<!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 `/Users/erlendbasso/.cargo/registry/src/github.com-1ecc6299db9ec823/nalgebra-0.32.1/src/linalg/col_piv_qr.rs`."><meta name="keywords" content="rust, rustlang, rust-lang"><title>col_piv_qr.rs - source</title><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../static.files/SourceSerif4-Regular-1f7d512b176f0f72.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../static.files/FiraSans-Regular-018c141bf0843ffd.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../static.files/FiraSans-Medium-8f9a781e4970d388.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../static.files/SourceCodePro-Regular-562dcc5011b6de7d.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../static.files/SourceSerif4-Bold-124a1ca42af929b6.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../static.files/SourceCodePro-Semibold-d899c5a5c4aeb14a.ttf.woff2"><link rel="stylesheet" href="../../../static.files/normalize-76eba96aa4d2e634.css"><link rel="stylesheet" href="../../../static.files/rustdoc-6827029ac823cab7.css" id="mainThemeStyle"><link rel="stylesheet" id="themeStyle" href="../../../static.files/light-ebce58d0a40c3431.css"><link rel="stylesheet" disabled href="../../../static.files/dark-f23faae4a2daf9a6.css"><link rel="stylesheet" disabled href="../../../static.files/ayu-8af5e100b21cd173.css"><script id="default-settings" ></script><script src="../../../static.files/storage-d43fa987303ecbbb.js"></script><script defer src="../../../static.files/source-script-5cf2e01a42cc9858.js"></script><script defer src="../../../source-files.js"></script><script defer src="../../../static.files/main-c55e1eb52e1886b4.js"></script><noscript><link rel="stylesheet" href="../../../static.files/noscript-13285aec31fa243e.css"></noscript><link rel="icon" href="https://nalgebra.org/img/favicon.ico"></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"></nav><main><div class="width-limiter"><nav class="sub"><a class="sub-logo-container" href="../../../nalgebra/index.html"><img class="rust-logo" src="../../../static.files/rust-logo-151179464ae7ed46.svg" alt="logo"></a><form class="search-form"><span></span><input class="search-input" name="search" aria-label="Run search in the documentation" 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="../../../static.files/wheel-5ec35bf9ca753509.svg"></a></div></form></nav><section id="main-content" class="content"><div class="example-wrap"><pre class="src-line-numbers"><a href="#1" id="1">1</a>
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</pre><pre class="rust"><code><span class="kw">use </span>num::Zero;
<span class="attr">#[cfg(feature = <span class="string">"serde-serialize-no-std"</span>)]
</span><span class="kw">use </span>serde::{Deserialize, Serialize};
<span class="kw">use </span><span class="kw">crate</span>::allocator::{Allocator, Reallocator};
<span class="kw">use </span><span class="kw">crate</span>::base::{Const, DefaultAllocator, Matrix, OMatrix, OVector, Unit};
<span class="kw">use </span><span class="kw">crate</span>::constraint::{SameNumberOfRows, ShapeConstraint};
<span class="kw">use </span><span class="kw">crate</span>::dimension::{Dim, DimMin, DimMinimum};
<span class="kw">use </span><span class="kw">crate</span>::storage::StorageMut;
<span class="kw">use </span><span class="kw">crate</span>::ComplexField;
<span class="kw">use </span><span class="kw">crate</span>::geometry::Reflection;
<span class="kw">use </span><span class="kw">crate</span>::linalg::{householder, PermutationSequence};
<span class="kw">use </span>std::mem::MaybeUninit;
<span class="doccomment">/// The QR decomposition (with column pivoting) of a general matrix.
</span><span class="attr">#[cfg_attr(feature = <span class="string">"serde-serialize-no-std"</span>, derive(Serialize, Deserialize))]
#[cfg_attr(
feature = <span class="string">"serde-serialize-no-std"</span>,
serde(bound(serialize = <span class="string">"DefaultAllocator: Allocator<T, R, C> +
Allocator<T, DimMinimum<R, C>>,
OMatrix<T, R, C>: Serialize,
PermutationSequence<DimMinimum<R, C>>: Serialize,
OVector<T, DimMinimum<R, C>>: Serialize"</span>))
)]
#[cfg_attr(
feature = <span class="string">"serde-serialize-no-std"</span>,
serde(bound(deserialize = <span class="string">"DefaultAllocator: Allocator<T, R, C> +
Allocator<T, DimMinimum<R, C>>,
OMatrix<T, R, C>: Deserialize<'de>,
PermutationSequence<DimMinimum<R, C>>: Deserialize<'de>,
OVector<T, DimMinimum<R, C>>: Deserialize<'de>"</span>))
)]
#[derive(Clone, Debug)]
</span><span class="kw">pub struct </span>ColPivQR<T: ComplexField, R: DimMin<C>, C: Dim>
<span class="kw">where
</span>DefaultAllocator: Allocator<T, R, C>
+ Allocator<T, DimMinimum<R, C>>
+ Allocator<(usize, usize), DimMinimum<R, C>>,
{
col_piv_qr: OMatrix<T, R, C>,
p: PermutationSequence<DimMinimum<R, C>>,
diag: OVector<T, DimMinimum<R, C>>,
}
<span class="kw">impl</span><T: ComplexField, R: DimMin<C>, C: Dim> Copy <span class="kw">for </span>ColPivQR<T, R, C>
<span class="kw">where
</span>DefaultAllocator: Allocator<T, R, C>
+ Allocator<T, DimMinimum<R, C>>
+ Allocator<(usize, usize), DimMinimum<R, C>>,
OMatrix<T, R, C>: Copy,
PermutationSequence<DimMinimum<R, C>>: Copy,
OVector<T, DimMinimum<R, C>>: Copy,
{
}
<span class="kw">impl</span><T: ComplexField, R: DimMin<C>, C: Dim> ColPivQR<T, R, C>
<span class="kw">where
</span>DefaultAllocator: Allocator<T, R, C>
+ Allocator<T, R>
+ Allocator<T, DimMinimum<R, C>>
+ Allocator<(usize, usize), DimMinimum<R, C>>,
{
<span class="doccomment">/// Computes the `ColPivQR` decomposition using householder reflections.
</span><span class="kw">pub fn </span>new(<span class="kw-2">mut </span>matrix: OMatrix<T, R, C>) -> <span class="self">Self </span>{
<span class="kw">let </span>(nrows, ncols) = matrix.shape_generic();
<span class="kw">let </span>min_nrows_ncols = nrows.min(ncols);
<span class="kw">let </span><span class="kw-2">mut </span>p = PermutationSequence::identity_generic(min_nrows_ncols);
<span class="kw">if </span>min_nrows_ncols.value() == <span class="number">0 </span>{
<span class="kw">return </span>ColPivQR {
col_piv_qr: matrix,
p,
diag: Matrix::zeros_generic(min_nrows_ncols, Const::<<span class="number">1</span>>),
};
}
<span class="kw">let </span><span class="kw-2">mut </span>diag = Matrix::uninit(min_nrows_ncols, Const::<<span class="number">1</span>>);
<span class="kw">for </span>i <span class="kw">in </span><span class="number">0</span>..min_nrows_ncols.value() {
<span class="kw">let </span>piv = matrix.view_range(i.., i..).icamax_full();
<span class="kw">let </span>col_piv = piv.<span class="number">1 </span>+ i;
matrix.swap_columns(i, col_piv);
p.append_permutation(i, col_piv);
diag[i] =
MaybeUninit::new(householder::clear_column_unchecked(<span class="kw-2">&mut </span>matrix, i, <span class="number">0</span>, <span class="prelude-val">None</span>));
}
<span class="comment">// Safety: diag is now fully initialized.
</span><span class="kw">let </span>diag = <span class="kw">unsafe </span>{ diag.assume_init() };
ColPivQR {
col_piv_qr: matrix,
p,
diag,
}
}
<span class="doccomment">/// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>r(<span class="kw-2">&</span><span class="self">self</span>) -> OMatrix<T, DimMinimum<R, C>, C>
<span class="kw">where
</span>DefaultAllocator: Allocator<T, DimMinimum<R, C>, C>,
{
<span class="kw">let </span>(nrows, ncols) = <span class="self">self</span>.col_piv_qr.shape_generic();
<span class="kw">let </span><span class="kw-2">mut </span>res = <span class="self">self
</span>.col_piv_qr
.rows_generic(<span class="number">0</span>, nrows.min(ncols))
.upper_triangle();
res.set_partial_diagonal(<span class="self">self</span>.diag.iter().map(|e| T::from_real(e.clone().modulus())));
res
}
<span class="doccomment">/// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
///
/// This is usually faster than `r` but consumes `self`.
</span><span class="attr">#[inline]
</span><span class="kw">pub fn </span>unpack_r(<span class="self">self</span>) -> OMatrix<T, DimMinimum<R, C>, C>
<span class="kw">where
</span>DefaultAllocator: Reallocator<T, R, C, DimMinimum<R, C>, C>,
{
<span class="kw">let </span>(nrows, ncols) = <span class="self">self</span>.col_piv_qr.shape_generic();
<span class="kw">let </span><span class="kw-2">mut </span>res = <span class="self">self
</span>.col_piv_qr
.resize_generic(nrows.min(ncols), ncols, T::zero());
res.fill_lower_triangle(T::zero(), <span class="number">1</span>);
res.set_partial_diagonal(<span class="self">self</span>.diag.iter().map(|e| T::from_real(e.clone().modulus())));
res
}
<span class="doccomment">/// Computes the orthogonal matrix `Q` of this decomposition.
</span><span class="attr">#[must_use]
</span><span class="kw">pub fn </span>q(<span class="kw-2">&</span><span class="self">self</span>) -> OMatrix<T, R, DimMinimum<R, C>>
<span class="kw">where
</span>DefaultAllocator: Allocator<T, R, DimMinimum<R, C>>,
{
<span class="kw">let </span>(nrows, ncols) = <span class="self">self</span>.col_piv_qr.shape_generic();
<span class="comment">// NOTE: we could build the identity matrix and call q_mul on it.
// Instead we don't so that we take in account the matrix sparseness.
</span><span class="kw">let </span><span class="kw-2">mut </span>res = Matrix::identity_generic(nrows, nrows.min(ncols));
<span class="kw">let </span>dim = <span class="self">self</span>.diag.len();
<span class="kw">for </span>i <span class="kw">in </span>(<span class="number">0</span>..dim).rev() {
<span class="kw">let </span>axis = <span class="self">self</span>.col_piv_qr.view_range(i.., i);
<span class="comment">// TODO: sometimes, the axis might have a zero magnitude.
</span><span class="kw">let </span>refl = Reflection::new(Unit::new_unchecked(axis), T::zero());
<span class="kw">let </span><span class="kw-2">mut </span>res_rows = res.view_range_mut(i.., i..);
refl.reflect_with_sign(<span class="kw-2">&mut </span>res_rows, <span class="self">self</span>.diag[i].clone().signum());
}
res
}
<span class="doccomment">/// Retrieves the column permutation of this decomposition.
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>p(<span class="kw-2">&</span><span class="self">self</span>) -> <span class="kw-2">&</span>PermutationSequence<DimMinimum<R, C>> {
<span class="kw-2">&</span><span class="self">self</span>.p
}
<span class="doccomment">/// Unpacks this decomposition into its two matrix factors.
</span><span class="kw">pub fn </span>unpack(
<span class="self">self</span>,
) -> (
OMatrix<T, R, DimMinimum<R, C>>,
OMatrix<T, DimMinimum<R, C>, C>,
PermutationSequence<DimMinimum<R, C>>,
)
<span class="kw">where
</span>DimMinimum<R, C>: DimMin<C, Output = DimMinimum<R, C>>,
DefaultAllocator: Allocator<T, R, DimMinimum<R, C>>
+ Reallocator<T, R, C, DimMinimum<R, C>, C>
+ Allocator<(usize, usize), DimMinimum<R, C>>,
{
(<span class="self">self</span>.q(), <span class="self">self</span>.r(), <span class="self">self</span>.p)
}
<span class="attr">#[doc(hidden)]
</span><span class="kw">pub fn </span>col_piv_qr_internal(<span class="kw-2">&</span><span class="self">self</span>) -> <span class="kw-2">&</span>OMatrix<T, R, C> {
<span class="kw-2">&</span><span class="self">self</span>.col_piv_qr
}
<span class="doccomment">/// Multiplies the provided matrix by the transpose of the `Q` matrix of this decomposition.
</span><span class="kw">pub fn </span>q_tr_mul<R2: Dim, C2: Dim, S2>(<span class="kw-2">&</span><span class="self">self</span>, rhs: <span class="kw-2">&mut </span>Matrix<T, R2, C2, S2>)
<span class="kw">where
</span>S2: StorageMut<T, R2, C2>,
{
<span class="kw">let </span>dim = <span class="self">self</span>.diag.len();
<span class="kw">for </span>i <span class="kw">in </span><span class="number">0</span>..dim {
<span class="kw">let </span>axis = <span class="self">self</span>.col_piv_qr.view_range(i.., i);
<span class="kw">let </span>refl = Reflection::new(Unit::new_unchecked(axis), T::zero());
<span class="kw">let </span><span class="kw-2">mut </span>rhs_rows = rhs.rows_range_mut(i..);
refl.reflect_with_sign(<span class="kw-2">&mut </span>rhs_rows, <span class="self">self</span>.diag[i].clone().signum().conjugate());
}
}
}
<span class="kw">impl</span><T: ComplexField, D: DimMin<D, Output = D>> ColPivQR<T, D, D>
<span class="kw">where
</span>DefaultAllocator:
Allocator<T, D, D> + Allocator<T, D> + Allocator<(usize, usize), DimMinimum<D, D>>,
{
<span class="doccomment">/// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
///
/// Returns `None` if `self` is not invertible.
</span><span class="attr">#[must_use = <span class="string">"Did you mean to use solve_mut()?"</span>]
</span><span class="kw">pub fn </span>solve<R2: Dim, C2: Dim, S2>(
<span class="kw-2">&</span><span class="self">self</span>,
b: <span class="kw-2">&</span>Matrix<T, R2, C2, S2>,
) -> <span class="prelude-ty">Option</span><OMatrix<T, R2, C2>>
<span class="kw">where
</span>S2: StorageMut<T, R2, C2>,
ShapeConstraint: SameNumberOfRows<R2, D>,
DefaultAllocator: Allocator<T, R2, C2>,
{
<span class="kw">let </span><span class="kw-2">mut </span>res = b.clone_owned();
<span class="kw">if </span><span class="self">self</span>.solve_mut(<span class="kw-2">&mut </span>res) {
<span class="prelude-val">Some</span>(res)
} <span class="kw">else </span>{
<span class="prelude-val">None
</span>}
}
<span class="doccomment">/// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
///
/// If the decomposed matrix is not invertible, this returns `false` and its input `b` is
/// overwritten with garbage.
</span><span class="kw">pub fn </span>solve_mut<R2: Dim, C2: Dim, S2>(<span class="kw-2">&</span><span class="self">self</span>, b: <span class="kw-2">&mut </span>Matrix<T, R2, C2, S2>) -> bool
<span class="kw">where
</span>S2: StorageMut<T, R2, C2>,
ShapeConstraint: SameNumberOfRows<R2, D>,
{
<span class="macro">assert_eq!</span>(
<span class="self">self</span>.col_piv_qr.nrows(),
b.nrows(),
<span class="string">"ColPivQR solve matrix dimension mismatch."
</span>);
<span class="macro">assert!</span>(
<span class="self">self</span>.col_piv_qr.is_square(),
<span class="string">"ColPivQR solve: unable to solve a non-square system."
</span>);
<span class="self">self</span>.q_tr_mul(b);
<span class="kw">let </span>solved = <span class="self">self</span>.solve_upper_triangular_mut(b);
<span class="self">self</span>.p.inv_permute_rows(b);
solved
}
<span class="comment">// TODO: duplicate code from the `solve` module.
</span><span class="kw">fn </span>solve_upper_triangular_mut<R2: Dim, C2: Dim, S2>(
<span class="kw-2">&</span><span class="self">self</span>,
b: <span class="kw-2">&mut </span>Matrix<T, R2, C2, S2>,
) -> bool
<span class="kw">where
</span>S2: StorageMut<T, R2, C2>,
ShapeConstraint: SameNumberOfRows<R2, D>,
{
<span class="kw">let </span>dim = <span class="self">self</span>.col_piv_qr.nrows();
<span class="kw">for </span>k <span class="kw">in </span><span class="number">0</span>..b.ncols() {
<span class="kw">let </span><span class="kw-2">mut </span>b = b.column_mut(k);
<span class="kw">for </span>i <span class="kw">in </span>(<span class="number">0</span>..dim).rev() {
<span class="kw">let </span>coeff;
<span class="kw">unsafe </span>{
<span class="kw">let </span>diag = <span class="self">self</span>.diag.vget_unchecked(i).clone().modulus();
<span class="kw">if </span>diag.is_zero() {
<span class="kw">return </span><span class="bool-val">false</span>;
}
coeff = b.vget_unchecked(i).clone().unscale(diag);
<span class="kw-2">*</span>b.vget_unchecked_mut(i) = coeff.clone();
}
b.rows_range_mut(..i)
.axpy(-coeff, <span class="kw-2">&</span><span class="self">self</span>.col_piv_qr.view_range(..i, i), T::one());
}
}
<span class="bool-val">true
</span>}
<span class="doccomment">/// Computes the inverse of the decomposed matrix.
///
/// Returns `None` if the decomposed matrix is not invertible.
</span><span class="attr">#[must_use]
</span><span class="kw">pub fn </span>try_inverse(<span class="kw-2">&</span><span class="self">self</span>) -> <span class="prelude-ty">Option</span><OMatrix<T, D, D>> {
<span class="macro">assert!</span>(
<span class="self">self</span>.col_piv_qr.is_square(),
<span class="string">"ColPivQR inverse: unable to compute the inverse of a non-square matrix."
</span>);
<span class="comment">// TODO: is there a less naive method ?
</span><span class="kw">let </span>(nrows, ncols) = <span class="self">self</span>.col_piv_qr.shape_generic();
<span class="kw">let </span><span class="kw-2">mut </span>res = OMatrix::identity_generic(nrows, ncols);
<span class="kw">if </span><span class="self">self</span>.solve_mut(<span class="kw-2">&mut </span>res) {
<span class="prelude-val">Some</span>(res)
} <span class="kw">else </span>{
<span class="prelude-val">None
</span>}
}
<span class="doccomment">/// Indicates if the decomposed matrix is invertible.
</span><span class="attr">#[must_use]
</span><span class="kw">pub fn </span>is_invertible(<span class="kw-2">&</span><span class="self">self</span>) -> bool {
<span class="macro">assert!</span>(
<span class="self">self</span>.col_piv_qr.is_square(),
<span class="string">"ColPivQR: unable to test the invertibility of a non-square matrix."
</span>);
<span class="kw">for </span>i <span class="kw">in </span><span class="number">0</span>..<span class="self">self</span>.diag.len() {
<span class="kw">if </span><span class="self">self</span>.diag[i].is_zero() {
<span class="kw">return </span><span class="bool-val">false</span>;
}
}
<span class="bool-val">true
</span>}
<span class="doccomment">/// Computes the determinant of the decomposed matrix.
</span><span class="attr">#[must_use]
</span><span class="kw">pub fn </span>determinant(<span class="kw-2">&</span><span class="self">self</span>) -> T {
<span class="kw">let </span>dim = <span class="self">self</span>.col_piv_qr.nrows();
<span class="macro">assert!</span>(
<span class="self">self</span>.col_piv_qr.is_square(),
<span class="string">"ColPivQR determinant: unable to compute the determinant of a non-square matrix."
</span>);
<span class="kw">let </span><span class="kw-2">mut </span>res = T::one();
<span class="kw">for </span>i <span class="kw">in </span><span class="number">0</span>..dim {
res <span class="kw-2">*</span>= <span class="kw">unsafe </span>{ <span class="self">self</span>.diag.vget_unchecked(i).clone() };
}
res * <span class="self">self</span>.p.determinant()
}
}
</code></pre></div>
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