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Document and refine `decompose` module

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master
Andrew Cassidy 12 months ago
parent e943345693
commit 6831027573
2 changed files with 209 additions and 63 deletions
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269
src/decompose.rs
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@ -1,79 +1,93 @@
use crate::util::checked_inv;
use crate::Matrix;
use crate::Vector;
use crate::{Matrix, Vector};
use num_traits::real::Real;
use num_traits::One;
use std::iter::{Product, Sum};
use std::ops::{Mul, Neg, Not};
/// The parity of an [LU decomposition](LUDecomposition). In other words, how many times the
/// source matrix has to have rows swapped before the decomposition takes place
#[derive(Copy, Clone, Debug, PartialEq)]
pub struct LUDecomposition<T: Copy, const N: usize> {
pub lu: Matrix<T, N, N>,
pub idx: Vector<usize, N>,
pub parity: T,
pub enum Parity {
Even,
Odd,
}
impl<T: Copy + Default, const N: usize> LUDecomposition<T, N>
impl<T> Mul<T> for Parity
where
T: Real + Default + Sum + Product,
T: Neg<Output = T> + One,
{
#[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();
type Output = T;
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
fn mul(self, rhs: T) -> Self::Output {
rhs * match self {
Parity::Even => T::one(),
Parity::Odd => -T::one(),
}
}
}
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),
})?;
impl Not for Parity {
type Output = Parity;
// 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
}
fn not(self) -> Self::Output {
match self {
Parity::Even => Parity::Odd,
Parity::Odd => Parity::Even,
}
}
}
let pivot = lu[(k, k)];
if pivot.abs() < T::epsilon() {
// if the pivot is zero, the matrix is singular
return None;
};
/// The result of the [LU decomposition](LUDecomposable::lu) of a matrix.
///
/// This struct provides a convenient way to reuse one LU decomposition to solve multiple
/// matrix equations. You likely do not need to worry about its contents.
///
/// See [LU decomposition](https://en.wikipedia.org/wiki/LU_decomposition)
/// on wikipedia for more information
#[derive(Copy, Clone, Debug, PartialEq)]
pub struct LUDecomposition<T: Copy, const N: usize> {
/// The $L$ and $U$ matrices combined into one
///
/// for example if
///
/// $ U = [[u_{11}, u_{12}, cdots, u_{1n} ],
/// [0, u_{22}, cdots, u_{2n} ],
/// [vdots, vdots, ddots, vdots ],
/// [0, 0, cdots, u_{mn} ]] $
/// and
/// $ L = [[1, 0, cdots, 0 ],
/// [l_{21}, 1, cdots, 0 ],
/// [vdots, vdots, ddots, vdots ],
/// [l_{m1}, l_{m2}, cdots, 1 ]] $,
/// then
/// $ LU = [[u_{11}, u_{12}, cdots, u_{1n} ],
/// [l_{21}, u_{22}, cdots, u_{2n} ],
/// [vdots, vdots, ddots, vdots ],
/// [l_{m1}, l_{m2}, cdots, u_{mn} ]] $
///
/// note that the diagonals of the $L$ matrix are always 1, so no information is lost
pub lu: Matrix<T, N, N>,
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)]);
}
}
}
/// The indices of the permutation matrix $P$, such that $PxxA$ = $LxxU$
///
/// The permutation matrix rearranges the rows of the original matrix in order to produce
/// the LU decomposition. This makes calculation simpler, but makes the result
/// (known as an LUP decomposition) no longer unique
pub idx: Vector<usize, N>,
return Some(Self { lu, idx, parity });
}
/// The parity of the decomposition.
pub parity: Parity,
}
impl<T: Copy + Default, const N: usize> LUDecomposition<T, N>
where
T: Real + Default + Sum + Product,
{
/// Solve for $x$ in $M xx x = b$, where $M$ is the original matrix this is a decomposition of.
///
/// This is equivalent to [`LUDecomposable::solve`] while allowing the LU decomposition
/// to be reused
#[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);
@ -81,7 +95,7 @@ where
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
// it will become the index of the first non-vanishing element of b
let mut ii = 0usize;
for i in 0..N {
// forward substitution using L
@ -107,14 +121,25 @@ where
}))
}
/// Calculate the determinant $|M|$ of the matrix $M$.
/// If the matrix is singular, the determinant is 0.
///
/// This is equivalent to [`LUDecomposable::det`] while allowing the LU decomposition
/// to be reused
pub fn det(&self) -> T {
self.parity * self.lu.diagonals().product()
}
/// Calculate the inverse of the original matrix, such that $MxxM^{-1} = I$
///
/// This is equivalent to [`Matrix::inverse`] while allowing the LU decomposition to be reused
#[must_use]
pub fn inverse(&self) -> Matrix<T, N, N> {
return self.solve(&Matrix::<T, N, N>::identity());
}
/// Separate the $L$ and $U$ sides of the $LU$ matrix.
/// See [the `lu` field](LUDecomposition::lu) for more information
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
@ -131,19 +156,83 @@ where
}
}
/// A Matrix that can be decomposed into an upper and lower diagonal matrix,
/// known as an [LU Decomposition](LUDecomposition).
///
/// See [LU decomposition](https://en.wikipedia.org/wiki/LU_decomposition)
/// on wikipedia for more information
pub trait LUDecomposable<T, const N: usize>
where
T: Copy + Default + Real + Product + Sum,
{
/// return this matrix's [`LUDecomposition`], or [`None`] if the matrix is singular.
/// This can be used to solve for multiple results
///
/// ```
/// # use vector_victor::decompose::LUDecomposable;
/// # use vector_victor::{Matrix, Vector};
/// let m = Matrix::new([[1.0,3.0],[2.0,4.0]]);
/// let lu = m.lu().expect("Cannot decompose a signular matrix");
///
/// let b = Vector::vec([7.0,10.0]);
/// assert_eq!(lu.solve(&b), Vector::vec([1.0,2.0]));
///
/// let c = Vector::vec([10.0, 14.0]);
/// assert_eq!(lu.solve(&c), Vector::vec([1.0,3.0]));
///
/// ```
#[must_use]
fn lu(&self) -> Option<LUDecomposition<T, N>>;
/// Calculate the inverse of the matrix, such that $MxxM^{-1} = I$, or [`None`] if the matrix is singular.
///
/// ```
/// # use vector_victor::decompose::LUDecomposable;
/// # use vector_victor::Matrix;
/// let m = Matrix::new([[1.0,3.0],[2.0,4.0]]);
/// let mi = m.inverse().expect("Cannot invert a singular matrix");
///
/// assert_eq!(mi, Matrix::new([[-2.0, 1.5],[1.0, -0.5]]), "unexpected inverse matrix");
///
/// // multiplying a matrix by its inverse yields the identity matrix
/// assert_eq!(m.mmul(&mi), Matrix::identity())
/// ```
#[must_use]
fn inverse(&self) -> Option<Matrix<T, N, N>>;
/// Calculate the determinant $|M|$ of the matrix $M$.
/// If the matrix is singular, the determinant is 0
#[must_use]
fn det(&self) -> T;
/// Solve for $x$ in $M xx x = b$, where $M$ is the original matrix this is a decomposition of.
///
/// ```
/// # use vector_victor::decompose::LUDecomposable;
/// # use vector_victor::{Matrix, Vector};
///
/// let m = Matrix::new([[1.0,3.0],[2.0,4.0]]);
/// let b = Vector::vec([7.0,10.0]);
/// let x = m.solve(&b).expect("Cannot solve a singular matrix");
///
/// assert_eq!(x, Vector::vec([1.0,2.0]), "x = [1,2]");
/// assert_eq!(m.mmul(&x), b, "Mx = b");
/// ```
///
/// $x$ does not need to be a column-vector, it can also be a 2D matrix. For example,
/// the following is another way to calculate the [`inverse`] by solving for the identity matrix $I$.
///
/// ```
/// # use vector_victor::decompose::LUDecomposable;
/// # use vector_victor::{Matrix, Vector};
///
/// let m = Matrix::new([[1.0,3.0],[2.0,4.0]]);
/// let i = Matrix::<f64,2,2>::identity();
/// let mi = m.solve(&i).expect("Cannot solve a singular matrix");
///
/// assert_eq!(mi, Matrix::new([[-2.0, 1.5],[1.0, -0.5]]));
/// assert_eq!(m.mmul(&mi), i, "M x M^-1 = I");
/// ```
#[must_use]
fn solve<const M: usize>(&self, b: &Matrix<T, N, M>) -> Option<Matrix<T, N, M>>;
}
@ -153,7 +242,63 @@ where
T: Copy + Default + Real + Sum + Product,
{
fn lu(&self) -> Option<LUDecomposition<T, N>> {
LUDecomposition::decompose(self)
// Implementation from Numerical Recipes §2.3
let mut lu = self.clone();
let mut idx: Vector<usize, N> = (0..N).collect();
let mut parity = Parity::Even;
let mut vv: Vector<T, N> = self
.rows()
.map(|row| {
let m = row.elements().cloned().reduce(|acc, x| acc.max(x.abs()))?;
checked_inv(m)
})
.collect::<Option<_>>()?; // get the inverse max abs value in each row
// for each column in the matrix...
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; // swap parity
vv[ipivot] = vv[k] // interchange scale factor
}
// select our pivot, which is now on the diagonal
let pivot = lu[(k, k)];
if pivot.abs() < T::epsilon() {
// if the pivot is zero, the matrix is singular
return None;
};
// for each element in the column k below the diagonal...
// this is called outer product Gaussian elimination
for i in (k + 1)..N {
// divide by the pivot element
lu[(i, k)] = lu[(i, k)] / pivot;
// for each element in the column k below the diagonal...
for j in (k + 1)..N {
// reduce remaining submatrix
lu[(i, j)] = lu[(i, j)] - (lu[(i, k)] * lu[(k, j)]);
}
}
}
return Some(LUDecomposition { lu, idx, parity });
}
fn inverse(&self) -> Option<Matrix<T, N, N>> {

3
tests/decompose.rs
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@ -7,6 +7,7 @@ use num_traits::real::Real;
use num_traits::Zero;
use std::fmt::Debug;
use std::iter::{Product, Sum};
use vector_victor::decompose::Parity::Even;
use vector_victor::decompose::{LUDecomposable, LUDecomposition};
use vector_victor::{Matrix, Vector};
@ -25,7 +26,7 @@ fn test_lu_identity<S: Default + Approx + Real + Debug + Product + Sum, const M:
(0..M).eq(idx.elements().cloned()),
"Incorrect permutation matrix",
);
assert_approx!(parity, S::one(), "Incorrect permutation parity");
assert_eq!(parity, Even, "Incorrect permutation parity");
// Check determinant calculation which uses LU decomposition
assert_approx!(

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