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<!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/householder.rs`."><meta name="keywords" content="rust, rustlang, rust-lang"><title>householder.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="kw">use </span>matrix::{Matrix, BaseMatrix, BaseMatrixMut, Column, ColumnMut};
<span class="kw">use </span>vector::Vector;
<span class="kw">use </span>utils;
<span class="kw">use </span>libnum::Float;
<span class="doccomment">/// An efficient representation of a Householder reflection,
/// also known as Householder matrix or elementary reflector.
///
/// Mathematically, it has the form
/// H := I - τ v vᵀ,
/// with τ = 2 / (vᵀv).
///
/// Given a vector `x`, it is possible to choose `v` such that
/// Hx = a e1,
/// where a is a constant and e1 is the standard unit vector
/// whose elements are zero except the first, which is 1.
///
/// The implementation here is largely based upon the contents
/// of Chapter 5.1 (Householder and Givens Transformations)
/// in Matrix Computations, 4th Ed, Golub and Van Loan,
/// but with modifications that among other things makes
/// the implementation compliant with LAPACK.
</span><span class="kw">pub struct </span>HouseholderReflection&lt;T&gt; {
v: Vector&lt;T&gt;,
tau: T
}
<span class="kw">impl</span>&lt;T: Float&gt; HouseholderReflection&lt;T&gt; {
<span class="doccomment">/// Compute the Householder reflection which will zero out
/// all elements in the vector `x` except the first.
</span><span class="kw">pub fn </span>compute(x: Vector&lt;T&gt;) -&gt; HouseholderReflection&lt;T&gt; {
<span class="comment">// The following code is loosely based on notes in
// Applied Numerical Linear Algebra by Demmel,
// Matrix Computations 4th Ed by Golub &amp; Van Loan,
// as well as LAPACK documentation.
//
// From Demmel, we have that we can choose the vector
// v = [ x1 + sign(x1) norm(x) ]
// [ x[2:] ]
// as our Householder vector (the choice of sign in v(1) avoids
// cancellation issues which would lead to reduced accuracy in
// certain corner cases). However, we must divide v by
// v1 so that the first element of v is 1. Propagating these
// changes into τ leads to the below code.
// Note that if x[2:] == 0 (norm is identically zero),
// we explicitly set τ = 0 since x is already a multiple of
// the unit vector e1 (and we avoid potential division by zero).
</span><span class="kw">let </span>m = x.size();
<span class="kw">if </span>m &gt; <span class="number">0 </span>{
<span class="kw">let </span>sigma = utils::dot(<span class="kw-2">&amp;</span>x.data()[<span class="number">1 </span>..], <span class="kw-2">&amp;</span>x.data()[<span class="number">1 </span>..]);
<span class="kw">let </span>x0 = x[<span class="number">0</span>];
<span class="kw">let </span>tau;
<span class="kw">let </span><span class="kw-2">mut </span>v = x;
<span class="kw">if </span>sigma == T::zero() {
<span class="comment">// The vector is already a multiple of e1, the unit vector for which
// 1 is the first element and all other elements are zero.
</span>tau = T::zero();
} <span class="kw">else </span>{
<span class="kw">let </span>x_norm = T::sqrt(x0 * x0 + sigma);
<span class="comment">// This choice avoids accuracy issues related
// to cancellation
// (see e.g. Demmel, Applied Numerical Linear Algebra).
</span><span class="kw">let </span>v0 = <span class="kw">if </span>x0 &gt; T::zero() { x0 + x_norm }
<span class="kw">else </span>{ x0 - x_norm };
<span class="comment">// Normalize the Householder vector v so that
// its first element is 1.
</span><span class="kw">let </span>two = T::from(<span class="number">2</span>).unwrap();
tau = two * v0 * v0 / (v0 * v0 + sigma);
v[<span class="number">0</span>] = v0;
v = v / v0;
}
HouseholderReflection {
v: v,
tau: tau
}
} <span class="kw">else </span>{
<span class="comment">// x is an empty vector, so just use it as the
// Householder vector
</span>HouseholderReflection {
v: x,
tau: T::zero()
}
}
}
<span class="doccomment">/// Left-multiplies the given matrix by this Householder reflection.
///
/// More precisely, let `H` denote this Householder reflection matrix,
/// and let `A` be a dimensionally compatible matrix. Then
/// this function computes the product `HA` and stores the result
/// back in `A`.
///
/// The user must provide a buffer of size `A.cols()` which is used
/// to store intermediate results.
</span><span class="kw">pub fn </span>buffered_left_multiply_into&lt;M&gt;(<span class="kw-2">&amp;</span><span class="self">self</span>, matrix: <span class="kw-2">&amp;mut </span>M, buffer: <span class="kw-2">&amp;mut </span>[T])
<span class="kw">where </span>M: BaseMatrixMut&lt;T&gt;
{
<span class="kw">use </span>internal_utils::{transpose_gemv, ger};
<span class="macro">assert!</span>(buffer.len() == matrix.cols());
<span class="comment">// Recall that the Householder reflection is represented by
// H = I - τ v vᵀ,
//
// which means that the product HA can be computed as
//
// HA = A - (τ v) (vᵀ A) = A - (τ v) (Aᵀ v)ᵀ,
//
// which constitutes a (transposed) matrix-vector product`
// u = Aᵀ v and a rank-1 update A &lt;- A - τ v uᵀ
//
// Performing both the matrix-vector product and the
// rank-1 update can actually be performed without
// allocating any additional memory, but this would access
// the data in the matrix column-by-column, which is inefficient.
// Instead, we will use the provided buffer to hold the result of the
// matrix-vector product.
</span><span class="kw">let </span><span class="kw-2">ref </span>v = <span class="self">self</span>.v.data();
<span class="kw">let </span><span class="kw-2">mut </span>u = buffer;
<span class="comment">// u = A^T v
</span>transpose_gemv(matrix, v, u);
<span class="comment">// A &lt;- A - τ v uᵀ
</span>ger(matrix, - <span class="self">self</span>.tau, v, u);
}
<span class="kw">pub fn </span>as_vector(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; <span class="kw-2">&amp;</span>Vector&lt;T&gt; {
<span class="kw-2">&amp;</span><span class="self">self</span>.v
}
<span class="kw">pub fn </span>into_vector(<span class="self">self</span>) -&gt; Vector&lt;T&gt; {
<span class="self">self</span>.v
}
<span class="kw">pub fn </span>from_parameters(v: Vector&lt;T&gt;, tau: T) -&gt; HouseholderReflection&lt;T&gt; {
HouseholderReflection {
v: v,
tau: tau
}
}
<span class="kw">pub fn </span>tau(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; T {
<span class="self">self</span>.tau
}
<span class="kw">pub fn </span>store_in_col(<span class="kw-2">&amp;</span><span class="self">self</span>, col: <span class="kw-2">&amp;mut </span>ColumnMut&lt;T&gt;) {
<span class="kw">let </span>m = col.rows();
<span class="macro">assert!</span>(m == <span class="self">self</span>.v.size());
<span class="kw">if </span>m &gt; <span class="number">0 </span>{
<span class="comment">// The first element is implicitly 1, so make sure we don&#39;t
// touch it
</span><span class="kw">let </span><span class="kw-2">mut </span>slice_after_first = col.sub_slice_mut([<span class="number">1</span>, <span class="number">0</span>], m - <span class="number">1</span>, <span class="number">1</span>);
<span class="kw">let </span><span class="kw-2">mut </span>col_after_first = slice_after_first.col_mut(<span class="number">0</span>);
col_after_first.clone_from_slice(<span class="kw-2">&amp;</span><span class="self">self</span>.as_vector().data()[<span class="number">1</span>..]);
}
}
}
<span class="doccomment">/// An efficient representation for a composition of
/// Householder transformations.
///
/// This means that `HouseholderComposition` represents
/// an operator `Q` of the form
///
/// ```text
/// Q = Q_1 * Q_2 * ... * Q_p
/// ```
///
/// as explained in the documentation for
/// [HouseholderQr](struct.HouseholderQr.html).
</span><span class="attribute">#[derive(Debug, Clone)]
</span><span class="kw">pub struct </span>HouseholderComposition&lt;<span class="lifetime">&#39;a</span>, T&gt; <span class="kw">where </span>T: <span class="lifetime">&#39;a </span>{
storage: <span class="kw-2">&amp;</span><span class="lifetime">&#39;a </span>Matrix&lt;T&gt;,
tau: <span class="kw-2">&amp;</span><span class="lifetime">&#39;a </span>[T]
}
<span class="doccomment">/// Instantiates a HouseholderComposition with the given
/// storage and vector of tau values.
///
/// Note: This function is deliberately not exported to
/// the public API. This means that users cannot create
/// a HouseholderComposition by themselves, which is desirable
/// because we want to have the freedom to change details
/// of the internal representation if necessary.
</span><span class="kw">pub fn </span>create_composition&lt;<span class="lifetime">&#39;a</span>, T&gt;(storage: <span class="kw-2">&amp;</span><span class="lifetime">&#39;a </span>Matrix&lt;T&gt;, tau: <span class="kw-2">&amp;</span><span class="lifetime">&#39;a </span>[T])
-&gt; HouseholderComposition&lt;<span class="lifetime">&#39;a</span>, T&gt;
{
HouseholderComposition {
storage: storage,
tau: tau
}
}
<span class="kw">impl</span>&lt;<span class="lifetime">&#39;a</span>, T&gt; HouseholderComposition&lt;<span class="lifetime">&#39;a</span>, T&gt; <span class="kw">where </span>T: Float {
<span class="doccomment">/// Given a matrix `A` of compatible dimensions, computes
/// the product `A &lt;- QA`, storing the result in `A`.
</span><span class="kw">pub fn </span>left_multiply_into&lt;X&gt;(<span class="kw-2">&amp;</span><span class="self">self</span>, matrix: <span class="kw-2">&amp;mut </span>X)
<span class="kw">where </span>X: BaseMatrixMut&lt;T&gt;
{
<span class="kw">use </span>std::cmp::min;
<span class="kw">let </span>m = <span class="self">self</span>.storage.rows();
<span class="kw">let </span>n = <span class="self">self</span>.storage.cols();
<span class="kw">let </span>p = min(m, n);
<span class="kw">let </span>q = matrix.cols();
<span class="macro">assert!</span>(matrix.rows() == m, <span class="string">&quot;Matrix does not have compatible dimensions.&quot;</span>);
<span class="kw">let </span><span class="kw-2">mut </span>house_buffer = Vec::with_capacity(m);
<span class="kw">let </span><span class="kw-2">mut </span>multiply_buffer = <span class="macro">vec!</span>[T::zero(); q];
<span class="kw">for </span>j <span class="kw">in </span>(<span class="number">0 </span>.. p).rev() {
house_buffer.resize(m - j, T::zero());
<span class="kw">let </span>storage_block = <span class="self">self</span>.storage.sub_slice([j, j], m - j, n - j);
<span class="kw">let </span><span class="kw-2">mut </span>matrix_block = matrix.sub_slice_mut([j, <span class="number">0</span>], m - j, q);
<span class="kw">let </span>house = load_house_from_col(<span class="kw-2">&amp;</span>storage_block.col(<span class="number">0</span>),
<span class="self">self</span>.tau[j], house_buffer);
house.buffered_left_multiply_into(<span class="kw-2">&amp;mut </span>matrix_block,
<span class="kw-2">&amp;mut </span>multiply_buffer);
house_buffer = house.into_vector().into_vec();
}
}
<span class="doccomment">/// Computes the first k columns of the implicitly
/// stored matrix `Q`.
///
/// # Panics
/// - `k` must be less than or equal to `m`, the number
/// of rows of `Q`.
</span><span class="kw">pub fn </span>first_k_columns(<span class="kw-2">&amp;</span><span class="self">self</span>, k: usize) -&gt; Matrix&lt;T&gt; {
<span class="kw">use </span>std::cmp::min;
<span class="kw">let </span>m = <span class="self">self</span>.storage.rows();
<span class="kw">let </span>n = <span class="self">self</span>.storage.cols();
<span class="kw">let </span>p = min(m, n);
<span class="macro">assert!</span>(k &lt;= <span class="self">self</span>.storage.rows(),
<span class="string">&quot;k cannot exceed m, the number of rows of Q&quot;</span>);
<span class="comment">// Let Q_k = Q[:, 1:k], the first k rows of Q
</span><span class="kw">let </span><span class="kw-2">mut </span>q_k = Matrix::from_fn(m, k, |row, col| {
<span class="kw">if </span>row == col { T::one()}
<span class="kw">else </span>{ T::zero() }
});
<span class="comment">// This is almost identical to left_multiply_into,
// but we can use the sparsity of the identity matrix
// to reduce the number of operations
// (note the size of the &quot;q_k_block&quot;)
</span><span class="kw">let </span><span class="kw-2">mut </span>buffer = Vec::with_capacity(m);
<span class="kw">let </span><span class="kw-2">mut </span>multiply_buffer = Vec::with_capacity(k);
<span class="kw">for </span>j <span class="kw">in </span>(<span class="number">0 </span>.. min(p, k)).rev() {
buffer.resize(m - j, T::zero());
multiply_buffer.resize(k - j, T::zero());
<span class="kw">let </span>storage_block = <span class="self">self</span>.storage.sub_slice([j, j], m - j, n - j);
<span class="kw">let </span><span class="kw-2">mut </span>q_k_block = q_k.sub_slice_mut([j, j], m - j, k - j);
<span class="kw">let </span>house = load_house_from_col(<span class="kw-2">&amp;</span>storage_block.col(<span class="number">0</span>),
<span class="self">self</span>.tau[j], buffer);
house.buffered_left_multiply_into(<span class="kw-2">&amp;mut </span>q_k_block,
<span class="kw-2">&amp;mut </span>multiply_buffer);
buffer = house.into_vector().into_vec();
}
q_k
}
}
<span class="kw">fn </span>load_house_from_col&lt;T: Float&gt;(col: <span class="kw-2">&amp;</span>Column&lt;T&gt;, tau: T, buffer: Vec&lt;T&gt;)
-&gt; HouseholderReflection&lt;T&gt; {
<span class="kw">let </span><span class="kw-2">mut </span>v = buffer;
col.clone_into_slice(<span class="kw-2">&amp;mut </span>v);
<span class="comment">// First element is implicitly 1 regardless of
// whatever is stored in the column.
</span><span class="kw">if let </span><span class="prelude-val">Some</span>(first_element) = v.get_mut(<span class="number">0</span>) {
<span class="kw-2">*</span>first_element = T::one();
}
HouseholderReflection::from_parameters(Vector::new(v), tau)
}
<span class="attribute">#[cfg(test)]
</span><span class="kw">mod </span>tests {
<span class="kw">use </span>vector::Vector;
<span class="kw">use </span>matrix::{Matrix, BaseMatrix};
<span class="kw">use </span><span class="kw">super</span>::HouseholderReflection;
<span class="kw">use </span><span class="kw">super</span>::create_composition;
<span class="kw">pub fn </span>house_as_matrix(house: HouseholderReflection&lt;f64&gt;)
-&gt; Matrix&lt;f64&gt;
{
<span class="kw">let </span>m = house.v.size();
<span class="kw">let </span>v = Matrix::new(m, <span class="number">1</span>, house.v.into_vec());
<span class="kw">let </span>v_t = v.transpose();
Matrix::identity(m) - v * v_t * house.tau
}
<span class="kw">fn </span>verify_house(x: Vector&lt;f64&gt;, house: HouseholderReflection&lt;f64&gt;) {
<span class="kw">let </span>m = x.size();
<span class="macro">assert!</span>(m &gt; <span class="number">0</span>);
<span class="kw">let </span>house = house_as_matrix(house);
<span class="kw">let </span>y = house.clone() * x.clone();
<span class="comment">// Check that y[1 ..] is approximately zero
</span><span class="kw">let </span>z = Vector::new(y.data().iter().skip(<span class="number">1</span>).cloned().collect::&lt;Vec&lt;<span class="kw">_</span>&gt;&gt;());
<span class="macro">assert_vector_eq!</span>(z, Vector::zeros(m - <span class="number">1</span>), comp = float, eps = <span class="number">1e-12</span>);
<span class="comment">// Check that applying the Householder transformation again
// recovers the original vector (since H = H^T = inv(H))
</span><span class="kw">let </span>w = house * y;
<span class="macro">assert_vector_eq!</span>(x, w, comp = float);
}
<span class="attribute">#[test]
</span><span class="kw">fn </span>compute_empty_vector() {
<span class="kw">let </span>x: Vector&lt;f64&gt; = <span class="macro">vector!</span>[];
<span class="kw">let </span>house = HouseholderReflection::compute(x.clone());
<span class="macro">assert_scalar_eq!</span>(house.tau, <span class="number">0.0</span>);
<span class="macro">assert_vector_eq!</span>(house.v, x.clone());
}
<span class="attribute">#[test]
</span><span class="kw">fn </span>compute_single_element_vector() {
<span class="kw">let </span>x = <span class="macro">vector!</span>[<span class="number">2.0</span>];
<span class="kw">let </span>house = HouseholderReflection::compute(x.clone());
<span class="macro">assert_scalar_eq!</span>(house.tau, <span class="number">0.0</span>);
}
<span class="attribute">#[test]
</span><span class="kw">fn </span>compute_examples() {
{
<span class="kw">let </span>x = <span class="macro">vector!</span>[<span class="number">1.0</span>, <span class="number">0.0</span>, <span class="number">0.0</span>];
<span class="kw">let </span>house = HouseholderReflection::compute(x.clone());
verify_house(x, house);
}
{
<span class="kw">let </span>x = <span class="macro">vector!</span>[-<span class="number">1.0</span>, <span class="number">0.0</span>, <span class="number">0.0</span>];
<span class="kw">let </span>house = HouseholderReflection::compute(x.clone());
verify_house(x, house);
}
{
<span class="kw">let </span>x = <span class="macro">vector!</span>[<span class="number">3.0</span>, -<span class="number">2.0</span>, <span class="number">5.0</span>];
<span class="kw">let </span>house = HouseholderReflection::compute(x.clone());
verify_house(x, house);
}
}
<span class="attribute">#[test]
</span><span class="kw">fn </span>householder_reflection_left_multiply() {
<span class="kw">let </span><span class="kw-2">mut </span>x = <span class="macro">matrix!</span>[ <span class="number">0.0</span>, <span class="number">1.0</span>, <span class="number">2.0</span>, <span class="number">3.0</span>;
<span class="number">4.0</span>, <span class="number">5.0</span>, <span class="number">6.0</span>, <span class="number">7.0</span>;
<span class="number">8.0</span>, <span class="number">9.0</span>, <span class="number">10.0</span>, <span class="number">11.0</span>;
<span class="number">12.0</span>, <span class="number">13.0</span>, <span class="number">14.0</span>, <span class="number">15.0 </span>];
<span class="comment">// The provided data is rather rubbish, but
// the result should still hold
</span><span class="kw">let </span>h = HouseholderReflection {
tau: <span class="number">0.06666666666666667</span>,
v: <span class="macro">vector!</span>[<span class="number">1.0</span>, <span class="number">2.0</span>, <span class="number">3.0</span>, <span class="number">4.0</span>]
};
<span class="kw">let </span><span class="kw-2">mut </span>buffer = <span class="macro">vec!</span>[<span class="number">0.0</span>; <span class="number">4</span>];
h.buffered_left_multiply_into(<span class="kw-2">&amp;mut </span>x, <span class="kw-2">&amp;mut </span>buffer);
<span class="kw">let </span>expected = <span class="macro">matrix!</span>[ -<span class="number">5.3333</span>, -<span class="number">5.0000</span>, -<span class="number">4.6667</span>, -<span class="number">4.3333</span>;
-<span class="number">6.6667</span>, -<span class="number">7.0000</span>, -<span class="number">7.3333</span>, -<span class="number">7.6667</span>;
-<span class="number">8.0000</span>, -<span class="number">9.0000</span>,-<span class="number">10.0000</span>, -<span class="number">11.0000</span>;
-<span class="number">9.3333</span>, -<span class="number">11.0000</span>,-<span class="number">12.6667</span>, -<span class="number">14.3333</span>];
<span class="macro">assert_matrix_eq!</span>(x, expected, comp = abs, tol = <span class="number">1e-3</span>);
}
<span class="attribute">#[test]
</span><span class="kw">fn </span>householder_composition_left_multiply() {
<span class="kw">let </span>storage = <span class="macro">matrix!</span>[ <span class="number">5.0</span>, <span class="number">3.0</span>, <span class="number">2.0</span>;
<span class="number">2.0</span>, <span class="number">1.0</span>, <span class="number">3.0</span>;
-<span class="number">2.0</span>, <span class="number">3.0</span>, -<span class="number">2.0</span>];
<span class="kw">let </span>tau = <span class="macro">vec!</span>[<span class="number">2.0</span>/<span class="number">9.0</span>, <span class="number">1.0 </span>/ <span class="number">5.0</span>, <span class="number">2.0</span>];
<span class="comment">// `q` is a manually computed matrix representation
// of the Householder composition stored implicitly in
// `storage` and `tau. We leave it here to make writing
// further tests easier
// let q = matrix![7.0/9.0, -28.0/45.0, 4.0/45.0;
// -4.0/9.0, - 4.0/ 9.0, 7.0/ 9.0;
// 4.0/9.0, 29.0/45.0, 28.0/45.0];
</span><span class="kw">let </span>composition = create_composition(<span class="kw-2">&amp;</span>storage, <span class="kw-2">&amp;</span>tau);
{
<span class="comment">// Square
</span><span class="kw">let </span><span class="kw-2">mut </span>x = <span class="macro">matrix!</span>[<span class="number">4.0</span>, <span class="number">5.0</span>, -<span class="number">3.0</span>;
<span class="number">2.0</span>, -<span class="number">1.0</span>, -<span class="number">3.0</span>;
<span class="number">1.0</span>, <span class="number">3.0</span>, <span class="number">5.0</span>];
composition.left_multiply_into(<span class="kw-2">&amp;mut </span>x);
<span class="kw">let </span>expected = <span class="macro">matrix!</span>[ <span class="number">88.0</span>/<span class="number">45.0</span>, <span class="number">43.0</span>/<span class="number">9.0</span>, -<span class="number">1.0</span>/<span class="number">45.0</span>;
-<span class="number">17.0</span>/ <span class="number">9.0</span>, <span class="number">5.0</span>/<span class="number">9.0</span>, <span class="number">59.0</span>/ <span class="number">9.0</span>;
<span class="number">166.0</span>/<span class="number">45.0</span>, <span class="number">31.0</span>/<span class="number">9.0</span>, -<span class="number">7.0</span>/<span class="number">45.0</span>];
<span class="macro">assert_matrix_eq!</span>(x, expected, comp = float, eps = <span class="number">1e-15</span>);
}
{
<span class="comment">// Tall
</span><span class="kw">let </span><span class="kw-2">mut </span>x = <span class="macro">matrix!</span>[ <span class="number">4.0</span>, <span class="number">5.0</span>;
<span class="number">3.0</span>, <span class="number">2.0</span>;
-<span class="number">1.0</span>,-<span class="number">2.0</span>];
composition.left_multiply_into(<span class="kw-2">&amp;mut </span>x);
<span class="kw">let </span>expected = <span class="macro">matrix!</span>[<span class="number">52.0</span>/<span class="number">45.0</span>, <span class="number">37.0</span>/<span class="number">15.0</span>;
-<span class="number">35.0</span>/ <span class="number">9.0</span>, -<span class="number">14.0</span>/ <span class="number">3.0</span>;
<span class="number">139.0</span>/<span class="number">45.0</span>, <span class="number">34.0</span>/<span class="number">15.0</span>];
<span class="macro">assert_matrix_eq!</span>(x, expected, comp = float, eps = <span class="number">1e-15</span>);
}
{
<span class="comment">// Short
</span><span class="kw">let </span><span class="kw-2">mut </span>x = <span class="macro">matrix!</span>[ <span class="number">4.0</span>, <span class="number">5.0</span>, <span class="number">2.0</span>, -<span class="number">5.0</span>;
<span class="number">3.0</span>, <span class="number">2.0</span>, <span class="number">1.0</span>, <span class="number">1.0</span>;
-<span class="number">1.0</span>, -<span class="number">2.0</span>, <span class="number">0.0</span>, -<span class="number">5.0</span>];
composition.left_multiply_into(<span class="kw-2">&amp;mut </span>x);
<span class="kw">let </span>expected = <span class="macro">matrix!</span>[<span class="number">52.0</span>/<span class="number">45.0</span>, <span class="number">37.0</span>/<span class="number">15.0</span>, <span class="number">14.0</span>/<span class="number">15.0</span>, -<span class="number">223.0</span>/<span class="number">45.0</span>;
-<span class="number">35.0</span>/ <span class="number">9.0</span>, -<span class="number">14.0</span>/ <span class="number">3.0</span>, -<span class="number">4.0</span>/ <span class="number">3.0</span>, -<span class="number">19.0</span>/ <span class="number">9.0</span>;
<span class="number">139.0</span>/<span class="number">45.0</span>, <span class="number">34.0</span>/<span class="number">15.0</span>, <span class="number">23.0</span>/<span class="number">15.0</span>, -<span class="number">211.0</span>/<span class="number">45.0</span>];
<span class="macro">assert_matrix_eq!</span>(x, expected, comp = float, eps = <span class="number">1e-15</span>);
}
}
<span class="attribute">#[test]
</span><span class="kw">fn </span>householder_composition_first_k_columns() {
<span class="kw">let </span>storage = <span class="macro">matrix!</span>[ <span class="number">5.0</span>, <span class="number">3.0</span>, <span class="number">2.0</span>;
<span class="number">2.0</span>, <span class="number">1.0</span>, <span class="number">3.0</span>;
-<span class="number">2.0</span>, <span class="number">3.0</span>, -<span class="number">2.0</span>];
<span class="kw">let </span>tau = <span class="macro">vec!</span>[<span class="number">2.0</span>/<span class="number">9.0</span>, <span class="number">1.0 </span>/ <span class="number">5.0</span>, <span class="number">2.0</span>];
<span class="kw">let </span>composition = create_composition(<span class="kw-2">&amp;</span>storage, <span class="kw-2">&amp;</span>tau);
<span class="comment">// This corresponds to the following `Q` matrix
</span><span class="kw">let </span>q = <span class="macro">matrix!</span>[<span class="number">7.0</span>/<span class="number">9.0</span>, -<span class="number">28.0</span>/<span class="number">45.0</span>, <span class="number">4.0</span>/<span class="number">45.0</span>;
-<span class="number">4.0</span>/<span class="number">9.0</span>, - <span class="number">4.0</span>/ <span class="number">9.0</span>, <span class="number">7.0</span>/ <span class="number">9.0</span>;
<span class="number">4.0</span>/<span class="number">9.0</span>, <span class="number">29.0</span>/<span class="number">45.0</span>, <span class="number">28.0</span>/<span class="number">45.0</span>];
{
<span class="comment">// First 0 columns
</span><span class="kw">let </span>q_k = composition.first_k_columns(<span class="number">0</span>);
<span class="macro">assert_eq!</span>(q_k.rows(), <span class="number">3</span>);
<span class="macro">assert_eq!</span>(q_k.cols(), <span class="number">0</span>);
}
{
<span class="comment">// First column
</span><span class="kw">let </span>q_k = composition.first_k_columns(<span class="number">1</span>);
<span class="macro">assert_matrix_eq!</span>(q_k, q.sub_slice([<span class="number">0</span>, <span class="number">0</span>], <span class="number">3</span>, <span class="number">1</span>),
comp = float);
}
{
<span class="comment">// First 2 columns
</span><span class="kw">let </span>q_k = composition.first_k_columns(<span class="number">2</span>);
<span class="macro">assert_matrix_eq!</span>(q_k, q.sub_slice([<span class="number">0</span>, <span class="number">0</span>], <span class="number">3</span>, <span class="number">2</span>),
comp = float);
}
{
<span class="comment">// First 3 columns
</span><span class="kw">let </span>q_k = composition.first_k_columns(<span class="number">3</span>);
<span class="macro">assert_matrix_eq!</span>(q_k, q.sub_slice([<span class="number">0</span>, <span class="number">0</span>], <span class="number">3</span>, <span class="number">3</span>),
comp = float);
}
}
}
</code></pre></div>
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