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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/permutation_matrix.rs`."><meta name="keywords" content="rust, rustlang, rust-lang"><title>permutation_matrix.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>std;

 <span class="kw">use </span>matrix::{Matrix, BaseMatrix, BaseMatrixMut};
 <span class="kw">use </span>vector::Vector;
 <span class="kw">use </span>error::{Error, ErrorKind};

 <span class="kw">use </span>libnum::Num;

 <span class="doccomment">/// An efficient implementation of a permutation matrix.
 ///
 /// # Examples
 /// ```
 /// # #[macro_use] extern crate rulinalg; fn main() {
 /// use rulinalg::matrix::PermutationMatrix;
 ///
 /// let ref x = matrix![1, 2, 3;
 ///                     4, 5, 6;
 ///                     7, 8, 9];
 ///
 /// // Swap the two first rows of x by left-multiplying a permutation matrix
 /// let expected = matrix![4, 5, 6;
 ///                        1, 2, 3;
 ///                        7, 8, 9];
 /// let mut p = PermutationMatrix::identity(3);
 /// p.swap_rows(0, 1);
 /// assert_eq!(expected, p * x);
 ///
 /// // Swap the two last columns of x by right-multiplying a permutation matrix
 /// let expected = matrix![1, 3, 2;
 ///                        4, 6, 5;
 ///                        7, 9, 8];
 /// let mut p = PermutationMatrix::identity(3);
 /// p.swap_rows(1, 2);
 /// assert_eq!(expected, x * p);
 ///
 /// // One can also construct the same permutation matrix directly
 /// // from an array representation.
 /// let ref p = PermutationMatrix::from_array(vec![0, 2, 1]).unwrap();
 /// assert_eq!(expected, x * p);
 ///
 /// // One may also obtain a full matrix representation of the permutation
 /// assert_eq!(p.as_matrix(), matrix![1, 0, 0;
 ///                                   0, 0, 1;
 ///                                   0, 1, 0]);
 ///
 /// // The inverse of a permutation matrix can efficiently be obtained
 /// let p_inv = p.inverse();
 ///
 /// // And permutations can be composed through multiplication
 /// assert_eq!(p * p_inv, PermutationMatrix::identity(3));
 /// # }
 /// ```
 ///
 /// # Rationale and complexity
 ///
 /// A [permutation matrix](https://en.wikipedia.org/wiki/Permutation_matrix)
 /// is a very special kind of matrix. It is essentially a matrix representation
 /// of the more general concept of a permutation. That is, an `n` x `n` permutation
 /// matrix corresponds to a permutation of ordered sets whose cardinality is `n`.
 /// In particular, given an `m` x `n` matrix `A` and an `m` x `m` permutation
 /// matrix `P`, the action of left-multiplying `A` by `P`, `PA`, corresponds
 /// to permuting the rows of `A` by the given permutation represented by `P`.
 /// Conversely, right-multiplication corresponds to column permutation.
 /// More precisely, given another permutation matrix `Q` of size `n` x `n`,
 /// then `AQ` is the corresponding permutation of the columns of `A`.
 ///
 /// Due to their unique structure, permutation matrices can be much more
 /// efficiently represented and applied than general matrices. Recall that
 /// for general matrices `X` and `Y` of size `m` x `m` and `n` x `n` respectively,
 /// the storage of `X` requires O(`m`&lt;sup&gt;2&lt;/sup&gt;) memory and the storage of
 /// `Y` requires O(`n`&lt;sup&gt;2&lt;/sup&gt;) memory. Ignoring for the moment the existence
 /// of Strassen&#39;s matrix multiplication algorithm and more theoretical alternatives,
 /// the multiplication `XA` requires O(`m`&lt;sup&gt;2&lt;/sup&gt;`n`) operations, and
 /// the multiplication `AY` requires O(`m``n`&lt;sup&gt;2&lt;/sup&gt;) operations.
 ///
 /// By contrast, the storage of `P` requires only O(`m`) memory, and
 /// the storage of `K` requires O(`n`) memory. Moreover, the products
 /// `PA` and `AK` both require merely O(`mn`) operations.
 ///
 /// # Representation
 /// A permutation of an ordered set of cardinality *n* is a map of the form
 ///
 /// ```text
 /// p: { 1, ..., n } -&gt; { 1, ..., n }.
 /// ```
 ///
 /// That is, for any index `i`, the permutation `p` sends `i` to some
 /// index `j = p(i)`, and hence the map may be represented as an array of integers
 /// of length *n*.
 ///
 /// By convention, an instance of `PermutationMatrix` represents row permutations.
 /// That is, the indices referred to above correspond to *row indices*,
 /// and the internal representation of a `PermutationMatrix` is an array
 /// describing how the permutation sends a row index `i` to a new row index
 /// `j` in the permuted matrix. Because of this internal representation, one can only
 /// efficiently swap *rows* of a `PermutationMatrix`.
 /// However, keep in mind that while this API only lets one swap individual rows
 /// of the permutation matrix itself, the right-multiplication of a general
 /// matrix with a permutation matrix will permute the columns of the general matrix,
 /// and so in practice this restriction is insignificant.
 ///
 </span><span class="attribute">#[derive(Debug, PartialEq, Eq, Clone)]
 </span><span class="kw">pub struct </span>PermutationMatrix&lt;T&gt; {
     <span class="comment">// A permutation matrix of dimensions NxN is represented as a permutation of the rows
     // of an NxM matrix for any M.
     </span>perm: Vec&lt;usize&gt;,

     <span class="comment">// Currently, we need to let PermutationMatrix be generic over T,
     // because BaseMatrixMut is.
     </span>marker: std::marker::PhantomData&lt;T&gt;
 }

 <span class="doccomment">/// Parity is the fact of being even or odd.
 </span><span class="attribute">#[derive(Debug, Copy, Clone, PartialEq, Eq)]
 </span><span class="kw">pub enum </span>Parity {
     <span class="doccomment">/// Even parity.
     </span>Even,
     <span class="doccomment">/// Odd parity.
     </span>Odd
 }

 <span class="kw">impl</span>&lt;T&gt; PermutationMatrix&lt;T&gt; {
     <span class="doccomment">/// The identity permutation.
     </span><span class="kw">pub fn </span>identity(n: usize) -&gt; <span class="self">Self </span>{
         PermutationMatrix {
             perm: (<span class="number">0 </span>.. n).collect(),
             marker: std::marker::PhantomData
         }
     }

     <span class="doccomment">/// Swaps rows i and j in the permutation matrix.
     </span><span class="kw">pub fn </span>swap_rows(<span class="kw-2">&amp;mut </span><span class="self">self</span>, i: usize, j: usize) {
         <span class="self">self</span>.perm.swap(i, j);
     }

     <span class="doccomment">/// The inverse of the permutation matrix.
     </span><span class="kw">pub fn </span>inverse(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; PermutationMatrix&lt;T&gt; {
         <span class="kw">let </span><span class="kw-2">mut </span>inv: Vec&lt;usize&gt; = <span class="macro">vec!</span>[<span class="number">0</span>; <span class="self">self</span>.size()];

         <span class="kw">for </span>(source, target) <span class="kw">in </span><span class="self">self</span>.perm.iter().cloned().enumerate() {
             inv[target] = source;
         }

         PermutationMatrix {
             perm: inv,
             marker: std::marker::PhantomData
         }
     }

     <span class="doccomment">/// The size of the permutation matrix.
     ///
     /// A permutation matrix is a square matrix, so `size()` is equal
     /// to both the number of rows, as well as the number of columns.
     </span><span class="kw">pub fn </span>size(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; usize {
         <span class="self">self</span>.perm.len()
     }

     <span class="doccomment">/// Constructs a `PermutationMatrix` from an array.
     ///
     /// # Errors
     /// The supplied N-length array must satisfy the following:
     ///
     /// - Each element must be in the half-open range [0, N).
     /// - Each element must be unique.
     </span><span class="kw">pub fn </span>from_array&lt;A: Into&lt;Vec&lt;usize&gt;&gt;&gt;(array: A) -&gt; <span class="prelude-ty">Result</span>&lt;PermutationMatrix&lt;T&gt;, Error&gt; {
         <span class="kw">let </span>p = PermutationMatrix {
             perm: array.into(),
             marker: std::marker::PhantomData
         };
         validate_permutation(<span class="kw-2">&amp;</span>p.perm).map(|<span class="kw">_</span>| p)
     }

     <span class="doccomment">/// Constructs a `PermutationMatrix` from an array, without checking the validity of
     /// the supplied permutation.
     ///
     /// # Safety
     /// The supplied N-length array must satisfy the following:
     ///
     /// - Each element must be in the half-open range [0, N).
     /// - Each element must be unique.
     ///
     /// Note that while this function *itself* is technically safe
     /// regardless of the input array, passing an incorrect permutation matrix
     /// may cause undefined behavior in other methods of `PermutationMatrix`.
     /// As such, it may be difficult to debug. The user is strongly
     /// encouraged to use the safe function `from_array`, which for
     /// most real world applications only incurs a minor
     /// or even insignificant performance hit as a result of the
     /// extra validation.
     </span><span class="kw">pub unsafe fn </span>from_array_unchecked&lt;A: Into&lt;Vec&lt;usize&gt;&gt;&gt;(array: A) -&gt; PermutationMatrix&lt;T&gt; {
         <span class="kw">let </span>p = PermutationMatrix {
             perm: array.into(),
             marker: std::marker::PhantomData
         };
         p
     }

     <span class="doccomment">/// Maps the given row index into the resulting row index in the permuted matrix.
     ///
     /// More specifically, if the permutation sends row `i` to `j`, then
     /// `map_row(i)` returns `j`.
     ///
     /// # Examples
     ///
     /// ```rust
     /// use rulinalg::matrix::PermutationMatrix;
     /// let mut p = PermutationMatrix::&lt;u32&gt;::identity(3);
     /// p.swap_rows(1, 2);
     /// assert_eq!(p.map_row(1), 2);
     /// ```
     </span><span class="kw">pub fn </span>map_row(<span class="kw-2">&amp;</span><span class="self">self</span>, row_index: usize) -&gt; usize {
         <span class="self">self</span>.perm[row_index]
     }

     <span class="doccomment">/// Computes the parity of the permutation (even- or oddness).
     </span><span class="kw">pub fn </span>parity(<span class="kw-2">mut </span><span class="self">self</span>) -&gt; Parity {
         <span class="comment">// As it happens, permute_by_swap effectively decomposes
         // each disjoint cycle in the permutation into a series
         // of transpositions. The result is that the whole permutation
         // is effectively decomposed into a series of
         // transpositions.
         // Hence, if we start out by assuming that the permutation
         // is even and simply flip this variable every time a swap
         // (transposition) is performed, we&#39;ll have the result by
         // the end of the procedure.
         </span><span class="kw">let </span><span class="kw-2">mut </span>is_even = <span class="bool-val">true</span>;
         permute_by_swap(<span class="kw-2">&amp;mut </span><span class="self">self</span>.perm, |<span class="kw">_</span>, <span class="kw">_</span>| is_even = !is_even);

         <span class="kw">if </span>is_even {
             Parity::Even
         } <span class="kw">else </span>{
             Parity::Odd
         }
     }
 }

 <span class="kw">impl</span>&lt;T: Num&gt; PermutationMatrix&lt;T&gt; {
     <span class="doccomment">/// The permutation matrix in an equivalent full matrix representation.
     </span><span class="kw">pub fn </span>as_matrix(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; Matrix&lt;T&gt; {
         Matrix::from_fn(<span class="self">self</span>.size(), <span class="self">self</span>.size(), |i, j|
             <span class="kw">if </span><span class="self">self</span>.perm[i] == j {
                 T::one()
             } <span class="kw">else </span>{
                 T::zero()
             }
         )
     }

     <span class="doccomment">/// Computes the determinant of the permutation matrix.
     ///
     /// The determinant of a permutation matrix is always
     /// +1 (if the permutation is even) or
     /// -1 (if the permutation is odd).
     </span><span class="kw">pub fn </span>det(<span class="self">self</span>) -&gt; T {
         <span class="kw">let </span>parity = <span class="self">self</span>.parity();
         <span class="kw">match </span>parity {
             Parity::Even =&gt; T::one(),
             Parity::Odd  =&gt; T::zero() - T::one()
         }
     }
 }

 <span class="kw">impl</span>&lt;T&gt; PermutationMatrix&lt;T&gt; {
     <span class="doccomment">/// Permutes the rows of the given matrix in-place.
     ///
     /// # Panics
     ///
     /// - The number of rows in the matrix is not equal to
     ///   the size of the permutation matrix.
     </span><span class="kw">pub fn </span>permute_rows_in_place&lt;M&gt;(<span class="kw-2">mut </span><span class="self">self</span>, matrix: <span class="kw-2">&amp;mut </span>M) <span class="kw">where </span>M: BaseMatrixMut&lt;T&gt; {
         validate_permutation_left_mul_dimensions(<span class="kw-2">&amp;</span><span class="self">self</span>, matrix);
         permute_by_swap(<span class="kw-2">&amp;mut </span><span class="self">self</span>.perm, |i, j| matrix.swap_rows(i, j));
     }

     <span class="doccomment">/// Permutes the columns of the given matrix in-place.
     ///
     /// # Panics
     ///
     /// - The number of columns in the matrix is not equal to
     ///   the size of the permutation matrix.
     </span><span class="kw">pub fn </span>permute_cols_in_place&lt;M&gt;(<span class="kw-2">mut </span><span class="self">self</span>, matrix: <span class="kw-2">&amp;mut </span>M) <span class="kw">where </span>M: BaseMatrixMut&lt;T&gt; {
         validate_permutation_right_mul_dimensions(matrix, <span class="kw-2">&amp;</span><span class="self">self</span>);
         <span class="comment">// Note: it _may_ be possible to increase cache efficiency
         // of this routine by swapping elements in each row individually
         // (since matrices are row major), but this would mean augmenting
         // permute_by_swap in such a way that the original permutation can
         // be recovered, which includes a little bit of additional work.
         // Moreover, it would mean having to work with signed indices
         // instead of unsigned (although temporarily casting would be sufficient),
         // which may or may not complicate matters.
         // For now, it was deemed significantly simpler and probably good enough
         // to just swap whole columns instead.
         </span>permute_by_swap(<span class="kw-2">&amp;mut </span><span class="self">self</span>.perm, |i, j| matrix.swap_cols(i, j));
     }

     <span class="doccomment">/// Permutes the elements of the given vector in-place.
     ///
     /// # Panics
     ///
     /// - The size of the vector is not equal to the size of
     ///   the permutation matrix.
     </span><span class="kw">pub fn </span>permute_vector_in_place(<span class="kw-2">mut </span><span class="self">self</span>, vector: <span class="kw-2">&amp;mut </span>Vector&lt;T&gt;) {
         validate_permutation_vector_dimensions(<span class="kw-2">&amp;</span><span class="self">self</span>, vector);
         permute_by_swap(<span class="kw-2">&amp;mut </span><span class="self">self</span>.perm, |i, j| vector.mut_data().swap(i, j));
     }
 }

 <span class="kw">impl</span>&lt;T: Clone&gt; PermutationMatrix&lt;T&gt; {
     <span class="doccomment">/// Permutes the rows of the given `source_matrix` and stores the
     /// result in `buffer`.
     ///
     /// # Panics
     ///
     /// - The number of rows in the source matrix is not equal to
     ///   the size of the permutation matrix.
     /// - The dimensions of the source matrix and the buffer
     ///   are not identical.
     </span><span class="kw">pub fn </span>permute_rows_into_buffer&lt;X, Y&gt;(<span class="kw-2">&amp;</span><span class="self">self</span>, source_matrix: <span class="kw-2">&amp;</span>X, buffer: <span class="kw-2">&amp;mut </span>Y)
         <span class="kw">where </span>X: BaseMatrix&lt;T&gt;, Y: BaseMatrixMut&lt;T&gt; {
         <span class="macro">assert!</span>(source_matrix.rows() == buffer.rows()
                 &amp;&amp; source_matrix.cols() == buffer.cols(),
                 <span class="string">&quot;Source and target matrix must have equal dimensions.&quot;</span>);
         validate_permutation_left_mul_dimensions(<span class="self">self</span>, source_matrix);
         <span class="kw">for </span>(source_row, target_row_index) <span class="kw">in </span>source_matrix.row_iter()
                                                            .zip(<span class="self">self</span>.perm.iter()
                                                                          .cloned()) {
             buffer.row_mut(target_row_index)
                   .raw_slice_mut()
                   .clone_from_slice(source_row.raw_slice());
         }
     }

     <span class="doccomment">/// Permutes the columns of the given `source_matrix` and stores the
     /// result in `buffer`.
     ///
     /// # Panics
     ///
     /// - The number of columns in the source matrix is not equal to
     ///   the size of the permutation matrix.
     /// - The dimensions of the source matrix and the buffer
     ///   are not identical.
     </span><span class="kw">pub fn </span>permute_cols_into_buffer&lt;X, Y&gt;(<span class="kw-2">&amp;</span><span class="self">self</span>, source_matrix: <span class="kw-2">&amp;</span>X, target_matrix: <span class="kw-2">&amp;mut </span>Y)
         <span class="kw">where </span>X: BaseMatrix&lt;T&gt;, Y: BaseMatrixMut&lt;T&gt; {
         <span class="macro">assert!</span>(source_matrix.rows() == target_matrix.rows()
                 &amp;&amp; source_matrix.cols() == target_matrix.cols(),
                 <span class="string">&quot;Source and target matrix must have equal dimensions.&quot;</span>);
         validate_permutation_right_mul_dimensions(source_matrix, <span class="self">self</span>);

         <span class="comment">// Permute columns in one row at a time for (presumably) better cache performance
         </span><span class="kw">for </span>(row_index, source_row) <span class="kw">in </span>source_matrix.row_iter()
                                                            .map(|r| r.raw_slice())
                                                            .enumerate() {
             <span class="kw">let </span>target_row = target_matrix.row_mut(row_index).raw_slice_mut();
             <span class="kw">for </span>(source_element, target_col) <span class="kw">in </span>source_row.iter().zip(<span class="self">self</span>.perm.iter().cloned()) {
                 target_row[target_col] = source_element.clone();
             }
         }
     }

     <span class="doccomment">/// Permutes the elements of the given `source_vector` and stores the
     /// result in `buffer`.
     ///
     /// # Panics
     ///
     /// - The size of the source vector is not equal to the
     ///   size of the permutation matrix.
     /// - The dimensions of the source vector and the buffer
     ///   are not identical.
     </span><span class="kw">pub fn </span>permute_vector_into_buffer(
         <span class="kw-2">&amp;</span><span class="self">self</span>,
         source_vector: <span class="kw-2">&amp;</span>Vector&lt;T&gt;,
         buffer: <span class="kw-2">&amp;mut </span>Vector&lt;T&gt;
     ) {
         <span class="macro">assert!</span>(source_vector.size() == buffer.size(),
                <span class="string">&quot;Source and target vector must have equal dimensions.&quot;</span>);
         validate_permutation_vector_dimensions(<span class="self">self</span>, buffer);
         <span class="kw">for </span>(source_element, target_index) <span class="kw">in </span>source_vector.data()
                                                            .iter()
                                                            .zip(<span class="self">self</span>.perm.iter().cloned()) {
             buffer[target_index] = source_element.clone();
         }
     }

     <span class="doccomment">/// Computes the composition of `self` with the given `source_perm`
     /// and stores the result in `buffer`.
     ///
     /// # Panics
     ///
     /// - The size of the permutation matrix (self) is not equal to the
     ///   size of the source permutation matrix.
     </span><span class="kw">pub fn </span>compose_into_buffer(
         <span class="kw-2">&amp;</span><span class="self">self</span>,
         source_perm: <span class="kw-2">&amp;</span>PermutationMatrix&lt;T&gt;,
         buffer: <span class="kw-2">&amp;mut </span>PermutationMatrix&lt;T&gt;
     ) {
         <span class="macro">assert!</span>(source_perm.size() == buffer.size(),
             <span class="string">&quot;Source and target permutation matrix must have equal dimensions.&quot;</span>);
         <span class="kw">let </span>left = <span class="self">self</span>;
         <span class="kw">let </span>right = source_perm;
         <span class="kw">for </span>i <span class="kw">in </span><span class="number">0 </span>.. <span class="self">self</span>.perm.len() {
             buffer.perm[i] = left.perm[right.perm[i]];
         }
     }
 }

 <span class="kw">impl</span>&lt;T&gt; From&lt;PermutationMatrix&lt;T&gt;&gt; <span class="kw">for </span>Vec&lt;usize&gt; {
     <span class="kw">fn </span>from(p: PermutationMatrix&lt;T&gt;) -&gt; Vec&lt;usize&gt; {
         p.perm
     }
 }

 <span class="kw">impl</span>&lt;<span class="lifetime">&#39;a</span>, T&gt; Into&lt;<span class="kw-2">&amp;</span><span class="lifetime">&#39;a </span>[usize]&gt; <span class="kw">for </span><span class="kw-2">&amp;</span><span class="lifetime">&#39;a </span>PermutationMatrix&lt;T&gt; {
     <span class="kw">fn </span>into(<span class="self">self</span>) -&gt; <span class="kw-2">&amp;</span><span class="lifetime">&#39;a </span>[usize] {
         <span class="kw-2">&amp;</span><span class="self">self</span>.perm
     }
 }

 <span class="kw">fn </span>validate_permutation_vector_dimensions&lt;T&gt;(p: <span class="kw-2">&amp;</span>PermutationMatrix&lt;T&gt;, v: <span class="kw-2">&amp;</span>Vector&lt;T&gt;) {
     <span class="macro">assert!</span>(p.size() == v.size(),
             <span class="string">&quot;Permutation matrix and Vector dimensions are not compatible.&quot;</span>);
 }


 <span class="kw">fn </span>validate_permutation_left_mul_dimensions&lt;T, M&gt;(p: <span class="kw-2">&amp;</span>PermutationMatrix&lt;T&gt;, rhs: <span class="kw-2">&amp;</span>M)
     <span class="kw">where </span>M: BaseMatrix&lt;T&gt; {
      <span class="macro">assert!</span>(p.size() == rhs.rows(),
             <span class="string">&quot;Permutation matrix and right-hand side matrix dimensions
              are not compatible.&quot;</span>);
 }

 <span class="kw">fn </span>validate_permutation_right_mul_dimensions&lt;T, M&gt;(lhs: <span class="kw-2">&amp;</span>M, p: <span class="kw-2">&amp;</span>PermutationMatrix&lt;T&gt;)
     <span class="kw">where </span>M: BaseMatrix&lt;T&gt; {
      <span class="macro">assert!</span>(lhs.cols() == p.size(),
             <span class="string">&quot;Left-hand side matrix and permutation matrix dimensions
              are not compatible.&quot;</span>);
 }

 <span class="kw">fn </span>validate_permutation(perm: <span class="kw-2">&amp;</span>[usize]) -&gt; <span class="prelude-ty">Result</span>&lt;(), Error&gt; {
     <span class="comment">// Recall that a permutation array of size n is valid if:
     // 1. All elements are in the range [0, n)
     // 2. All elements are unique

     </span><span class="kw">let </span>n = perm.len();

     <span class="comment">// Here we use a vector of boolean. If memory usage or performance
     // is ever an issue, we could replace the vector with a bit vector
     // (from e.g. the bit-vec crate), which would cut memory usage
     // to 1/8, and likely improve performance due to more data
     // fitting in cache.
     </span><span class="kw">let </span><span class="kw-2">mut </span>visited = <span class="macro">vec!</span>[<span class="bool-val">false</span>; n];
     <span class="kw">for </span>p <span class="kw">in </span>perm.iter().cloned() {
         <span class="kw">if </span>p &lt; n {
             visited[p] = <span class="bool-val">true</span>;
         }
         <span class="kw">else </span>{
             <span class="kw">return </span><span class="prelude-val">Err</span>(Error::new(ErrorKind::InvalidPermutation,
                 <span class="string">&quot;Supplied permutation array contains elements out of bounds.&quot;</span>));
         }
     }
     <span class="kw">let </span>all_unique = visited.iter().all(|x| x.clone());
     <span class="kw">if </span>all_unique {
         <span class="prelude-val">Ok</span>(())
     } <span class="kw">else </span>{
         <span class="prelude-val">Err</span>(Error::new(ErrorKind::InvalidPermutation,
             <span class="string">&quot;Supplied permutation array contains duplicate elements.&quot;</span>))
     }
 }

 <span class="doccomment">/// Applies the permutation by swapping elements in an abstract
 /// container.
 ///
 /// The permutation is applied by calls to `swap(i, j)` for indices
 /// `i` and `j`.
 ///
 /// # Complexity
 ///
 /// - O(1) memory usage.
 /// - O(n) worst case number of calls to `swap`.
 </span><span class="kw">fn </span>permute_by_swap&lt;S&gt;(perm: <span class="kw-2">&amp;mut </span>[usize], <span class="kw-2">mut </span>swap: S) <span class="kw">where </span>S: FnMut(usize, usize) -&gt; () {
     <span class="comment">// Please see https://en.wikipedia.org/wiki/Cyclic_permutation
     // for some explanation to the terminology used here.
     // Some useful resources I found on the internet:
     //
     // https://blog.merovius.de/2014/08/12/applying-permutation-in-constant.html
     // http://stackoverflow.com/questions/16501424/algorithm-to-apply-permutation-in-constant-memory-space
     //
     // A fundamental property of permutations on finite sets is that
     // any such permutation can be decomposed into a collection of
     // cycles on disjoint orbits.
     //
     // An observation is thus that given a permutation P,
     // we can trace out the cycle that includes index i
     // by starting at i and moving to P[i] recursively.
     </span><span class="kw">for </span>i <span class="kw">in </span><span class="number">0 </span>.. perm.len() {
         <span class="kw">let </span><span class="kw-2">mut </span>target = perm[i];
         <span class="kw">while </span>i != target {
             <span class="comment">// When resolving a cycle, we resolve each index in the cycle
             // by repeatedly moving the current item into the target position,
             // and item in the target position into the current position.
             // By repeating this until we hit the start index,
             // we effectively resolve the entire cycle.
             </span><span class="kw">let </span>new_target = perm[target];
             swap(i, target);
             perm[target] = target;
             target = new_target;
         }
         perm[i] = i;
     }
 }

 <span class="attribute">#[cfg(test)]
 </span><span class="kw">mod </span>tests {
     <span class="kw">use </span>matrix::Matrix;
     <span class="kw">use </span>vector::Vector;
     <span class="kw">use super</span>::{PermutationMatrix, Parity};
     <span class="kw">use super</span>::{permute_by_swap, validate_permutation};

     <span class="kw">use </span>itertools::Itertools;

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>swap_rows() {
         <span class="kw">let </span><span class="kw-2">mut </span>p = PermutationMatrix::&lt;u64&gt;::identity(<span class="number">4</span>);
         p.swap_rows(<span class="number">0</span>, <span class="number">3</span>);
         p.swap_rows(<span class="number">1</span>, <span class="number">3</span>);

         <span class="kw">let </span>expected_permutation = PermutationMatrix::from_array(<span class="macro">vec!</span>[<span class="number">3</span>, <span class="number">0</span>, <span class="number">2</span>, <span class="number">1</span>]).unwrap();
         <span class="macro">assert_eq!</span>(p, expected_permutation);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>as_matrix() {
         <span class="kw">let </span>p = PermutationMatrix::from_array(<span class="macro">vec!</span>[<span class="number">2</span>, <span class="number">1</span>, <span class="number">0</span>, <span class="number">3</span>]).unwrap();
         <span class="kw">let </span>expected_matrix: Matrix&lt;u32&gt; = <span class="macro">matrix!</span>[<span class="number">0</span>, <span class="number">0</span>, <span class="number">1</span>, <span class="number">0</span>;
                                                    <span class="number">0</span>, <span class="number">1</span>, <span class="number">0</span>, <span class="number">0</span>;
                                                    <span class="number">1</span>, <span class="number">0</span>, <span class="number">0</span>, <span class="number">0</span>;
                                                    <span class="number">0</span>, <span class="number">0</span>, <span class="number">0</span>, <span class="number">1</span>];

         <span class="macro">assert_matrix_eq!</span>(expected_matrix, p.as_matrix());
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>from_array() {
         <span class="kw">let </span>array = <span class="macro">vec!</span>[<span class="number">1</span>, <span class="number">0</span>, <span class="number">3</span>, <span class="number">2</span>];
         <span class="kw">let </span>p = PermutationMatrix::&lt;u32&gt;::from_array(array.clone()).unwrap();
         <span class="kw">let </span>p_as_array: Vec&lt;usize&gt; = p.into();
         <span class="macro">assert_eq!</span>(p_as_array, array);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>from_array_unchecked() {
         <span class="kw">let </span>array = <span class="macro">vec!</span>[<span class="number">1</span>, <span class="number">0</span>, <span class="number">3</span>, <span class="number">2</span>];
         <span class="kw">let </span>p = <span class="kw">unsafe </span>{ PermutationMatrix::&lt;u32&gt;::from_array_unchecked(array.clone()) };
         <span class="kw">let </span>p_as_array: Vec&lt;usize&gt; = p.into();
         <span class="macro">assert_eq!</span>(p_as_array, array);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>from_array_invalid() {
         <span class="macro">assert!</span>(PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">0</span>, <span class="number">1</span>, <span class="number">3</span>]).is_err());
         <span class="macro">assert!</span>(PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">0</span>, <span class="number">0</span>]).is_err());
         <span class="macro">assert!</span>(PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">3</span>, <span class="number">0</span>, <span class="number">1</span>]).is_err());
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>vec_from_permutation() {
         <span class="kw">let </span>source_vec = <span class="macro">vec!</span>[<span class="number">0</span>, <span class="number">2</span>, <span class="number">1</span>];
         <span class="kw">let </span>p = PermutationMatrix::&lt;u32&gt;::from_array(source_vec.clone()).unwrap();
         <span class="kw">let </span>vec = Vec::from(p);
         <span class="macro">assert_eq!</span>(<span class="kw-2">&amp;</span>source_vec, <span class="kw-2">&amp;</span>vec);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>into_slice_ref() {
         <span class="kw">let </span>source_vec = <span class="macro">vec!</span>[<span class="number">0</span>, <span class="number">2</span>, <span class="number">1</span>];
         <span class="kw">let </span><span class="kw-2">ref </span>p = PermutationMatrix::&lt;u32&gt;::from_array(source_vec.clone()).unwrap();
         <span class="kw">let </span>p_as_slice_ref: <span class="kw-2">&amp;</span>[usize] = p.into();
         <span class="macro">assert_eq!</span>(source_vec.as_slice(), p_as_slice_ref);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>map_row() {
         <span class="kw">let </span>p = PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">0</span>, <span class="number">2</span>, <span class="number">1</span>]).unwrap();
         <span class="macro">assert_eq!</span>(p.map_row(<span class="number">0</span>), <span class="number">0</span>);
         <span class="macro">assert_eq!</span>(p.map_row(<span class="number">1</span>), <span class="number">2</span>);
         <span class="macro">assert_eq!</span>(p.map_row(<span class="number">2</span>), <span class="number">1</span>);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>inverse() {
         <span class="kw">let </span>p = PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">1</span>, <span class="number">2</span>, <span class="number">0</span>]).unwrap();
         <span class="kw">let </span>expected_inverse = PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">2</span>, <span class="number">0</span>, <span class="number">1</span>]).unwrap();
         <span class="macro">assert_eq!</span>(p.inverse(), expected_inverse);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>parity() {
         {
             <span class="kw">let </span>p = PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">1</span>, <span class="number">0</span>, <span class="number">3</span>, <span class="number">2</span>]).unwrap();
             <span class="macro">assert_eq!</span>(p.parity(), Parity::Even);
         }

         {
             <span class="kw">let </span>p = PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">4</span>, <span class="number">2</span>, <span class="number">3</span>, <span class="number">1</span>, <span class="number">0</span>, <span class="number">5</span>]).unwrap();
             <span class="macro">assert_eq!</span>(p.parity(), Parity::Odd);
         }
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>det() {
         {
             <span class="kw">let </span>p = PermutationMatrix::&lt;i32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">1</span>, <span class="number">0</span>, <span class="number">3</span>, <span class="number">2</span>]).unwrap();
             <span class="macro">assert_eq!</span>(p.det(), <span class="number">1</span>);
         }

         {
             <span class="kw">let </span>p = PermutationMatrix::&lt;i32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">4</span>, <span class="number">2</span>, <span class="number">3</span>, <span class="number">1</span>, <span class="number">0</span>, <span class="number">5</span>]).unwrap();
             <span class="macro">assert_eq!</span>(p.det(), -<span class="number">1</span>);
         }
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>permute_by_swap_on_empty_array() {
         <span class="kw">let </span><span class="kw-2">mut </span>x = Vec::&lt;char&gt;::new();
         <span class="kw">let </span><span class="kw-2">mut </span>permutation_array = Vec::new();
         permute_by_swap(<span class="kw-2">&amp;mut </span>permutation_array, |i, j| x.swap(i, j));
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>permute_by_swap_on_arbitrary_array() {
         <span class="kw">let </span><span class="kw-2">mut </span>x = <span class="macro">vec!</span>[<span class="string">&#39;a&#39;</span>, <span class="string">&#39;b&#39;</span>, <span class="string">&#39;c&#39;</span>, <span class="string">&#39;d&#39;</span>];
         <span class="kw">let </span><span class="kw-2">mut </span>permutation_array = <span class="macro">vec!</span>[<span class="number">0</span>, <span class="number">2</span>, <span class="number">3</span>, <span class="number">1</span>];

         permute_by_swap(<span class="kw-2">&amp;mut </span>permutation_array, |i, j| x.swap(i, j));
         <span class="macro">assert_eq!</span>(x, <span class="macro">vec!</span>[<span class="string">&#39;a&#39;</span>, <span class="string">&#39;d&#39;</span>, <span class="string">&#39;b&#39;</span>, <span class="string">&#39;c&#39;</span>]);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>permute_by_swap_identity_on_arbitrary_array() {
         <span class="kw">let </span><span class="kw-2">mut </span>x = <span class="macro">vec!</span>[<span class="string">&#39;a&#39;</span>, <span class="string">&#39;b&#39;</span>, <span class="string">&#39;c&#39;</span>, <span class="string">&#39;d&#39;</span>];
         <span class="kw">let </span><span class="kw-2">mut </span>permutation_array = <span class="macro">vec!</span>[<span class="number">0</span>, <span class="number">1</span>, <span class="number">2</span>, <span class="number">3</span>];
         permute_by_swap(<span class="kw-2">&amp;mut </span>permutation_array, |i, j| x.swap(i, j));
         <span class="macro">assert_eq!</span>(x, <span class="macro">vec!</span>[<span class="string">&#39;a&#39;</span>, <span class="string">&#39;b&#39;</span>, <span class="string">&#39;c&#39;</span>, <span class="string">&#39;d&#39;</span>]);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>compose_into_buffer() {
         <span class="kw">let </span>p = PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">2</span>, <span class="number">1</span>, <span class="number">0</span>]).unwrap();
         <span class="kw">let </span>q = PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">1</span>, <span class="number">0</span>, <span class="number">2</span>]).unwrap();
         <span class="kw">let </span>pq_expected = PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">1</span>, <span class="number">2</span>, <span class="number">0</span>]).unwrap();
         <span class="kw">let </span>qp_expected = PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">2</span>, <span class="number">0</span>, <span class="number">1</span>]).unwrap();

         {
             <span class="kw">let </span><span class="kw-2">mut </span>pq = PermutationMatrix::identity(<span class="number">3</span>);
             p.compose_into_buffer(<span class="kw-2">&amp;</span>q, <span class="kw-2">&amp;mut </span>pq);
             <span class="macro">assert_eq!</span>(pq, pq_expected);
         }

         {
             <span class="kw">let </span><span class="kw-2">mut </span>qp = PermutationMatrix::identity(<span class="number">3</span>);
             q.compose_into_buffer(<span class="kw-2">&amp;</span>p, <span class="kw-2">&amp;mut </span>qp);
             <span class="macro">assert_eq!</span>(qp, qp_expected);
         }
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>compose_regression() {
         <span class="comment">// At some point during development, this example failed to
         // give the expected results
         </span><span class="kw">let </span>p = PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">1</span>, <span class="number">2</span>, <span class="number">0</span>]).unwrap();
         <span class="kw">let </span>q = PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">2</span>, <span class="number">0</span>, <span class="number">1</span>]).unwrap();
         <span class="kw">let </span>pq_expected = PermutationMatrix::&lt;u32&gt;::from_array(<span class="macro">vec!</span>[<span class="number">0</span>, <span class="number">1</span>, <span class="number">2</span>]).unwrap();

         <span class="kw">let </span><span class="kw-2">mut </span>pq = PermutationMatrix::identity(<span class="number">3</span>);
         p.compose_into_buffer(<span class="kw-2">&amp;</span>q, <span class="kw-2">&amp;mut </span>pq);
         <span class="macro">assert_eq!</span>(pq, pq_expected);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>permute_rows_into_buffer() {
         <span class="kw">let </span>x = <span class="macro">matrix!</span>[ <span class="number">0</span>;
                          <span class="number">1</span>;
                          <span class="number">2</span>;
                          <span class="number">3</span>];
         <span class="kw">let </span>p = PermutationMatrix::from_array(<span class="macro">vec!</span>[<span class="number">2</span>, <span class="number">1</span>, <span class="number">3</span>, <span class="number">0</span>]).unwrap();
         <span class="kw">let </span><span class="kw-2">mut </span>output = Matrix::zeros(<span class="number">4</span>, <span class="number">1</span>);
         p.permute_rows_into_buffer(<span class="kw-2">&amp;</span>x, <span class="kw-2">&amp;mut </span>output);
         <span class="macro">assert_matrix_eq!</span>(output, <span class="macro">matrix!</span>[ <span class="number">3</span>; <span class="number">1</span>; <span class="number">0</span>; <span class="number">2</span>]);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>permute_rows_in_place() {
         <span class="kw">let </span><span class="kw-2">mut </span>x = <span class="macro">matrix!</span>[ <span class="number">0</span>;
                          <span class="number">1</span>;
                          <span class="number">2</span>;
                          <span class="number">3</span>];
         <span class="kw">let </span>p = PermutationMatrix::from_array(<span class="macro">vec!</span>[<span class="number">2</span>, <span class="number">1</span>, <span class="number">3</span>, <span class="number">0</span>]).unwrap();
         p.permute_rows_in_place(<span class="kw-2">&amp;mut </span>x);
         <span class="macro">assert_matrix_eq!</span>(x, <span class="macro">matrix!</span>[ <span class="number">3</span>; <span class="number">1</span>; <span class="number">0</span>; <span class="number">2</span>]);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>permute_cols_into_buffer() {
         <span class="kw">let </span>x = <span class="macro">matrix!</span>[ <span class="number">0</span>, <span class="number">1</span>, <span class="number">2</span>, <span class="number">3</span>];
         <span class="kw">let </span>p = PermutationMatrix::from_array(<span class="macro">vec!</span>[<span class="number">2</span>, <span class="number">1</span>, <span class="number">3</span>, <span class="number">0</span>]).unwrap();
         <span class="kw">let </span><span class="kw-2">mut </span>output = Matrix::zeros(<span class="number">1</span>, <span class="number">4</span>);
         p.permute_cols_into_buffer(<span class="kw-2">&amp;</span>x, <span class="kw-2">&amp;mut </span>output);
         <span class="macro">assert_matrix_eq!</span>(output, <span class="macro">matrix!</span>[ <span class="number">3</span>, <span class="number">1</span>, <span class="number">0</span>, <span class="number">2</span>]);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>permute_cols_in_place() {
         <span class="kw">let </span><span class="kw-2">mut </span>x = <span class="macro">matrix!</span>[ <span class="number">0</span>, <span class="number">1</span>, <span class="number">2</span>, <span class="number">3</span>];
         <span class="kw">let </span>p = PermutationMatrix::from_array(<span class="macro">vec!</span>[<span class="number">2</span>, <span class="number">1</span>, <span class="number">3</span>, <span class="number">0</span>]).unwrap();
         p.permute_cols_in_place(<span class="kw-2">&amp;mut </span>x);
         <span class="macro">assert_matrix_eq!</span>(x, <span class="macro">matrix!</span>[ <span class="number">3</span>, <span class="number">1</span>, <span class="number">0</span>, <span class="number">2</span>]);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>permute_vector_into_buffer() {
         <span class="kw">let </span>x = <span class="macro">vector!</span>[ <span class="number">0</span>, <span class="number">1</span>, <span class="number">2</span>, <span class="number">3</span>];
         <span class="kw">let </span>p = PermutationMatrix::from_array(<span class="macro">vec!</span>[<span class="number">2</span>, <span class="number">1</span>, <span class="number">3</span>, <span class="number">0</span>]).unwrap();
         <span class="kw">let </span><span class="kw-2">mut </span>output = Vector::zeros(<span class="number">4</span>);
         p.permute_vector_into_buffer(<span class="kw-2">&amp;</span>x, <span class="kw-2">&amp;mut </span>output);
         <span class="macro">assert_vector_eq!</span>(output, <span class="macro">vector!</span>[ <span class="number">3</span>, <span class="number">1</span>, <span class="number">0</span>, <span class="number">2</span>]);
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>permute_vector_in_place() {
         <span class="kw">let </span><span class="kw-2">mut </span>x = <span class="macro">vector!</span>[ <span class="number">0</span>, <span class="number">1</span>, <span class="number">2</span>, <span class="number">3</span>];
         <span class="kw">let </span>p = PermutationMatrix::from_array(<span class="macro">vec!</span>[<span class="number">2</span>, <span class="number">1</span>, <span class="number">3</span>, <span class="number">0</span>]).unwrap();
         p.permute_vector_in_place(<span class="kw-2">&amp;mut </span>x);
         <span class="macro">assert_vector_eq!</span>(x, <span class="macro">vector!</span>[ <span class="number">3</span>, <span class="number">1</span>, <span class="number">0</span>, <span class="number">2</span>]);
     }

     <span class="kw">use </span>quickcheck::{Arbitrary, Gen};

     <span class="comment">// In order to write property tests for the validation of a permutation,
     // we need to be able to generate arbitrary (valid) permutations.
     </span><span class="attribute">#[derive(Debug, Clone, PartialEq, Eq)]
     </span><span class="kw">struct </span>ValidPermutationArray(<span class="kw">pub </span>Vec&lt;usize&gt;);

     <span class="kw">impl </span>Arbitrary <span class="kw">for </span>ValidPermutationArray {
         <span class="kw">fn </span>arbitrary&lt;G: Gen&gt;(g: <span class="kw-2">&amp;mut </span>G) -&gt; <span class="self">Self </span>{
             <span class="kw">let </span>upper_bound = g.size();
             <span class="kw">let </span><span class="kw-2">mut </span>array = (<span class="number">0 </span>.. upper_bound).collect::&lt;Vec&lt;usize&gt;&gt;();
             g.shuffle(<span class="kw-2">&amp;mut </span>array);
             ValidPermutationArray(array)
         }
     }

     <span class="comment">// We also want to be able to generate invalid permutations for
     // the same reasons
     </span><span class="attribute">#[derive(Debug, Clone, PartialEq, Eq)]
     </span><span class="kw">struct </span>InvalidPermutationArray(<span class="kw">pub </span>Vec&lt;usize&gt;);

     <span class="kw">impl </span>Arbitrary <span class="kw">for </span>InvalidPermutationArray {
         <span class="kw">fn </span>arbitrary&lt;G: Gen&gt;(g: <span class="kw-2">&amp;mut </span>G) -&gt; <span class="self">Self </span>{
             <span class="comment">// Take an arbitrary valid permutation and mutate it so that
             // it is invalid
             </span><span class="kw">let </span><span class="kw-2">mut </span>permutation_array = ValidPermutationArray::arbitrary(g).<span class="number">0</span>;
             <span class="kw">let </span>n = permutation_array.len();

             <span class="comment">// There are two essential sources of invalidity:
             // 1. Duplicate elements
             // 2. Indices out of bounds
             // We want to have either or both

             </span><span class="kw">let </span>should_have_duplicates = g.gen::&lt;bool&gt;();
             <span class="kw">let </span>should_have_out_of_bounds = !should_have_duplicates || g.gen::&lt;bool&gt;();
             <span class="macro">assert!</span>(should_have_duplicates || should_have_out_of_bounds);

             <span class="kw">if </span>should_have_out_of_bounds {
                 <span class="kw">let </span>num_out_of_bounds_rounds = g.gen_range::&lt;usize&gt;(<span class="number">1</span>, n);
                 <span class="kw">for _ in </span><span class="number">0 </span>.. num_out_of_bounds_rounds {
                     <span class="kw">let </span>interior_index = g.gen_range::&lt;usize&gt;(<span class="number">0</span>, n);
                     <span class="kw">let </span>exterior_index = n + g.gen::&lt;usize&gt;();
                     permutation_array[interior_index] = exterior_index;
                 }
             }

             <span class="kw">if </span>should_have_duplicates {
                 <span class="kw">let </span>num_duplicates = g.gen_range::&lt;usize&gt;(<span class="number">1</span>, n);
                 <span class="kw">for _ in </span><span class="number">0 </span>.. num_duplicates {
                     <span class="kw">let </span>interior_index = g.gen_range::&lt;usize&gt;(<span class="number">0</span>, n);
                     <span class="kw">let </span>duplicate_value = permutation_array[interior_index];
                     permutation_array.push(duplicate_value);
                 }
             }

             <span class="comment">// The duplicates are placed at the end, so we perform
             // an additional shuffle to end up with a more or less
             // arbitrary invalid permutation
             </span>g.shuffle(<span class="kw-2">&amp;mut </span>permutation_array);
             InvalidPermutationArray(permutation_array)
         }
     }

     <span class="kw">impl</span>&lt;T: Send + Clone + <span class="lifetime">&#39;static</span>&gt; Arbitrary <span class="kw">for </span>PermutationMatrix&lt;T&gt; {
         <span class="kw">fn </span>arbitrary&lt;G: Gen&gt;(g: <span class="kw-2">&amp;mut </span>G) -&gt; <span class="self">Self </span>{
             <span class="kw">let </span>ValidPermutationArray(array) = ValidPermutationArray::arbitrary(g);
             PermutationMatrix::from_array(array)
                 .expect(<span class="string">&quot;The generated permutation array should always be valid.&quot;</span>)
         }
     }

     <span class="macro">quickcheck! </span>{
         <span class="kw">fn </span>property_validate_permutation_is_ok_for_valid_input(array: ValidPermutationArray) -&gt; bool {
             validate_permutation(<span class="kw-2">&amp;</span>array.<span class="number">0</span>).is_ok()
         }
     }

     <span class="macro">quickcheck! </span>{
         <span class="kw">fn </span>property_validate_permutation_is_err_for_invalid_input(array: InvalidPermutationArray) -&gt; bool {
             validate_permutation(<span class="kw-2">&amp;</span>array.<span class="number">0</span>).is_err()
         }
     }

     <span class="macro">quickcheck! </span>{
         <span class="kw">fn </span>property_identity_has_identity_array(size: usize) -&gt; bool {
             <span class="kw">let </span>p = PermutationMatrix::&lt;u64&gt;::identity(size);
             <span class="kw">let </span>p_as_array: Vec&lt;usize&gt; = p.into();
             <span class="kw">let </span>expected = (<span class="number">0 </span>.. size).collect::&lt;Vec&lt;usize&gt;&gt;();
             p_as_array == expected
         }
     }

     <span class="macro">quickcheck! </span>{
         <span class="kw">fn </span>property_dim_is_equal_to_array_dimensions(array: ValidPermutationArray) -&gt; bool {
             <span class="kw">let </span>ValidPermutationArray(array) = array;
             <span class="kw">let </span>n = array.len();
             <span class="kw">let </span>p = PermutationMatrix::&lt;u32&gt;::from_array(array).unwrap();
             p.size() == n
         }
     }

     <span class="macro">quickcheck! </span>{
         <span class="kw">fn </span>property_inverse_of_inverse_is_original(p: PermutationMatrix&lt;u32&gt;) -&gt; bool {
             p == p.inverse().inverse()
         }
     }

     <span class="macro">quickcheck! </span>{
         <span class="kw">fn </span>property_inverse_composes_to_identity(p: PermutationMatrix&lt;u32&gt;) -&gt; bool {
             <span class="comment">// Recall that P * P_inv = I and P_inv * P = I
             </span><span class="kw">let </span>n = p.size();
             <span class="kw">let </span>pinv = p.inverse();
             <span class="kw">let </span><span class="kw-2">mut </span>p_pinv_composition = PermutationMatrix::identity(n);
             <span class="kw">let </span><span class="kw-2">mut </span>pinv_p_composition = PermutationMatrix::identity(n);
             p.compose_into_buffer(<span class="kw-2">&amp;</span>pinv, <span class="kw-2">&amp;mut </span>p_pinv_composition);
             pinv.compose_into_buffer(<span class="kw-2">&amp;</span>p, <span class="kw-2">&amp;mut </span>pinv_p_composition);
             <span class="macro">assert_eq!</span>(p_pinv_composition, PermutationMatrix::identity(n));
             <span class="macro">assert_eq!</span>(pinv_p_composition, PermutationMatrix::identity(n));
             <span class="bool-val">true
         </span>}
     }

     <span class="macro">quickcheck! </span>{
         <span class="kw">fn </span>property_identity_parity_is_even(n: usize) -&gt; bool {
             <span class="kw">let </span>p = PermutationMatrix::&lt;u32&gt;::identity(n);
             p.parity() ==  Parity::Even
         }
     }

     <span class="macro">quickcheck! </span>{
         <span class="kw">fn </span>property_parity_agrees_with_parity_of_inversions(p: PermutationMatrix&lt;u32&gt;) -&gt; bool {
             <span class="kw">let </span>array: <span class="kw-2">&amp;</span>[usize] = (<span class="kw-2">&amp;</span>p).into();
             <span class="kw">let </span>num_inversions = array.iter().cloned().enumerate()
                                       .cartesian_product(array.iter().cloned().enumerate())
                                       .filter(|<span class="kw-2">&amp;</span>((i, permuted_i), (j, permuted_j))|
                                         <span class="comment">// This is simply the definition of an inversion
                                         </span>i &lt; j &amp;&amp; permuted_i &gt; permuted_j
                                       )
                                       .count();
             <span class="comment">// Recall that the parity of the number of inversions in the
             // permutation is equal to the parity of the permutation
             </span><span class="kw">let </span>parity = <span class="kw">if </span>num_inversions % <span class="number">2 </span>== <span class="number">0 </span>{
                 Parity::Even
             } <span class="kw">else </span>{
                 Parity::Odd
             };

             parity == p.clone().parity()
         }
     }
 }
 </code></pre></div>
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