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

 <span class="kw">use </span>std::any::Any;
 <span class="kw">use </span>std::cmp;

 <span class="kw">use </span>libnum::{Float, Zero, One};

 <span class="kw">use </span>matrix::decomposition::Decomposition;

 <span class="doccomment">/// Result of unpacking an instance of
 /// [PartialPivLu](struct.PartialPivLu.html).
 </span><span class="attribute">#[derive(Debug, Clone)]
 </span><span class="kw">pub struct </span>LUP&lt;T&gt; {
     <span class="doccomment">/// The lower triangular matrix in the decomposition.
     </span><span class="kw">pub </span>l: Matrix&lt;T&gt;,
     <span class="doccomment">/// The upper triangular matrix in the decomposition.
     </span><span class="kw">pub </span>u: Matrix&lt;T&gt;,
     <span class="doccomment">/// The permutation matrix in the decomposition.
     </span><span class="kw">pub </span>p: PermutationMatrix&lt;T&gt;
 }

 <span class="doccomment">/// LU decomposition with partial pivoting.
 ///
 /// For any square matrix A, there exist a permutation matrix
 /// `P`, a lower triangular matrix `L` and an upper triangular
 /// matrix `U` such that
 ///
 /// ```text
 /// PA = LU.
 /// ```
 ///
 /// However, due to the way partial pivoting algorithms work,
 /// LU decomposition with partial pivoting is in general
 /// *only numerically stable for well-conditioned invertible matrices*.
 ///
 /// That said, partial pivoting is sufficient in the vast majority
 /// of practical applications, and it is also the fastest of the
 /// pivoting schemes in existence.
 ///
 ///
 /// # Applications
 ///
 /// Given a matrix `x`, computing the LU(P) decomposition is simple:
 ///
 /// ```
 /// use rulinalg::matrix::decomposition::{PartialPivLu, LUP, Decomposition};
 /// use rulinalg::matrix::Matrix;
 ///
 /// let x = Matrix::&lt;f64&gt;::identity(4);
 ///
 /// // The matrix is consumed and its memory
 /// // re-purposed for the decomposition
 /// let lu = PartialPivLu::decompose(x).expect(&quot;Matrix is invertible.&quot;);
 ///
 /// // See below for applications
 /// // ...
 ///
 /// // The factors L, U and P can be obtained by unpacking the
 /// // decomposition, for example by destructuring as seen here
 /// let LUP { l, u, p } = lu.unpack();
 ///
 /// ```
 ///
 /// ## Solving linear systems
 ///
 /// Arguably the most common use case of LU decomposition
 /// is the computation of solutions to (multiple) linear systems
 /// that share the same coefficient matrix.
 ///
 /// ```
 /// # #[macro_use] extern crate rulinalg;
 /// # use rulinalg::matrix::decomposition::PartialPivLu;
 /// # use rulinalg::matrix::Matrix;
 /// # fn main() {
 /// # let x = Matrix::identity(4);
 /// # let lu = PartialPivLu::decompose(x).unwrap();
 /// let b = vector![3.0, 4.0, 2.0, 1.0];
 /// let y = lu.solve(b)
 ///           .expect(&quot;Matrix is invertible.&quot;);
 /// assert_vector_eq!(y, vector![3.0, 4.0, 2.0, 1.0], comp = float);
 ///
 /// // We can efficiently solve multiple such systems
 /// let c = vector![0.0, 0.0, 0.0, 0.0];
 /// let z = lu.solve(c).unwrap();
 /// assert_vector_eq!(z, vector![0.0, 0.0, 0.0, 0.0], comp = float);
 /// # }
 /// ```
 ///
 /// ## Computing the inverse of a matrix
 ///
 /// The LU decomposition provides a convenient way to obtain
 /// the inverse of the decomposed matrix. However, please keep
 /// in mind that explicitly computing the inverse of a matrix
 /// is *usually* a bad idea. In many cases, one might instead simply
 /// solve multiple systems using `solve`.
 ///
 /// For example, a common misconception is that when one needs
 /// to solve multiple linear systems `Ax = b` for different `b`,
 /// one should pre-compute the inverse of the matrix for efficiency.
 /// In fact, this is practically never a good idea! A far more efficient
 /// and accurate method is to perform the LU decomposition once, and
 /// then solve each system as shown in the examples of the previous
 /// subsection.
 ///
 /// That said, there are definitely cases where an explicit inverse is
 /// needed. In these cases, the inverse can easily be obtained
 /// through the `inverse()` method.
 ///
 /// # Computing the determinant of a matrix
 ///
 /// Once the LU decomposition has been obtained, computing
 /// the determinant of the decomposed matrix is a very cheap
 /// operation.
 ///
 /// ```
 /// # #[macro_use] extern crate rulinalg;
 /// # use rulinalg::matrix::decomposition::PartialPivLu;
 /// # use rulinalg::matrix::Matrix;
 /// # fn main() {
 /// # let x = Matrix::&lt;f64&gt;::identity(4);
 /// # let lu = PartialPivLu::decompose(x).unwrap();
 /// assert_eq!(lu.det(), 1.0);
 /// # }
 /// ```
 </span><span class="attribute">#[derive(Debug, Clone)]
 </span><span class="kw">pub struct </span>PartialPivLu&lt;T&gt; {
     lu: Matrix&lt;T&gt;,
     p: PermutationMatrix&lt;T&gt;
 }

 <span class="kw">impl</span>&lt;T: Clone + One + Zero&gt; Decomposition <span class="kw">for </span>PartialPivLu&lt;T&gt; {
     <span class="kw">type </span>Factors = LUP&lt;T&gt;;

     <span class="kw">fn </span>unpack(<span class="self">self</span>) -&gt; LUP&lt;T&gt; {
         <span class="kw">use </span>internal_utils::nullify_lower_triangular_part;
         <span class="kw">let </span>l = unit_lower_triangular_part(<span class="kw-2">&amp;</span><span class="self">self</span>.lu);
         <span class="kw">let </span><span class="kw-2">mut </span>u = <span class="self">self</span>.lu;
         nullify_lower_triangular_part(<span class="kw-2">&amp;mut </span>u);

         LUP {
             l: l,
             u: u,
             p: <span class="self">self</span>.p
         }
     }
 }

 <span class="kw">impl</span>&lt;T: <span class="lifetime">&#39;static </span>+ Float&gt; PartialPivLu&lt;T&gt; {
     <span class="doccomment">/// Performs the decomposition.
     ///
     /// # Panics
     ///
     /// The matrix must be square.
     ///
     /// # Errors
     ///
     /// An error will be returned if the matrix
     /// is singular to working precision (badly conditioned).
     </span><span class="kw">pub fn </span>decompose(matrix: Matrix&lt;T&gt;) -&gt; <span class="prelude-ty">Result</span>&lt;<span class="self">Self</span>, Error&gt; {
         <span class="kw">let </span>n = matrix.cols;
         <span class="macro">assert!</span>(matrix.rows == n, <span class="string">&quot;Matrix must be square for LU decomposition.&quot;</span>);
         <span class="kw">let </span><span class="kw-2">mut </span>lu = matrix;
         <span class="kw">let </span><span class="kw-2">mut </span>p = PermutationMatrix::identity(n);

         <span class="kw">for </span>index <span class="kw">in </span><span class="number">0</span>..n {
             <span class="kw">let </span><span class="kw-2">mut </span>curr_max_idx = index;
             <span class="kw">let </span><span class="kw-2">mut </span>curr_max = lu[[curr_max_idx, curr_max_idx]];

             <span class="kw">for </span>i <span class="kw">in </span>(curr_max_idx+<span class="number">1</span>)..n {
                 <span class="kw">if </span>lu[[i, index]].abs() &gt; curr_max.abs() {
                     curr_max = lu[[i, index]];
                     curr_max_idx = i;
                 }
             }
             <span class="kw">if </span>curr_max.abs() &lt; T::epsilon() {
                 <span class="kw">return </span><span class="prelude-val">Err</span>(Error::new(ErrorKind::DivByZero,
                     <span class="string">&quot;The matrix is too ill-conditioned for
                      LU decomposition with partial pivoting.&quot;</span>));
             }

             lu.swap_rows(index, curr_max_idx);
             p.swap_rows(index, curr_max_idx);

             gaussian_elimination(<span class="kw-2">&amp;mut </span>lu, index);
         }
         <span class="prelude-val">Ok</span>(PartialPivLu {
             lu: lu,
             p: p.inverse()
         })
     }
 }

 <span class="comment">// TODO: Remove Any bound (cannot for the time being, since
 // back substitution uses Any bound)
 </span><span class="kw">impl</span>&lt;T&gt; PartialPivLu&lt;T&gt; <span class="kw">where </span>T: Any + Float {
     <span class="doccomment">/// Solves the linear system `Ax = b`.
     ///
     /// Here, `A` is the decomposed matrix satisfying
     /// `PA = LU`. Note that this method is particularly
     /// well suited to solving multiple such linear systems
     /// involving the same `A` but different `b`.
     ///
     /// # Errors
     ///
     /// If the matrix is very ill-conditioned, the function
     /// might fail to obtain the solution to the system.
     ///
     /// # Panics
     ///
     /// The right-hand side vector `b` must have compatible size.
     ///
     /// # Examples
     ///
     /// ```
     /// # #[macro_use] extern crate rulinalg;
     /// # use rulinalg::matrix::decomposition::PartialPivLu;
     /// # use rulinalg::matrix::Matrix;
     /// # fn main() {
     /// let x = Matrix::identity(4);
     /// let lu = PartialPivLu::decompose(x).unwrap();
     /// let b = vector![3.0, 4.0, 2.0, 1.0];
     /// let y = lu.solve(b)
     ///           .expect(&quot;Matrix is invertible.&quot;);
     /// assert_vector_eq!(y, vector![3.0, 4.0, 2.0, 1.0], comp = float);
     /// # }
     /// ```
     </span><span class="kw">pub fn </span>solve(<span class="kw-2">&amp;</span><span class="self">self</span>, b: Vector&lt;T&gt;) -&gt; <span class="prelude-ty">Result</span>&lt;Vector&lt;T&gt;, Error&gt; {
         <span class="macro">assert!</span>(b.size() == <span class="self">self</span>.lu.rows(),
             <span class="string">&quot;Right-hand side vector must have compatible size.&quot;</span>);
         <span class="comment">// Note that applying p here implicitly incurs a clone.
         // TODO: Is it possible to avoid the clone somehow?
         // To my knowledge, applying a permutation matrix
         // in-place in O(n) time requires O(n) storage for bookkeeping.
         // However, we might be able to get by with something like
         // O(n log n) for the permutation as the forward/backward
         // substitution algorithms are O(n^2), if this helps us
         // avoid the memory overhead.
         </span><span class="kw">let </span>b = lu_forward_substitution(<span class="kw-2">&amp;</span><span class="self">self</span>.lu, <span class="kw-2">&amp;</span><span class="self">self</span>.p * b);
         back_substitution(<span class="kw-2">&amp;</span><span class="self">self</span>.lu, b)
     }

     <span class="doccomment">/// Computes the inverse of the matrix which this LUP decomposition
     /// represents.
     ///
     /// # Errors
     /// The inversion might fail if the matrix is very ill-conditioned.
     </span><span class="kw">pub fn </span>inverse(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; <span class="prelude-ty">Result</span>&lt;Matrix&lt;T&gt;, Error&gt; {
         <span class="kw">let </span>n = <span class="self">self</span>.lu.rows();
         <span class="kw">let </span><span class="kw-2">mut </span>inv = Matrix::zeros(n, n);
         <span class="kw">let </span><span class="kw-2">mut </span>e = Vector::zeros(n);

         <span class="comment">// To compute the inverse of a matrix A, note that
         // we can simply solve the system
         // AX = I,
         // where X is the inverse of A, and I is the identity
         // matrix of appropriate dimension.
         //
         // Note that this is not optimal in terms of performance,
         // and there is likely significant potential for improvement.
         //
         // A more performant technique is usually to compute the
         // triangular inverse of each of the L and U triangular matrices,
         // but this again requires efficient algorithms (blocked/recursive)
         // to invert triangular matrices, which at this point
         // we do not have available.

         // Solve for each column of the inverse matrix
         </span><span class="kw">for </span>i <span class="kw">in </span><span class="number">0 </span>.. n {
             e[i] = T::one();

             <span class="kw">let </span>col = <span class="macro">try!</span>(<span class="self">self</span>.solve(e));

             <span class="kw">for </span>j <span class="kw">in </span><span class="number">0 </span>.. n {
                 inv[[j, i]] = col[j];
             }

             e = col.apply(<span class="kw-2">&amp;</span>|<span class="kw">_</span>| T::zero());
         }

         <span class="prelude-val">Ok</span>(inv)
     }

     <span class="doccomment">/// Computes the determinant of the decomposed matrix.
     ///
     /// Note that the determinant of an empty matrix is considered
     /// to be equal to 1.
     </span><span class="kw">pub fn </span>det(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; T {
         <span class="comment">// Recall that the determinant of a triangular matrix
         // is the product of its diagonal entries. Also,
         // the determinant of L is implicitly 1.
         </span><span class="kw">let </span>u_det = <span class="self">self</span>.lu.diag().fold(T::one(), |x, <span class="kw-2">&amp;</span>y| x * y);
         <span class="comment">// Note that the determinant of P is equal to the
         // determinant of P^T, so we don&#39;t have to invert it
         </span><span class="kw">let </span>p_det = <span class="self">self</span>.p.clone().det();
         p_det * u_det
     }
 }

 <span class="doccomment">/// Result of unpacking an instance of
 /// [FullPivLu](struct.FullPivLu.html).
 ///
 /// PAQ = LU
 </span><span class="attribute">#[derive(Debug, Clone)]
 </span><span class="kw">pub struct </span>LUPQ&lt;T&gt; {
     <span class="doccomment">/// The lower triangular matrix in the decomposition.
     </span><span class="kw">pub </span>l: Matrix&lt;T&gt;,

     <span class="doccomment">/// The upper triangular matrix in the decomposition.
     </span><span class="kw">pub </span>u: Matrix&lt;T&gt;,

     <span class="doccomment">/// The row-exchange permutation matrix in the decomposition.
     </span><span class="kw">pub </span>p: PermutationMatrix&lt;T&gt;,

     <span class="doccomment">/// The column-exchange permutation matrix in the decomposition.
     </span><span class="kw">pub </span>q: PermutationMatrix&lt;T&gt;
 }

 <span class="doccomment">/// LU decomposition with complete pivoting.
 ///
 /// For any square matrix A, there exist two permutation matrices
 /// `P` and `Q`, a lower triangular matrix `L` and an upper triangular
 /// matrix `U` such that
 ///
 /// ```text
 /// PAQ = LU.
 /// ```
 ///
 /// Unlike the LU decomposition computed with partial pivoting, this
 /// decomposition is stable for singular matrices. It is also a rank-
 /// revealing decomposition.
 ///
 /// See [PartialPivLu](decomposition/struct.PartialPivLu.html) for
 /// applications of LU decompositions in general.
 </span><span class="attribute">#[derive(Debug, Clone)]
 </span><span class="kw">pub struct </span>FullPivLu&lt;T&gt; {
     lu: Matrix&lt;T&gt;,
     p: PermutationMatrix&lt;T&gt;,
     q: PermutationMatrix&lt;T&gt;
 }

 <span class="kw">impl</span>&lt;T: Clone + One + Zero&gt; Decomposition <span class="kw">for </span>FullPivLu&lt;T&gt; {
     <span class="kw">type </span>Factors = LUPQ&lt;T&gt;;

     <span class="kw">fn </span>unpack(<span class="self">self</span>) -&gt; LUPQ&lt;T&gt; {
         <span class="kw">use </span>internal_utils::nullify_lower_triangular_part;
         <span class="kw">let </span>l = unit_lower_triangular_part(<span class="kw-2">&amp;</span><span class="self">self</span>.lu);
         <span class="kw">let </span><span class="kw-2">mut </span>u = <span class="self">self</span>.lu;
         nullify_lower_triangular_part(<span class="kw-2">&amp;mut </span>u);

         LUPQ {
             l: l,
             u: u,
             p: <span class="self">self</span>.p,
             q: <span class="self">self</span>.q,
         }
     }
 }

 <span class="kw">impl</span>&lt;T: <span class="lifetime">&#39;static </span>+ Float&gt; FullPivLu&lt;T&gt; {
     <span class="kw">fn </span>select_pivot(mat: <span class="kw-2">&amp;</span>Matrix&lt;T&gt;, index: usize) -&gt; (usize, usize, T) {
         <span class="kw">let </span><span class="kw-2">mut </span>piv_row = index;
         <span class="kw">let </span><span class="kw-2">mut </span>piv_col = index;
         <span class="kw">let </span><span class="kw-2">mut </span>piv_val = mat[[index,index]];

         <span class="kw">for </span>row <span class="kw">in </span>index..mat.rows() {
             <span class="kw">for </span>col <span class="kw">in </span>index..mat.cols() {
                 <span class="kw">let </span>val = mat[[row,col]];

                 <span class="kw">if </span>val.abs() &gt; piv_val.abs() {
                     piv_val = val;
                     piv_row = row;
                     piv_col = col;
                 }
             }
         }

         (piv_row, piv_col, piv_val)
     }

     <span class="doccomment">/// Performs the decomposition.
     </span><span class="kw">pub fn </span>decompose(matrix: Matrix&lt;T&gt;) -&gt; <span class="prelude-ty">Result</span>&lt;<span class="self">Self</span>, Error&gt; {
         <span class="macro">assert!</span>(
             matrix.rows() == matrix.cols(),
             <span class="string">&quot;Matrix must be square for LU decomposition.&quot;</span>);

         <span class="kw">let </span><span class="kw-2">mut </span>lu = matrix;

         <span class="kw">let </span>nrows = lu.rows();
         <span class="kw">let </span>ncols = lu.cols();
         <span class="kw">let </span>diag_size = cmp::min(nrows, ncols);

         <span class="kw">let </span><span class="kw-2">mut </span>p = PermutationMatrix::identity(nrows);
         <span class="kw">let </span><span class="kw-2">mut </span>q = PermutationMatrix::identity(ncols);

         <span class="kw">for </span>index <span class="kw">in </span><span class="number">0</span>..diag_size {
             <span class="comment">// Select the current pivot. This is the largest value in
             // the bottom right corner of the matrix, starting at
             // (index, index).
             </span><span class="kw">let </span>(piv_row, piv_col, piv_val) = FullPivLu::select_pivot(<span class="kw-2">&amp;</span>lu, index);

             <span class="kw">if </span>piv_val.abs() == T::zero() {
               <span class="kw">break</span>;
             }

             lu.swap_rows(index, piv_row);
             lu.swap_cols(index, piv_col);

             p.swap_rows(index, piv_row);

             <span class="comment">// This is a little misleading, but even though
             // we&#39;re calling swap_rows here, since q is applied on the
             // right to A (i.e. P * A * Q), the result is a column swap of A.
             </span>q.swap_rows(index, piv_col);

             <span class="comment">// We&#39;ve swapped the pivot row and column so that the pivot
             // ends up in the (index, index) position, so apply gaussian
             // elimination to the bottom-right corner.
             </span>gaussian_elimination(<span class="kw-2">&amp;mut </span>lu, index);
         }

         <span class="prelude-val">Ok</span>(FullPivLu {
             lu: lu,
             p: p.inverse(),
             q: q.inverse()
         })
     }
 }

 <span class="comment">// TODO: Remove Any bound (cannot for the time being, since
 // back substitution uses Any bound)
 </span><span class="kw">impl</span>&lt;T&gt; FullPivLu&lt;T&gt; <span class="kw">where </span>T: Any + Float {

     <span class="doccomment">/// Solves the linear system `Ax = b`.
     ///
     /// Here, `A` is the decomposed matrix satisfying
     /// `PAQ = LU`. Note that this method is particularly
     /// well suited to solving multiple such linear systems
     /// involving the same `A` but different `b`.
     ///
     /// # Errors
     ///
     /// If the matrix is very ill-conditioned, the function
     /// might fail to obtain the solution to the system.
     ///
     /// # Panics
     ///
     /// The right-hand side vector `b` must have compatible size.
     ///
     /// # Examples
     ///
     /// ```
     /// # #[macro_use] extern crate rulinalg;
     /// # use rulinalg::matrix::decomposition::FullPivLu;
     /// # use rulinalg::matrix::Matrix;
     /// # fn main() {
     /// let x = Matrix::identity(4);
     /// let lu = FullPivLu::decompose(x).unwrap();
     /// let b = vector![3.0, 4.0, 2.0, 1.0];
     /// let y = lu.solve(b)
     ///           .expect(&quot;Matrix is invertible.&quot;);
     /// assert_vector_eq!(y, vector![3.0, 4.0, 2.0, 1.0], comp = float);
     /// # }
     /// ```
     </span><span class="kw">pub fn </span>solve(<span class="kw-2">&amp;</span><span class="self">self</span>, b: Vector&lt;T&gt;) -&gt; <span class="prelude-ty">Result</span>&lt;Vector&lt;T&gt;, Error&gt; {
         <span class="macro">assert!</span>(b.size() == <span class="self">self</span>.lu.rows(),
             <span class="string">&quot;Right-hand side vector must have compatible size.&quot;</span>);

         <span class="kw">let </span>b = lu_forward_substitution(<span class="kw-2">&amp;</span><span class="self">self</span>.lu, <span class="kw-2">&amp;</span><span class="self">self</span>.p * b);
         back_substitution(<span class="kw-2">&amp;</span><span class="self">self</span>.lu, b).map(|x| <span class="kw-2">&amp;</span><span class="self">self</span>.q * x)
     }

     <span class="doccomment">/// Computes the inverse of the matrix which this LUP decomposition
     /// represents.
     ///
     /// # Errors
     /// The inversion might fail if the matrix is very ill-conditioned.
     /// The inversion fails if the matrix is not invertible.
     </span><span class="kw">pub fn </span>inverse(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; <span class="prelude-ty">Result</span>&lt;Matrix&lt;T&gt;, Error&gt; {
         <span class="kw">let </span>n = <span class="self">self</span>.lu.rows();
         <span class="kw">let </span><span class="kw-2">mut </span>inv = Matrix::zeros(n, n);
         <span class="kw">let </span><span class="kw-2">mut </span>e = Vector::zeros(n);

         <span class="kw">if </span>!<span class="self">self</span>.is_invertible() {
             <span class="kw">return </span><span class="prelude-val">Err</span>(
                 Error::new(
                     ErrorKind::DivByZero,
                     <span class="string">&quot;Non-invertible matrix found while attempting inversion&quot;</span>));
         }

         <span class="kw">for </span>i <span class="kw">in </span><span class="number">0 </span>.. n {
             e[i] = T::one();

             <span class="kw">let </span>col = <span class="macro">try!</span>(<span class="self">self</span>.solve(e));

             <span class="kw">for </span>j <span class="kw">in </span><span class="number">0 </span>.. n {
                 inv[[j, i]] = col[j];
             }

             e = col.apply(<span class="kw-2">&amp;</span>|<span class="kw">_</span>| T::zero());
         }

         <span class="prelude-val">Ok</span>(inv)
     }

     <span class="doccomment">/// Computes the determinant of the decomposed matrix.
     ///
     /// Empty matrices are considered to have a determinant of 1.0.
     ///
     /// # Panics
     /// If the underlying matrix is non-square.
     </span><span class="kw">pub fn </span>det(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; T {
         <span class="comment">// Recall that the determinant of a triangular matrix
         // is the product of its diagonal entries. Also,
         // the determinant of L is implicitly 1.
         </span><span class="kw">let </span>u_det = <span class="self">self</span>.lu.diag().fold(T::one(), |x, <span class="kw-2">&amp;</span>y| x * y);

         <span class="comment">// Note that the determinants of P and Q are equal to the
         // determinant of P^T and Q^T, so we don&#39;t have to invert them
         </span><span class="kw">let </span>p_det = <span class="self">self</span>.p.clone().det();
         <span class="kw">let </span>q_det = <span class="self">self</span>.q.clone().det();

         p_det * u_det * q_det
     }

     <span class="doccomment">/// Computes the rank of the decomposed matrix.
     /// # Examples
     ///
     /// ```
     /// # #[macro_use] extern crate rulinalg;
     /// # use rulinalg::matrix::decomposition::FullPivLu;
     /// # use rulinalg::matrix::Matrix;
     /// # fn main() {
     /// let x = matrix![1.0, 2.0, 3.0;
     ///                 4.0, 5.0, 6.0;
     ///                 5.0, 7.0, 9.0];
     /// let lu = FullPivLu::decompose(x).unwrap();
     /// assert_eq!(lu.rank(), 2);
     /// # }
     /// ```
     </span><span class="kw">pub fn </span>rank(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; usize {
         <span class="kw">let </span>eps = <span class="self">self</span>.epsilon();
         <span class="kw">let </span><span class="kw-2">mut </span>rank = <span class="number">0</span>;

         <span class="kw">for </span>d <span class="kw">in </span><span class="self">self</span>.lu.diag() {
             <span class="kw">if </span>d.abs() &gt; eps {
                 rank = rank + <span class="number">1</span>;
             } <span class="kw">else </span>{
                 <span class="kw">break</span>;
             }
         }

         rank
     }

     <span class="doccomment">/// Returns whether the matrix is invertible.
     ///
     /// Empty matrices are considered to be invertible for
     /// the sake of this function.
     ///
     /// # Examples
     ///
     /// ```
     /// # #[macro_use] extern crate rulinalg;
     /// # use rulinalg::matrix::decomposition::FullPivLu;
     /// # use rulinalg::matrix::Matrix;
     /// # fn main() {
     /// let x = Matrix::&lt;f64&gt;::identity(4);
     /// let lu = FullPivLu::decompose(x).unwrap();
     /// assert!(lu.is_invertible());
     ///
     /// let y = matrix![1.0, 2.0, 3.0;
     ///                 4.0, 5.0, 6.0;
     ///                 5.0, 7.0, 9.0];
     /// let lu = FullPivLu::decompose(y).unwrap();
     /// assert!(!lu.is_invertible());
     /// # }
     /// ```
     </span><span class="kw">pub fn </span>is_invertible(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; bool {
         <span class="kw">let </span>diag_size = cmp::min(<span class="self">self</span>.lu.rows(), <span class="self">self</span>.lu.cols());

         <span class="kw">if </span>diag_size &gt; <span class="number">0 </span>{
             <span class="kw">let </span>diag_last = diag_size - <span class="number">1</span>;
             <span class="kw">let </span>last =
                 <span class="kw">unsafe </span>{ <span class="self">self</span>.lu.get_unchecked([diag_last, diag_last]) };

             last.abs() &gt; <span class="self">self</span>.epsilon()
         } <span class="kw">else </span>{
             <span class="bool-val">true
         </span>}
     }

     <span class="kw">fn </span>epsilon(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; T {
         <span class="self">self</span>.lu.get([<span class="number">0</span>, <span class="number">0</span>]).unwrap_or(<span class="kw-2">&amp;</span>T::one()).abs() * T::epsilon()
     }
 }

 <span class="doccomment">/// Performs Gaussian elimination in the lower-right hand corner starting at
 /// (index, index).
 </span><span class="kw">fn </span>gaussian_elimination&lt;T: Float&gt;(lu: <span class="kw-2">&amp;mut </span>Matrix&lt;T&gt;, index: usize) {

     <span class="kw">let </span>piv_val = lu[[index, index]];

     <span class="kw">for </span>i <span class="kw">in </span>(index+<span class="number">1</span>)..lu.rows() {
         <span class="kw">let </span>mult = lu[[i, index]] / piv_val;

         lu[[i, index]] = mult;

         <span class="kw">for </span>j <span class="kw">in </span>(index+<span class="number">1</span>)..lu.cols() {
             lu[[i, j]] = lu[[i,j]] - mult<span class="kw-2">*</span>lu[[index, j]];
         }
     }
 }

 <span class="doccomment">/// Performs forward substitution using the LU matrix
 /// for which L has an implicit unit diagonal. That is,
 /// the strictly lower triangular part of LU corresponds
 /// to the strictly lower triangular part of L.
 ///
 /// This is equivalent to solving the system Lx = b.
 </span><span class="kw">fn </span>lu_forward_substitution&lt;T: Float&gt;(lu: <span class="kw-2">&amp;</span>Matrix&lt;T&gt;, b: Vector&lt;T&gt;) -&gt; Vector&lt;T&gt; {
     <span class="macro">assert!</span>(lu.rows() == lu.cols(), <span class="string">&quot;LU matrix must be square.&quot;</span>);
     <span class="macro">assert!</span>(b.size() == lu.rows(), <span class="string">&quot;LU matrix and RHS vector must be compatible.&quot;</span>);
     <span class="kw">let </span><span class="kw-2">mut </span>x = b;

     <span class="kw">for </span>(i, row) <span class="kw">in </span>lu.row_iter().enumerate().skip(<span class="number">1</span>) {
         <span class="comment">// Note that at time of writing we need raw_slice here for
         // auto-vectorization to kick in
         </span><span class="kw">let </span>adjustment = row.raw_slice()
                             .iter()
                             .take(i)
                             .cloned()
                             .zip(x.iter().cloned())
                             .fold(T::zero(), |sum, (l, x)| sum + l * x);

         x[i] = x[i] - adjustment;
     }
     x
 }

 <span class="kw">fn </span>unit_lower_triangular_part&lt;T, M&gt;(matrix: <span class="kw-2">&amp;</span>M) -&gt; Matrix&lt;T&gt;
     <span class="kw">where </span>T: Zero + One + Clone, M: BaseMatrix&lt;T&gt; {

     <span class="kw">let </span>m = matrix.rows();
     <span class="kw">let </span><span class="kw-2">mut </span>data = Vec::&lt;T&gt;::with_capacity(m * m);

     <span class="kw">for </span>(i, row) <span class="kw">in </span>matrix.row_iter().enumerate() {
         <span class="kw">for </span>element <span class="kw">in </span>row.iter().take(i).cloned() {
             data.push(element);
         }

         data.push(T::one());

         <span class="kw">for _ in </span>(i + <span class="number">1</span>) .. m {
             data.push(T::zero());
         }
     }

     Matrix::new(m, m, data)
 }


 <span class="kw">impl</span>&lt;T&gt; Matrix&lt;T&gt; <span class="kw">where </span>T: Any + Float
 {
     <span class="doccomment">/// Computes L, U, and P for LUP decomposition.
     ///
     /// Returns L,U, and P respectively.
     ///
     /// This function is deprecated.
     /// Please see [PartialPivLu](decomposition/struct.PartialPivLu.html)
     /// for a replacement.
     ///
     /// # Examples
     ///
     /// ```
     /// # #[macro_use] extern crate rulinalg; fn main() {
     /// use rulinalg::matrix::Matrix;
     ///
     /// let a = matrix![1.0, 2.0, 0.0;
     ///                 0.0, 3.0, 4.0;
     ///                 5.0, 1.0, 2.0];
     ///
     /// let (l, u, p) = a.lup_decomp().expect(&quot;This matrix should decompose!&quot;);
     /// # }
     /// ```
     ///
     /// # Panics
     ///
     /// - Matrix is not square.
     ///
     /// # Failures
     ///
     /// - Matrix cannot be LUP decomposed.
     </span><span class="attribute">#[deprecated]
     </span><span class="kw">pub fn </span>lup_decomp(<span class="self">self</span>) -&gt; <span class="prelude-ty">Result</span>&lt;(Matrix&lt;T&gt;, Matrix&lt;T&gt;, Matrix&lt;T&gt;), Error&gt; {
         <span class="kw">let </span>n = <span class="self">self</span>.cols;
         <span class="macro">assert!</span>(<span class="self">self</span>.rows == n, <span class="string">&quot;Matrix must be square for LUP decomposition.&quot;</span>);
         <span class="kw">let </span><span class="kw-2">mut </span>l = Matrix::&lt;T&gt;::zeros(n, n);
         <span class="kw">let </span><span class="kw-2">mut </span>u = <span class="self">self</span>;
         <span class="kw">let </span><span class="kw-2">mut </span>p = Matrix::&lt;T&gt;::identity(n);

         <span class="kw">for </span>index <span class="kw">in </span><span class="number">0</span>..n {
             <span class="kw">let </span><span class="kw-2">mut </span>curr_max_idx = index;
             <span class="kw">let </span><span class="kw-2">mut </span>curr_max = u[[curr_max_idx, curr_max_idx]];

             <span class="kw">for </span>i <span class="kw">in </span>(curr_max_idx+<span class="number">1</span>)..n {
                 <span class="kw">if </span>u[[i, index]].abs() &gt; curr_max.abs() {
                     curr_max = u[[i, index]];
                     curr_max_idx = i;
                 }
             }
             <span class="kw">if </span>curr_max.abs() &lt; T::epsilon() {
                 <span class="kw">return </span><span class="prelude-val">Err</span>(Error::new(ErrorKind::DivByZero,
                     <span class="string">&quot;Singular matrix found in LUP decomposition. \
                     A value in the diagonal of U == 0.0.&quot;</span>));
             }

             <span class="kw">if </span>curr_max_idx != index {
                 l.swap_rows(index, curr_max_idx);
                 u.swap_rows(index, curr_max_idx);
                 p.swap_rows(index, curr_max_idx);
             }
             l[[index, index]] = T::one();
             <span class="kw">for </span>i <span class="kw">in </span>(index+<span class="number">1</span>)..n {
                 <span class="kw">let </span>mult = u[[i, index]]/curr_max;
                 l[[i, index]] = mult;
                 u[[i, index]] = T::zero();
                 <span class="kw">for </span>j <span class="kw">in </span>(index+<span class="number">1</span>)..n {
                     u[[i, j]] = u[[i,j]] - mult<span class="kw-2">*</span>u[[index, j]];
                 }
             }
         }
         <span class="prelude-val">Ok</span>((l, u, p))
     }
 }

 <span class="attribute">#[cfg(test)]
 </span><span class="kw">mod </span>tests {
     <span class="kw">use </span>matrix::{Matrix, PermutationMatrix};
     <span class="kw">use </span>testsupport::{is_lower_triangular, is_upper_triangular};

     <span class="kw">use super</span>::{PartialPivLu, LUP, FullPivLu, LUPQ};
     <span class="kw">use </span>matrix::decomposition::Decomposition;

     <span class="attribute">#[allow(deprecated)]
     #[test]
     #[should_panic]
     </span><span class="kw">fn </span>test_non_square_lup_decomp() {
         <span class="kw">let </span>a: Matrix&lt;f64&gt; = Matrix::ones(<span class="number">2</span>, <span class="number">3</span>);

         <span class="kw">let _ </span>= a.lup_decomp();
     }

     <span class="attribute">#[allow(deprecated)]
     #[test]
     </span><span class="kw">fn </span>test_lup_decomp() {
         <span class="kw">use </span>error::ErrorKind;
         <span class="kw">let </span>a: Matrix&lt;f64&gt; = <span class="macro">matrix!</span>[
             <span class="number">1.</span>, <span class="number">2.</span>, <span class="number">3.</span>, <span class="number">4.</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">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="kw">match </span>a.lup_decomp() {
             <span class="prelude-val">Err</span>(e) =&gt; <span class="macro">assert!</span>(<span class="kw-2">*</span>e.kind() == ErrorKind::DivByZero),
             <span class="prelude-val">Ok</span>(<span class="kw">_</span>) =&gt; <span class="macro">panic!</span>()
         }
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>partial_piv_lu_decompose_arbitrary() {
         <span class="comment">// Since the LUP decomposition is not in general unique,
         // we can not test against factors directly, but
         // instead we must rely on the fact that the
         // matrices P, L and U together construct the
         // original matrix
         </span><span class="kw">let </span>x = <span class="macro">matrix!</span>[ -<span class="number">3.0</span>,   <span class="number">0.0</span>,   <span class="number">4.0</span>,   <span class="number">1.0</span>;
                         -<span class="number">12.0</span>,   <span class="number">5.0</span>,  <span class="number">17.0</span>,   <span class="number">1.0</span>;
                          <span class="number">15.0</span>,   <span class="number">0.0</span>, -<span class="number">18.0</span>,  -<span class="number">5.0</span>;
                           <span class="number">6.0</span>,  <span class="number">20.0</span>, -<span class="number">10.0</span>, -<span class="number">15.0 </span>];

         <span class="kw">let </span>LUP { l, u, p } = PartialPivLu::decompose(x.clone())
                                            .unwrap()
                                            .unpack();
         <span class="kw">let </span>y = p.inverse() * <span class="kw-2">&amp;</span>l * <span class="kw-2">&amp;</span>u;
         <span class="macro">assert_matrix_eq!</span>(x, y, comp = float);
         <span class="macro">assert!</span>(is_lower_triangular(<span class="kw-2">&amp;</span>l));
         <span class="macro">assert!</span>(is_upper_triangular(<span class="kw-2">&amp;</span>u));
     }

     <span class="attribute">#[test]
     </span><span class="kw">pub fn </span>partial_piv_lu_inverse_identity() {
         <span class="kw">let </span>lu = PartialPivLu::&lt;f64&gt; {
             lu: Matrix::identity(<span class="number">3</span>),
             p: PermutationMatrix::identity(<span class="number">3</span>)
         };

         <span class="kw">let </span>inv = lu.inverse().expect(<span class="string">&quot;Matrix is invertible.&quot;</span>);

         <span class="macro">assert_matrix_eq!</span>(inv, Matrix::identity(<span class="number">3</span>), comp = float);
     }

     <span class="attribute">#[test]
     </span><span class="kw">pub fn </span>partial_piv_lu_inverse_arbitrary_invertible_matrix() {
         <span class="kw">let </span>x = <span class="macro">matrix!</span>[<span class="number">5.0</span>, <span class="number">0.0</span>, <span class="number">0.0</span>, <span class="number">1.0</span>;
                         <span class="number">2.0</span>, <span class="number">2.0</span>, <span class="number">2.0</span>, <span class="number">1.0</span>;
                         <span class="number">4.0</span>, <span class="number">5.0</span>, <span class="number">5.0</span>, <span class="number">5.0</span>;
                         <span class="number">1.0</span>, <span class="number">6.0</span>, <span class="number">4.0</span>, <span class="number">5.0</span>];

         <span class="kw">let </span>inv = <span class="macro">matrix!</span>[<span class="number">1.85185185185185203e-01</span>,   <span class="number">1.85185185185185175e-01</span>, -<span class="number">7.40740740740740561e-02</span>, -<span class="number">1.02798428206033007e-17</span>;
                           <span class="number">1.66666666666666630e-01</span>,   <span class="number">6.66666666666666519e-01</span>, -<span class="number">6.66666666666666519e-01</span>,  <span class="number">4.99999999999999833e-01</span>;
                          -<span class="number">3.88888888888888840e-01</span>,   <span class="number">1.11111111111111174e-01</span>,  <span class="number">5.55555555555555358e-01</span>, -<span class="number">4.99999999999999833e-01</span>;
                           <span class="number">7.40740740740740838e-02</span>,  -<span class="number">9.25925925925925819e-01</span>,  <span class="number">3.70370370370370294e-01</span>,  <span class="number">5.13992141030165006e-17</span>];

         <span class="kw">let </span>lu = PartialPivLu::decompose(x).unwrap();

         <span class="macro">assert_matrix_eq!</span>(lu.inverse().unwrap(), inv, comp = float);
     }

     <span class="attribute">#[test]
     </span><span class="kw">pub fn </span>partial_piv_lu_det_identity() {
         <span class="kw">let </span>lu = PartialPivLu::&lt;f64&gt; {
             lu: Matrix::identity(<span class="number">3</span>),
             p: PermutationMatrix::identity(<span class="number">3</span>)
         };

         <span class="macro">assert_eq!</span>(lu.det(), <span class="number">1.0</span>);
     }

     <span class="attribute">#[test]
     </span><span class="kw">pub fn </span>partial_piv_lu_det_arbitrary_invertible_matrix() {
         <span class="kw">let </span>x = <span class="macro">matrix!</span>[ <span class="number">5.0</span>,  <span class="number">0.0</span>,  <span class="number">0.0</span>,  <span class="number">1.0</span>;
                          <span class="number">0.0</span>,  <span class="number">2.0</span>,  <span class="number">2.0</span>,  <span class="number">1.0</span>;
                         <span class="number">15.0</span>,  <span class="number">4.0</span>,  <span class="number">7.0</span>, <span class="number">10.0</span>;
                          <span class="number">5.0</span>,  <span class="number">2.0</span>, <span class="number">17.0</span>, <span class="number">32.0</span>];

         <span class="kw">let </span>lu = PartialPivLu::decompose(x).unwrap();

         <span class="kw">let </span>expected_det = <span class="number">149.99999999999997</span>;
         <span class="macro">assert_scalar_eq!</span>(lu.det(), expected_det, comp = float);
     }

     <span class="attribute">#[test]
     </span><span class="kw">pub fn </span>partial_piv_lu_solve_arbitrary_matrix() {
         <span class="kw">let </span>x = <span class="macro">matrix!</span>[ <span class="number">5.0</span>, <span class="number">0.0</span>, <span class="number">0.0</span>, <span class="number">1.0</span>;
                          <span class="number">2.0</span>, <span class="number">2.0</span>, <span class="number">2.0</span>, <span class="number">1.0</span>;
                          <span class="number">4.0</span>, <span class="number">5.0</span>, <span class="number">5.0</span>, <span class="number">5.0</span>;
                          <span class="number">1.0</span>, <span class="number">6.0</span>, <span class="number">4.0</span>, <span class="number">5.0 </span>];
         <span class="kw">let </span>b = <span class="macro">vector!</span>[<span class="number">9.0</span>, <span class="number">16.0</span>, <span class="number">49.0</span>, <span class="number">45.0</span>];
         <span class="kw">let </span>expected = <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>lu = PartialPivLu::decompose(x).unwrap();
         <span class="kw">let </span>y = lu.solve(b).unwrap();
         <span class="comment">// Need to up the tolerance to take into account
         // numerical error. Ideally there&#39;d be a more systematic
         // way to test this.
         </span><span class="macro">assert_vector_eq!</span>(y, expected, comp = ulp, tol = <span class="number">100</span>);
     }

     <span class="attribute">#[test]
     </span><span class="kw">pub fn </span>lu_forward_substitution() {
         <span class="kw">use </span><span class="kw">super</span>::lu_forward_substitution;

         {
             <span class="kw">let </span>lu: Matrix&lt;f64&gt; = <span class="macro">matrix!</span>[];
             <span class="kw">let </span>b = <span class="macro">vector!</span>[];
             <span class="kw">let </span>x = lu_forward_substitution(<span class="kw-2">&amp;</span>lu, b);
             <span class="macro">assert!</span>(x.size() == <span class="number">0</span>);
         }

         {
             <span class="kw">let </span>lu = <span class="macro">matrix!</span>[<span class="number">3.0</span>];
             <span class="kw">let </span>b = <span class="macro">vector!</span>[<span class="number">1.0</span>];
             <span class="kw">let </span>x = lu_forward_substitution(<span class="kw-2">&amp;</span>lu, b);
             <span class="macro">assert_eq!</span>(x, <span class="macro">vector!</span>[<span class="number">1.0</span>]);
         }

         {
             <span class="kw">let </span>lu = <span class="macro">matrix!</span>[<span class="number">3.0</span>, <span class="number">2.0</span>;
                              <span class="number">2.0</span>, <span class="number">2.0</span>];
             <span class="kw">let </span>b = <span class="macro">vector!</span>[<span class="number">1.0</span>, <span class="number">2.0</span>];
             <span class="kw">let </span>x = lu_forward_substitution(<span class="kw-2">&amp;</span>lu, b);
             <span class="macro">assert_eq!</span>(x, <span class="macro">vector!</span>[<span class="number">1.0</span>, <span class="number">0.0</span>]);
         }
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>full_piv_lu_decompose_arbitrary() {
         <span class="comment">// Since the LUP decomposition is not in general unique,
         // we can not test against factors directly, but
         // instead we must rely on the fact that the
         // matrices P, L and U together construct the
         // original matrix
         </span><span class="kw">let </span>x = <span class="macro">matrix!</span>[ -<span class="number">3.0</span>,   <span class="number">0.0</span>,   <span class="number">4.0</span>,   <span class="number">1.0</span>;
                         -<span class="number">12.0</span>,   <span class="number">5.0</span>,  <span class="number">17.0</span>,   <span class="number">1.0</span>;
                          <span class="number">15.0</span>,   <span class="number">0.0</span>, -<span class="number">18.0</span>,  -<span class="number">5.0</span>;
                           <span class="number">6.0</span>,  <span class="number">20.0</span>, -<span class="number">10.0</span>, -<span class="number">15.0 </span>];

         <span class="kw">let </span>LUPQ { l, u, p, q } = FullPivLu::decompose(x.clone())
                                            .unwrap()
                                            .unpack();

         <span class="kw">let </span>y = p.inverse() * <span class="kw-2">&amp;</span>l * <span class="kw-2">&amp;</span>u * q.inverse();

         <span class="macro">assert_matrix_eq!</span>(x, y, comp = float);
         <span class="macro">assert!</span>(is_lower_triangular(<span class="kw-2">&amp;</span>l));
         <span class="macro">assert!</span>(is_upper_triangular(<span class="kw-2">&amp;</span>u));
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>full_piv_lu_decompose_singular() {
         <span class="kw">let </span>x = <span class="macro">matrix!</span>[ -<span class="number">3.0</span>,   <span class="number">0.0</span>,   <span class="number">4.0</span>,   <span class="number">1.0</span>;
                         -<span class="number">12.0</span>,   <span class="number">5.0</span>,  <span class="number">17.0</span>,   <span class="number">1.0</span>;
                          <span class="number">15.0</span>,   <span class="number">0.0</span>, -<span class="number">18.0</span>,  -<span class="number">5.0</span>;
                          -<span class="number">6.0</span>,   <span class="number">0.0</span>,   <span class="number">8.0</span>,   <span class="number">2.0 </span>];

         <span class="kw">let </span>lu = FullPivLu::decompose(x.clone()).unwrap();

         <span class="macro">assert_eq!</span>(lu.rank(), <span class="number">3</span>);

         <span class="kw">let </span>LUPQ { l, u, p, q } = lu.unpack();

         <span class="kw">let </span>y = p.inverse() * <span class="kw-2">&amp;</span>l * <span class="kw-2">&amp;</span>u * q.inverse();

         <span class="macro">assert_matrix_eq!</span>(x, y, comp = float);
         <span class="macro">assert!</span>(is_lower_triangular(<span class="kw-2">&amp;</span>l));
         <span class="macro">assert!</span>(is_upper_triangular(<span class="kw-2">&amp;</span>u));
     }

     <span class="attribute">#[test]
     #[should_panic]
     </span><span class="kw">fn </span>full_piv_lu_decompose_rectangular() {
         <span class="kw">let </span>x = <span class="macro">matrix!</span>[ -<span class="number">3.0</span>,   <span class="number">0.0</span>,   <span class="number">4.0</span>;
                         -<span class="number">12.0</span>,   <span class="number">5.0</span>,  <span class="number">17.0</span>;
                          <span class="number">15.0</span>,   <span class="number">0.0</span>, -<span class="number">18.0</span>;
                          -<span class="number">6.0</span>,   <span class="number">0.0</span>,   <span class="number">20.0</span>];

         FullPivLu::decompose(x.clone()).unwrap();
     }

     <span class="attribute">#[test]
     </span><span class="kw">pub fn </span>full_piv_lu_solve_arbitrary_matrix() {
         <span class="kw">let </span>x = <span class="macro">matrix!</span>[ <span class="number">5.0</span>, <span class="number">0.0</span>, <span class="number">0.0</span>, <span class="number">1.0</span>;
                          <span class="number">2.0</span>, <span class="number">2.0</span>, <span class="number">2.0</span>, <span class="number">1.0</span>;
                          <span class="number">4.0</span>, <span class="number">5.0</span>, <span class="number">5.0</span>, <span class="number">5.0</span>;
                          <span class="number">1.0</span>, <span class="number">6.0</span>, <span class="number">4.0</span>, <span class="number">5.0 </span>];
         <span class="kw">let </span>b = <span class="macro">vector!</span>[<span class="number">9.0</span>, <span class="number">16.0</span>, <span class="number">49.0</span>, <span class="number">45.0</span>];
         <span class="kw">let </span>expected = <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>lu = FullPivLu::decompose(x).unwrap();
         <span class="kw">let </span>y = lu.solve(b).unwrap();

         <span class="comment">// Need to up the tolerance to take into account
         // numerical error. Ideally there&#39;d be a more systematic
         // way to test this.
         </span><span class="macro">assert_vector_eq!</span>(y, expected, comp = ulp, tol = <span class="number">100</span>);
     }

     <span class="attribute">#[test]
     </span><span class="kw">pub fn </span>full_piv_lu_inverse_arbitrary_invertible_matrix() {
         <span class="kw">let </span>x = <span class="macro">matrix!</span>[<span class="number">5.0</span>, <span class="number">0.0</span>, <span class="number">0.0</span>, <span class="number">1.0</span>;
                         <span class="number">2.0</span>, <span class="number">2.0</span>, <span class="number">2.0</span>, <span class="number">1.0</span>;
                         <span class="number">4.0</span>, <span class="number">5.0</span>, <span class="number">5.0</span>, <span class="number">5.0</span>;
                         <span class="number">1.0</span>, <span class="number">6.0</span>, <span class="number">4.0</span>, <span class="number">5.0</span>];

         <span class="kw">let </span>inv = <span class="macro">matrix!</span>[<span class="number">1.85185185185185203e-01</span>,   <span class="number">1.85185185185185175e-01</span>, -<span class="number">7.40740740740740561e-02</span>, -<span class="number">1.02798428206033007e-17</span>;
                           <span class="number">1.66666666666666630e-01</span>,   <span class="number">6.66666666666666519e-01</span>, -<span class="number">6.66666666666666519e-01</span>,  <span class="number">4.99999999999999833e-01</span>;
                          -<span class="number">3.88888888888888840e-01</span>,   <span class="number">1.11111111111111174e-01</span>,  <span class="number">5.55555555555555358e-01</span>, -<span class="number">4.99999999999999833e-01</span>;
                           <span class="number">7.40740740740740838e-02</span>,  -<span class="number">9.25925925925925819e-01</span>,  <span class="number">3.70370370370370294e-01</span>,  <span class="number">5.13992141030165006e-17</span>];

         <span class="kw">let </span>lu = FullPivLu::decompose(x).unwrap();

         <span class="macro">assert_matrix_eq!</span>(lu.inverse().unwrap(), inv, comp = float);
     }

     <span class="attribute">#[test]
     </span><span class="kw">pub fn </span>full_piv_lu_inverse_noninvertible() {
         <span class="kw">let </span>x = <span class="macro">matrix!</span>[<span class="number">5.0</span>, <span class="number">0.0</span>, <span class="number">1.0</span>;
                         <span class="number">4.0</span>, <span class="number">5.0</span>, <span class="number">5.0</span>;
                         <span class="number">9.0</span>, <span class="number">5.0</span>, <span class="number">6.0</span>];

         <span class="kw">let </span>lu = FullPivLu::decompose(x).unwrap();

         <span class="macro">assert!</span>(lu.inverse().is_err());
     }

     <span class="attribute">#[test]
     </span><span class="kw">pub fn </span>full_piv_lu_empty_matrix() {
         <span class="kw">use </span>matrix::base::BaseMatrix;

         <span class="kw">let </span>x = Matrix::from_fn(<span class="number">0</span>, <span class="number">0</span>, |<span class="kw">_</span>, <span class="kw">_</span>| <span class="number">0.0</span>);
         <span class="macro">assert_eq!</span>(x.rows(), <span class="number">0</span>);
         <span class="macro">assert_eq!</span>(x.cols(), <span class="number">0</span>);

         <span class="kw">let </span>lu = FullPivLu::decompose(x).unwrap();

         <span class="macro">assert!</span>(lu.is_invertible());
         <span class="macro">assert_eq!</span>(lu.rank(), <span class="number">0</span>);
         <span class="macro">assert_eq!</span>(lu.det(), <span class="number">1.0</span>);

         <span class="kw">let </span>inverse = lu.inverse().unwrap();
         <span class="macro">assert_eq!</span>(inverse.rows(), <span class="number">0</span>);
         <span class="macro">assert_eq!</span>(inverse.cols(), <span class="number">0</span>);

         <span class="kw">let </span>LUPQ { l, u, p, q } = lu.unpack();
         <span class="macro">assert_eq!</span>(l.rows(), <span class="number">0</span>);
         <span class="macro">assert_eq!</span>(l.cols(), <span class="number">0</span>);

         <span class="macro">assert_eq!</span>(u.rows(), <span class="number">0</span>);
         <span class="macro">assert_eq!</span>(u.cols(), <span class="number">0</span>);

         <span class="macro">assert_eq!</span>(p.size(), <span class="number">0</span>);
         <span class="macro">assert_eq!</span>(q.size(), <span class="number">0</span>);
     }
 }
 </code></pre></div>
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