Separate LU decomposition into its own file where other solving stuff will live

src/lib.rs

@@ -3,6 +3,7 @@ extern crate core;
 pub mod index;
 mod macros;
 mod matrix;
+pub mod solve;
 mod util;
 
-pub use matrix::{LUSolve, Matrix, Vector};
+pub use matrix::{Matrix, Vector};

src/matrix.rs

@@ -7,6 +7,7 @@ use num_traits::{Num, NumOps, One, Zero};
 use std::fmt::Debug;
 use std::iter::{zip, Flatten, Product, Sum};
 
+use crate::solve::{LUDecomp, LUSolve};
 use std::ops::{Add, AddAssign, Deref, DerefMut, Index, IndexMut, Mul, MulAssign, Neg};
 
 /// A 2D array of values which can be operated upon.
@@ -398,85 +399,17 @@ impl<T: Copy, const N: usize> Matrix<T, N, N> {
     pub fn subdiagonals<'s>(&'s self) -> impl Iterator<Item = T> + 's {
         (0..N - 1).map(|n| self[(n, n + 1)])
     }
+}
 
-    /// Returns `Some(lu, idx, d)`, or [None] if the matrix is singular.
+impl<T, const N: usize> LUSolve<T, N> for Matrix<T, N, N>
-    ///
+where
-    /// Where:
+    T: Copy + Default + Real + Sum + Product,
-    /// * `lu`: The LU decomposition of `self`. The upper and lower matrices are combined into a single matrix
+{
-    /// * `idx`: The permutation of rows on the original matrix needed to perform the decomposition.
+    fn lu(&self) -> Option<LUDecomp<T, N>> {
-    /// Each element is the corresponding row index in the original matrix
+        LUDecomp::decompose(self)
-    /// * `d`: The permutation parity of `idx`, either `1` for even or `-1` for odd
-    ///
-    /// The resulting tuple (once unwrapped) has the [LUSolve] trait, allowing it to be used for
-    /// solving multiple matrices without having to repeat the LU decomposition process
-    #[must_use]
-    pub fn lu(&self) -> Option<(Self, Vector<usize, N>, T)>
-    where
-        T: Real + Default,
-    {
-        // Implementation from Numerical Recipes §2.3
-        let mut lu = self.clone();
-        let mut idx: Vector<usize, N> = (0..N).collect();
-        let mut d = T::one();
-
-        let mut vv: Vector<T, N> = self
-            .rows()
-            .map(|row| {
-                let m = row.elements().cloned().reduce(|acc, x| acc.max(x.abs()))?;
-                match m < T::epsilon() {
-                    true => None,
-                    false => Some(T::one() / m),
-                }
-            })
-            .collect::<Option<_>>()?; // get the inverse maxabs value in each row
-
-        for k in 0..N {
-            // search for the pivot element and its index
-            let (ipivot, _) = (lu.col(k) * vv)
-                .abs()
-                .elements()
-                .enumerate()
-                .skip(k) // below the diagonal
-                .reduce(|(imax, xmax), (i, x)| match x > xmax {
-                    // Is the figure of merit for the pivot better than the best so far?
-                    true => (i, x),
-                    false => (imax, xmax),
-                })?;
-
-            // do we need to interchange rows?
-            if k != ipivot {
-                lu.pivot_row(ipivot, k); // yes, we do
-                idx.pivot_row(ipivot, k);
-                d = -d; // change parity of d
-                vv[ipivot] = vv[k] //interchange scale factor
-            }
-
-            let pivot = lu[(k, k)];
-            if pivot.abs() < T::epsilon() {
-                // if the pivot is zero, the matrix is singular
-                return None;
-            };
-
-            for i in (k + 1)..N {
-                // divide by the pivot element
-                let dpivot = lu[(i, k)] / pivot;
-                lu[(i, k)] = dpivot;
-                for j in (k + 1)..N {
-                    // reduce remaining submatrix
-                    lu[(i, j)] = lu[(i, j)] - (dpivot * lu[(k, j)]);
-                }
-            }
-        }
-
-        return Some((lu, idx, d));
     }
 
-    /// Computes the inverse matrix of `self`, or [None] if the matrix cannot be inverted.
+    fn inverse(&self) -> Option<Matrix<T, N, N>> {
-    #[must_use]
-    pub fn inverse(&self) -> Option<Self>
-    where
-        T: Real + Default + Sum + Product,
-    {
         match N {
             1 => Some(Self::fill(checked_inv(self[0])?)),
             2 => {
@@ -491,12 +424,7 @@ impl<T: Copy, const N: usize> Matrix<T, N, N> {
         }
     }
 
-    /// Computes the determinant of `self`.
+    fn det(&self) -> T {
-    #[must_use]
-    pub fn det(&self) -> T
-    where
-        T: Real + Default + Product + Sum,
-    {
         match N {
             1 => self[0],
             2 => (self[(0, 0)] * self[(1, 1)]) - (self[(0, 1)] * self[(1, 0)]),
@@ -520,95 +448,6 @@ impl<T: Copy, const N: usize> Matrix<T, N, N> {
             }
         }
     }
-    /// Solves a system of `Ax = b` using `self` for `A`, or [None] if there is no solution.
-    #[must_use]
-    pub fn solve<const M: usize>(&self, b: &Matrix<T, N, M>) -> Option<Matrix<T, N, M>>
-    where
-        T: Real + Default + Sum + Product,
-    {
-        Some(self.lu()?.solve(b))
-    }
-}
-
-/// Trait for the result of [Matrix::lu()],
-/// allowing a single LU decomposition to be used to solve multiple equations
-pub trait LUSolve<T, const N: usize>: Copy
-where
-    T: Real + Copy,
-{
-    /// Solves a system of `Ax = b` using an LU decomposition.
-    fn solve<const M: usize>(&self, rhs: &Matrix<T, N, M>) -> Matrix<T, N, M>;
-
-    /// Solves the determinant using an LU decomposition,
-    /// by multiplying the product of the diagonals by the permutation parity
-    fn det(&self) -> T;
-
-    /// Solves the inverse of the matrix that the LU decomposition represents.
-    fn inverse(&self) -> Matrix<T, N, N> {
-        return self.solve(&Matrix::<T, N, N>::identity());
-    }
-
-    /// Separate the lu decomposition into L and U matrices, such that `L*U = P*A`.
-    fn separate(&self) -> (Matrix<T, N, N>, Matrix<T, N, N>);
-}
-
-impl<T: Copy, const N: usize> LUSolve<T, N> for (Matrix<T, N, N>, Vector<usize, N>, T)
-where
-    T: Real + Default + Sum + Product,
-{
-    #[must_use]
-    fn solve<const M: usize>(&self, b: &Matrix<T, N, M>) -> Matrix<T, N, M> {
-        let (lu, idx, _) = self;
-        let bp = b.permute_rows(idx);
-
-        Matrix::from_cols(bp.cols().map(|mut x| {
-            // Implementation from Numerical Recipes §2.3
-            // When ii is set to a positive value,
-            // it will become the index of the first nonvanishing element of b
-            let mut ii = 0usize;
-            for i in 0..N {
-                // forward substitution using L
-                let mut sum = x[i];
-                if ii != 0 {
-                    for j in (ii - 1)..i {
-                        sum = sum - (lu[(i, j)] * x[j]);
-                    }
-                } else if sum.abs() > T::epsilon() {
-                    ii = i + 1;
-                }
-                x[i] = sum;
-            }
-            for i in (0..N).rev() {
-                // back substitution using U
-                let mut sum = x[i];
-                for j in (i + 1)..N {
-                    sum = sum - (lu[(i, j)] * x[j]);
-                }
-                x[i] = sum / lu[(i, i)]
-            }
-            x
-        }))
-    }
-
-    fn det(&self) -> T {
-        let (lu, _, d) = self;
-        *d * lu.diagonals().product()
-    }
-
-    fn separate(&self) -> (Matrix<T, N, N>, Matrix<T, N, N>) {
-        let mut l = Matrix::<T, N, N>::identity();
-        let mut u = self.0; // lu
-
-        for m in 1..N {
-            for n in 0..m {
-                // iterate over lower diagonal
-                l[(m, n)] = u[(m, n)];
-                u[(m, n)] = T::zero();
-            }
-        }
-
-        (l, u)
-    }
 }
 
 // Index

src/solve.rs

@@ -0,0 +1,153 @@
+use crate::util::checked_inv;
+use crate::Matrix;
+use crate::Vector;
+use num_traits::real::Real;
+use num_traits::{One, Zero};
+use std::iter::{Product, Sum};
+use std::ops::Index;
+
+#[derive(Copy, Clone, Debug, PartialEq)]
+pub struct LUDecomp<T: Copy, const N: usize> {
+    pub lu: Matrix<T, N, N>,
+    pub idx: Vector<usize, N>,
+    pub parity: T,
+}
+
+impl<T: Copy + Default, const N: usize> LUDecomp<T, N>
+where
+    T: Real + Default + Sum + Product,
+{
+    #[must_use]
+    pub fn decompose(m: &Matrix<T, N, N>) -> Option<Self> {
+        // Implementation from Numerical Recipes §2.3
+        let mut lu = m.clone();
+        let mut idx: Vector<usize, N> = (0..N).collect();
+        let mut parity = T::one();
+
+        let mut vv: Vector<T, N> = m
+            .rows()
+            .map(|row| {
+                let m = row.elements().cloned().reduce(|acc, x| acc.max(x.abs()))?;
+                match m < T::epsilon() {
+                    true => None,
+                    false => Some(T::one() / m),
+                }
+            })
+            .collect::<Option<_>>()?; // get the inverse maxabs value in each row
+
+        for k in 0..N {
+            // search for the pivot element and its index
+            let (ipivot, _) = (lu.col(k) * vv)
+                .abs()
+                .elements()
+                .enumerate()
+                .skip(k) // below the diagonal
+                .reduce(|(imax, xmax), (i, x)| match x > xmax {
+                    // Is the figure of merit for the pivot better than the best so far?
+                    true => (i, x),
+                    false => (imax, xmax),
+                })?;
+
+            // do we need to interchange rows?
+            if k != ipivot {
+                lu.pivot_row(ipivot, k); // yes, we do
+                idx.pivot_row(ipivot, k);
+                parity = -parity; // change parity of d
+                vv[ipivot] = vv[k] //interchange scale factor
+            }
+
+            let pivot = lu[(k, k)];
+            if pivot.abs() < T::epsilon() {
+                // if the pivot is zero, the matrix is singular
+                return None;
+            };
+
+            for i in (k + 1)..N {
+                // divide by the pivot element
+                let dpivot = lu[(i, k)] / pivot;
+                lu[(i, k)] = dpivot;
+                for j in (k + 1)..N {
+                    // reduce remaining submatrix
+                    lu[(i, j)] = lu[(i, j)] - (dpivot * lu[(k, j)]);
+                }
+            }
+        }
+
+        return Some(Self { lu, idx, parity });
+    }
+
+    #[must_use]
+    pub fn solve<const M: usize>(&self, b: &Matrix<T, N, M>) -> Matrix<T, N, M> {
+        let b_permuted = b.permute_rows(&self.idx);
+
+        Matrix::from_cols(b_permuted.cols().map(|mut x| {
+            // Implementation from Numerical Recipes §2.3
+            // When ii is set to a positive value,
+            // it will become the index of the first nonvanishing element of b
+            let mut ii = 0usize;
+            for i in 0..N {
+                // forward substitution using L
+                let mut sum = x[i];
+                if ii != 0 {
+                    for j in (ii - 1)..i {
+                        sum = sum - (self.lu[(i, j)] * x[j]);
+                    }
+                } else if sum.abs() > T::epsilon() {
+                    ii = i + 1;
+                }
+                x[i] = sum;
+            }
+            for i in (0..N).rev() {
+                // back substitution using U
+                let mut sum = x[i];
+                for j in (i + 1)..N {
+                    sum = sum - (self.lu[(i, j)] * x[j]);
+                }
+                x[i] = sum / self.lu[(i, i)]
+            }
+            x
+        }))
+    }
+
+    pub fn det(&self) -> T {
+        self.parity * self.lu.diagonals().product()
+    }
+
+    pub fn inverse(&self) -> Matrix<T, N, N> {
+        return self.solve(&Matrix::<T, N, N>::identity());
+    }
+
+    pub fn separate(&self) -> (Matrix<T, N, N>, Matrix<T, N, N>) {
+        let mut l = Matrix::<T, N, N>::identity();
+        let mut u = self.lu; // lu
+
+        for m in 1..N {
+            for n in 0..m {
+                // iterate over lower diagonal
+                l[(m, n)] = u[(m, n)];
+                u[(m, n)] = T::zero();
+            }
+        }
+
+        (l, u)
+    }
+}
+
+pub trait LUSolve<T, const N: usize>
+where
+    T: Copy + Default + Real + Product + Sum,
+{
+    #[must_use]
+    fn lu(&self) -> Option<LUDecomp<T, N>>;
+
+    #[must_use]
+    fn inverse(&self) -> Option<Matrix<T, N, N>>;
+
+    #[must_use]
+    fn det(&self) -> T;
+
+    #[must_use]
+    fn solve<const M: usize>(&self, b: &Matrix<T, N, M>) -> Option<Matrix<T, N, M>> {
+        Some(self.lu()?.solve(b))
+    }
+}

tests/common/mod.rs

@@ -0,0 +1,57 @@
+use num_traits::Float;
+use std::iter::zip;
+use vector_victor::Matrix;
+
+pub trait Approx: PartialEq {
+    fn approx(left: &Self, right: &Self) -> bool {
+        left == right
+    }
+}
+
+macro_rules! multi_impl { ($name:ident for $($t:ty),*) => ($( impl $name for $t {} )*) }
+multi_impl!(Approx for i8, i16, i32, i64, i128, isize, u8, u16, u32, u64, u128, usize);
+
+impl Approx for f32 {
+    fn approx(left: &f32, right: &f32) -> bool {
+        f32::abs(left - right) <= f32::epsilon()
+    }
+}
+
+impl Approx for f64 {
+    fn approx(left: &f64, right: &f64) -> bool {
+        f64::abs(left - right) <= f32::epsilon() as f64
+    }
+}
+
+impl<T: Copy + Approx, const M: usize, const N: usize> Approx for Matrix<T, M, N> {
+    fn approx(left: &Self, right: &Self) -> bool {
+        zip(left.elements(), right.elements()).all(|(l, r)| T::approx(l, r))
+    }
+}
+
+pub fn approx<T: Approx>(left: &T, right: &T) -> bool {
+    T::approx(left, right)
+}
+
+macro_rules! assert_approx {
+    ($left:expr, $right:expr $(,)?) => {
+        match (&$left, &$right) {
+            (_left_val, _right_val) => {
+                assert_approx!($left, $right, "Difference is less than epsilon")
+            }
+        }
+    };
+    ($left:expr, $right:expr, $($arg:tt)+) => {
+        match (&$left, &$right) {
+            (left_val, right_val) => {
+                pub fn approx<T: Approx>(left: &T, right: &T) -> bool {
+                    T::approx(left, right)
+                }
+
+                if !approx(left_val, right_val){
+                    assert_eq!(left_val, right_val, $($arg)+) // done this way to get nice errors
+                }
+            }
+        }
+    };
+}

tests/ops.rs

@@ -1,61 +1,16 @@
+#[macro_use]
+mod common;
+
+use common::Approx;
 use generic_parameterize::parameterize;
 use num_traits::real::Real;
 use num_traits::Zero;
 use std::fmt::Debug;
 use std::iter::{zip, Product, Sum};
 use std::ops;
-use vector_victor::{LUSolve, Matrix, Vector};
+use vector_victor::{Matrix, Vector};
 
-macro_rules! scalar_eq {
+#[parameterize(S = (i32, f32, f64, u32), M = [1,4], N = [1,4])]
-    ($left:expr, $right:expr $(,)?) => {
-        match (&$left, &$right) {
-            (_left_val, _right_val) => {
-                scalar_eq!($left, $right, "Difference is less than epsilon")
-            }
-        }
-    };
-    ($left:expr, $right:expr, $($arg:tt)+) => {
-        match (&$left, &$right) {
-            (left_val, right_val) => {
-                let epsilon = f32::epsilon() as f64;
-                let lf : f64 = (*left_val).into();
-                let rf : f64 = (*right_val).into();
-                let diff : f64 = (lf - rf).abs();
-                if diff >= epsilon {
-                    assert_eq!(left_val, right_val, $($arg)+) // done this way to get nice errors
-                }
-            }
-        }
-    };
-}
-
-macro_rules! matrix_eq {
-    ($left:expr, $right:expr $(,)?) => {
-        match (&$left, &$right) {
-            (_left_val, _right_val) => {
-                matrix_eq!($left, $right, "Difference is less than epsilon")
-            }
-        }
-    };
-    ($left:expr, $right:expr, $($arg:tt)+) => {
-        match (&$left, &$right) {
-            (left_val, right_val) => {
-                let epsilon = f32::epsilon() as f64;
-                for (l, r) in zip(left_val.elements(), right_val.elements()) {
-                    let lf : f64 = (*l).into();
-                    let rf : f64 = (*r).into();
-                    let diff : f64 = (lf - rf).abs();
-                    if diff >= epsilon {
-                        assert_eq!($left, $right, $($arg)+) // done this way to get nice errors
-                    }
-                }
-
-            }
-        }
-    };
-}
-
-#[parameterize(S = (i32, f32, u32), M = [1,4], N = [1,4])]
 #[test]
 fn test_add<S: Copy + From<u16> + PartialEq + Debug, const M: usize, const N: usize>()
 where
@@ -64,85 +19,7 @@ where
     let a = Matrix::<S, M, N>::fill(S::from(1));
     let b = Matrix::<S, M, N>::fill(S::from(3));
     let c: Matrix<S, M, N> = a + b;
-    for (i, ci) in c.elements().enumerate() {
+    for (_, ci) in c.elements().enumerate() {
         assert_eq!(*ci, S::from(4));
     }
 }
-
-#[parameterize(S = (f32, f64), M = [1,2,3,4])]
-#[test]
-fn test_lu_identity<S: Default + Real + Debug + Product + Sum + Into<f64>, const M: usize>() {
-    // let a: Matrix<f32, 3, 3> = Matrix::<f32, 3, 3>::identity();
-    let i = Matrix::<S, M, M>::identity();
-    let ones = Vector::<S, M>::fill(S::one());
-    let decomp = i.lu().expect("Singular matrix encountered");
-    let (lu, idx, d) = decomp;
-    assert_eq!(lu, i, "Incorrect LU decomposition");
-    assert!(
-        (0..M).eq(idx.elements().cloned()),
-        "Incorrect permutation matrix",
-    );
-    scalar_eq!(d, S::one(), "Incorrect permutation parity");
-    scalar_eq!(i.det(), S::one());
-    assert_eq!(i.inverse(), Some(i));
-    assert_eq!(i.solve(&ones), Some(ones));
-    assert_eq!(decomp.separate(), (i, i));
-}
-
-#[parameterize(S = (f32, f64), M = [2,3,4])]
-#[test]
-fn test_lu_singular<S: Default + Real + Debug + Product + Sum, const M: usize>() {
-    // let a: Matrix<f32, 3, 3> = Matrix::<f32, 3, 3>::identity();
-    let mut a = Matrix::<S, M, M>::zero();
-    let ones = Vector::<S, M>::fill(S::one());
-    a.set_row(0, &ones);
-
-    assert_eq!(a.lu(), None, "Matrix should be singular");
-    assert_eq!(a.det(), S::zero());
-    assert_eq!(a.inverse(), None);
-    assert_eq!(a.solve(&ones), None)
-}
-
-#[test]
-fn test_lu_2x2() {
-    let a = Matrix::new([[1.0, 2.0], [3.0, 0.0]]);
-    let decomp = a.lu().expect("Singular matrix encountered");
-    let (_lu, idx, _d) = decomp;
-    // the decomposition is non-unique, due to the combination of lu and idx.
-    // Instead of checking the exact value, we only check the results.
-    // Also check if they produce the same results with both methods, since the
-    // Matrix<> methods use shortcuts the decomposition methods don't
-
-    let (l, u) = decomp.separate();
-    matrix_eq!(l.mmul(&u), a.permute_rows(&idx));
-
-    scalar_eq!(a.det(), -6.0);
-    scalar_eq!(a.det(), decomp.det());
-
-    matrix_eq!(
-        a.inverse().unwrap(),
-        Matrix::new([[0.0, 2.0], [3.0, -1.0]]) * (1.0 / 6.0)
-    );
-    matrix_eq!(a.inverse().unwrap(), decomp.inverse());
-    matrix_eq!(a.inverse().unwrap().inverse().unwrap(), a)
-}
-
-#[test]
-fn test_lu_3x3() {
-    let a = Matrix::new([[1.0, -5.0, 8.0], [1.0, -2.0, 1.0], [2.0, -1.0, -4.0]]);
-    let decomp = a.lu().expect("Singular matrix encountered");
-    let (_lu, idx, _d) = decomp;
-
-    let (l, u) = decomp.separate();
-    matrix_eq!(l.mmul(&u), a.permute_rows(&idx));
-
-    scalar_eq!(a.det(), 3.0);
-    scalar_eq!(a.det(), decomp.det());
-
-    matrix_eq!(
-        a.inverse().unwrap(),
-        Matrix::new([[9.0, -28.0, 11.0], [6.0, -20.0, 7.0], [3.0, -9.0, 3.0]]) * (1.0 / 3.0)
-    );
-    matrix_eq!(a.inverse().unwrap(), decomp.inverse());
-    matrix_eq!(a.inverse().unwrap().inverse().unwrap(), a)
-}

tests/solve.rs

@@ -0,0 +1,116 @@
+#[macro_use]
+mod common;
+
+use common::Approx;
+use generic_parameterize::parameterize;
+use num_traits::real::Real;
+use num_traits::Zero;
+use std::fmt::Debug;
+use std::iter::{zip, Product, Sum};
+use vector_victor::solve::{LUDecomp, LUSolve};
+use vector_victor::{Matrix, Vector};
+
+#[parameterize(S = (f32, f64), M = [1,2,3,4])]
+#[test]
+/// The LU decomposition of the identity matrix should produce
+/// the identity matrix with no permutations and parity 1
+fn test_lu_identity<S: Default + Approx + Real + Debug + Product + Sum, const M: usize>() {
+    // let a: Matrix<f32, 3, 3> = Matrix::<f32, 3, 3>::identity();
+    let i = Matrix::<S, M, M>::identity();
+    let ones = Vector::<S, M>::fill(S::one());
+    let decomp = i.lu().expect("Singular matrix encountered");
+    let LUDecomp { lu, idx, parity } = decomp;
+    assert_eq!(lu, i, "Incorrect LU decomposition");
+    assert!(
+        (0..M).eq(idx.elements().cloned()),
+        "Incorrect permutation matrix",
+    );
+    assert_approx!(parity, S::one(), "Incorrect permutation parity");
+
+    // Check determinant calculation which uses LU decomposition
+    assert_approx!(
+        i.det(),
+        S::one(),
+        "Identity matrix should have determinant of 1"
+    );
+
+    // Check inverse calculation with uses LU decomposition
+    assert_eq!(
+        i.inverse(),
+        Some(i),
+        "Identity matrix should be its own inverse"
+    );
+    assert_eq!(
+        i.solve(&ones),
+        Some(ones),
+        "Failed to solve using identity matrix"
+    );
+
+    // Check triangle separation
+    assert_eq!(decomp.separate(), (i, i));
+}
+
+#[parameterize(S = (f32, f64), M = [2,3,4])]
+#[test]
+/// The LU decomposition of any singular matrix should be `None`
+fn test_lu_singular<S: Default + Real + Debug + Product + Sum, const M: usize>() {
+    // let a: Matrix<f32, 3, 3> = Matrix::<f32, 3, 3>::identity();
+    let mut a = Matrix::<S, M, M>::zero();
+    let ones = Vector::<S, M>::fill(S::one());
+    a.set_row(0, &ones);
+
+    assert_eq!(a.lu(), None, "Matrix should be singular");
+    assert_eq!(
+        a.det(),
+        S::zero(),
+        "Singular matrix should have determinant of zero"
+    );
+    assert_eq!(a.inverse(), None, "Singular matrix should have no inverse");
+    assert_eq!(
+        a.solve(&ones),
+        None,
+        "Singular matrix should not be solvable"
+    )
+}
+
+#[test]
+fn test_lu_2x2() {
+    let a = Matrix::new([[1.0, 2.0], [3.0, 0.0]]);
+    let decomp = a.lu().expect("Singular matrix encountered");
+    // the decomposition is non-unique, due to the combination of lu and idx.
+    // Instead of checking the exact value, we only check the results.
+    // Also check if they produce the same results with both methods, since the
+    // Matrix<> methods use shortcuts the decomposition methods don't
+
+    let (l, u) = decomp.separate();
+    assert_approx!(l.mmul(&u), a.permute_rows(&decomp.idx));
+
+    assert_approx!(a.det(), -6.0);
+    assert_approx!(a.det(), decomp.det());
+
+    assert_approx!(
+        a.inverse().unwrap(),
+        Matrix::new([[0.0, 2.0], [3.0, -1.0]]) * (1.0 / 6.0)
+    );
+    assert_approx!(a.inverse().unwrap(), decomp.inverse());
+    assert_approx!(a.inverse().unwrap().inverse().unwrap(), a)
+}
+
+#[test]
+fn test_lu_3x3() {
+    let a = Matrix::new([[1.0, -5.0, 8.0], [1.0, -2.0, 1.0], [2.0, -1.0, -4.0]]);
+    let decomp = a.lu().expect("Singular matrix encountered");
+
+    let (l, u) = decomp.separate();
+    assert_approx!(l.mmul(&u), a.permute_rows(&decomp.idx));
+
+    assert_approx!(a.det(), 3.0);
+    assert_approx!(a.det(), decomp.det());
+
+    assert_approx!(
+        a.inverse().unwrap(),
+        Matrix::new([[9.0, -28.0, 11.0], [6.0, -20.0, 7.0], [3.0, -9.0, 3.0]]) * (1.0 / 3.0)
+    );
+    assert_approx!(a.inverse().unwrap(), decomp.inverse());
+    assert_approx!(a.inverse().unwrap().inverse().unwrap(), a)
+}