.solve_upper_triangular_mut(b);</span>
<span class="comment">// }</span>
<span class="comment">//</span>
<span class="comment">// // FIXME: duplicate code from the `solve` module.</span>
<span class="comment">// fn solve_upper_triangular_mut<R2: Dim, C2: Dim, S2>(&self, b: &mut Matrix<N, R2, C2, S2>)</span>
<span class="comment">// where S2: StorageMut<N, R2, C2>,</span>
<span class="comment">// ShapeConstraint: SameNumberOfRows<R2, D> {</span>
<span class="comment">//</span>
<span class="comment">// let dim = self.uv.nrows();</span>
<span class="comment">//</span>
<span class="comment">// for k in 0 .. b.ncols() {</span>
<span class="comment">// let mut b = b.column_mut(k);</span>
<span class="comment">// for i in (0 .. dim).rev() {</span>
<span class="comment">// let coeff;</span>
<span class="comment">//</span>
<span class="comment">// unsafe {</span>
<span class="comment">// let diag = *self.diag.vget_unchecked(i);</span>
<span class="comment">// coeff = *b.vget_unchecked(i) / diag;</span>
<span class="comment">// *b.vget_unchecked_mut(i) = coeff;</span>
<span class="comment">// }</span>
<span class="comment">//</span>
<span class="comment">// b.rows_range_mut(.. i).axpy(-coeff, &self.uv.slice_range(.. i, i), N::one());</span>
<span class="comment">// }</span>
<span class="comment">// }</span>
<span class="comment">// }</span>
<span class="comment">//</span>
<span class="comment">// /// Computes the inverse of the decomposed matrix.</span>
<span class="comment">// pub fn inverse(&self) -> MatrixN<N, D> {</span>
<span class="comment">// assert!(self.uv.is_square(), "Bidiagonal inverse: unable to compute the inverse of a non-square matrix.");</span>
<span class="comment">//</span>
<span class="comment">// // FIXME: is there a less naive method ?</span>
<span class="comment">// let (nrows, ncols) = self.uv.data.shape();</span>
<span class="comment">// let mut res = MatrixN::identity_generic(nrows, ncols);</span>
<span class="comment">// self.solve_mut(&mut res);</span>
<span class="comment">// res</span>
<span class="comment">// }</span>
<span class="comment">//</span>
<span class="comment">// // /// Computes the determinant of the decomposed matrix.</span>
<span class="comment">// // pub fn determinant(&self) -> N {</span>
<span class="comment">// // let dim = self.uv.nrows();</span>
<span class="comment">// // assert!(self.uv.is_square(), "Bidiagonal determinant: unable to compute the determinant of a non-square matrix.");</span>
<span class="comment">//</span>
<span class="comment">// // let mut res = N::one();</span>
<span class="comment">// // for i in 0 .. dim {</span>
<span class="comment">// // res *= unsafe { *self.diag.vget_unchecked(i) };</span>
<span class="comment">// // }</span>
<span class="comment">//</span>
<span class="comment">// // res self.q_determinant()</span>
<span class="comment">// // }</span>
<span class="comment">// }</span>
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">ComplexField</span>, <span class="ident">R</span>: <span class="ident">DimMin</span><span class="op"><</span><span class="ident">C</span><span class="op">></span>, <span class="ident">C</span>: <span class="ident">Dim</span>, <span class="ident">S</span>: <span class="ident">Storage</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span> <span class="ident">Matrix</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">C</span>, <span class="ident">S</span><span class="op">></span>
<span class="kw">where</span>
<span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>: <span class="ident">DimSub</span><span class="op"><</span><span class="ident">U1</span><span class="op">></span>,
<span class="ident">DefaultAllocator</span>: <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">C</span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span><span class="op">></span>
<span class="op">+</span> <span class="ident">Allocator</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">DimDiff</span><span class="op"><</span><span class="ident">DimMinimum</span><span class="op"><</span><span class="ident">R</span>, <span class="ident">C</span><span class="op">></span>, <span class="ident">U1</span><span class="op">></span><span class="op">></span>,
{
<span class="doccomment">/// Computes the bidiagonalization using householder reflections.</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">bidiagonalize</span>(<span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">Bidiagonal</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">R</span>, <span class="ident">C</span><span class="op">></span> {
<span class="ident">Bidiagonal</span>::<span class="ident">new</span>(<span class="self">self</span>.<span class="ident">into_owned</span>())
}
}
</pre></div>
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