nalgebra/src/linalg/full_piv_lu.rs

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use alga::general::Real;
use core::{Matrix, MatrixN, MatrixMN, DefaultAllocator};
use dimension::{Dim, DimMin, DimMinimum};
use storage::{Storage, StorageMut};
use allocator::Allocator;
use constraint::{ShapeConstraint, SameNumberOfRows};
use linalg::lu;
use linalg::PermutationSequence;
/// LU decomposition with full pivoting.
pub struct FullPivLU<N: Real, R: DimMin<C>, C: Dim>
where DefaultAllocator: Allocator<N, R, C> +
Allocator<(usize, usize), DimMinimum<R, C>> {
lu: MatrixMN<N, R, C>,
p: PermutationSequence<DimMinimum<R, C>>,
q: PermutationSequence<DimMinimum<R, C>>
}
impl<N: Real, R: DimMin<C>, C: Dim> FullPivLU<N, R, C>
where DefaultAllocator: Allocator<N, R, C> +
Allocator<(usize, usize), DimMinimum<R, C>> {
/// This computes the matrixces `P, L, U` such that `P * matrix = LU`.
pub fn new(mut matrix: MatrixMN<N, R, C>) -> Self {
let (nrows, ncols) = matrix.data.shape();
let min_nrows_ncols = nrows.min(ncols);
let mut p = PermutationSequence::identity_generic(min_nrows_ncols);
let mut q = PermutationSequence::identity_generic(min_nrows_ncols);
if min_nrows_ncols.value() == 0 {
return FullPivLU { lu: matrix, p: p, q: q };
}
for i in 0 .. min_nrows_ncols.value() {
let piv = matrix.slice_range(i .., i ..).iamax_full();
let row_piv = piv.0 + i;
let col_piv = piv.1 + i;
let diag = matrix[(row_piv, col_piv)];
if diag.is_zero() {
// The remaining of the matrix is zero.
break;
}
matrix.swap_columns(i, col_piv);
q.append_permutation(i, col_piv);
if row_piv != i {
p.append_permutation(i, row_piv);
matrix.columns_range_mut(.. i).swap_rows(i, row_piv);
lu::gauss_step_swap(&mut matrix, diag, i, row_piv);
}
else {
lu::gauss_step(&mut matrix, diag, i);
}
}
FullPivLU { lu: matrix, p: p, q: q }
}
#[doc(hidden)]
pub fn lu_internal(&self) -> &MatrixMN<N, R, C> {
&self.lu
}
/// The lower triangular matrix of this decomposition.
#[inline]
pub fn l(&self) -> MatrixMN<N, R, DimMinimum<R, C>>
where DefaultAllocator: Allocator<N, R, DimMinimum<R, C>> {
let (nrows, ncols) = self.lu.data.shape();
let mut m = self.lu.columns_generic(0, nrows.min(ncols)).into_owned();
m.fill_upper_triangle(N::zero(), 1);
m.fill_diagonal(N::one());
m
}
/// The upper triangular matrix of this decomposition.
#[inline]
pub fn u(&self) -> MatrixMN<N, DimMinimum<R, C>, C>
where DefaultAllocator: Allocator<N, DimMinimum<R, C>, C> {
let (nrows, ncols) = self.lu.data.shape();
self.lu.rows_generic(0, nrows.min(ncols)).upper_triangle()
}
/// The row permutations of this decomposition.
#[inline]
pub fn p(&self) -> &PermutationSequence<DimMinimum<R, C>> {
&self.p
}
/// The q permutations of this decomposition.
#[inline]
pub fn q(&self) -> &PermutationSequence<DimMinimum<R, C>> {
&self.q
}
/// The two matrix of this decomposition and the permutation matrix: `(P, L, U, Q)`.
#[inline]
pub fn unpack(self) -> (PermutationSequence<DimMinimum<R, C>>,
MatrixMN<N, R, DimMinimum<R, C>>,
MatrixMN<N, DimMinimum<R, C>, C>,
PermutationSequence<DimMinimum<R, C>>)
where DefaultAllocator: Allocator<N, R, DimMinimum<R, C>> +
Allocator<N, DimMinimum<R, C>, C> {
// Use reallocation for either l or u.
let l = self.l();
let u = self.u();
let p = self.p;
let q = self.q;
(p, l, u, q)
}
}
impl<N: Real, D: DimMin<D, Output = D>> FullPivLU<N, D, D>
where DefaultAllocator: Allocator<N, D, D> +
Allocator<(usize, usize), D> {
/// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
///
/// Retuns `None` if the decomposed matrix is not invertible.
pub fn solve<R2: Dim, C2: Dim, S2>(&self, b: &Matrix<N, R2, C2, S2>) -> Option<MatrixMN<N, R2, C2>>
where S2: StorageMut<N, R2, C2>,
ShapeConstraint: SameNumberOfRows<R2, D>,
DefaultAllocator: Allocator<N, R2, C2> {
let mut res = b.clone_owned();
if self.solve_mut(&mut res) {
Some(res)
}
else {
None
}
}
/// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
///
/// If the decomposed matrix is not invertible, this returns `false` and its input `b` is
/// left unchanged.
pub fn solve_mut<R2: Dim, C2: Dim, S2>(&self, b: &mut Matrix<N, R2, C2, S2>) -> bool
where S2: StorageMut<N, R2, C2>,
ShapeConstraint: SameNumberOfRows<R2, D> {
assert_eq!(self.lu.nrows(), b.nrows(), "FullPivLU solve matrix dimension mismatch.");
assert!(self.lu.is_square(), "FullPivLU solve: unable to solve a non-square system.");
if self.is_invertible() {
self.p.permute_rows(b);
self.lu.solve_lower_triangular_with_diag_mut(b, N::one());
self.lu.solve_upper_triangular_mut(b);
self.q.inv_permute_rows(b);
true
}
else {
false
}
}
/// Computes the inverse of the decomposed matrix.
///
/// Returns `None` if the decomposed matrix is not invertible.
pub fn try_inverse(&self) -> Option<MatrixN<N, D>> {
assert!(self.lu.is_square(), "FullPivLU inverse: unable to compute the inverse of a non-square matrix.");
let (nrows, ncols) = self.lu.data.shape();
let mut res = MatrixN::identity_generic(nrows, ncols);
if self.solve_mut(&mut res) {
Some(res)
}
else {
None
}
}
/// Indicates if the decomposed matrix is invertible.
pub fn is_invertible(&self) -> bool {
assert!(self.lu.is_square(), "FullPivLU: unable to test the invertibility of a non-square matrix.");
let dim = self.lu.nrows();
!self.lu[(dim - 1, dim - 1)].is_zero()
}
/// Computes the determinant of the decomposed matrix.
pub fn determinant(&self) -> N {
assert!(self.lu.is_square(), "FullPivLU determinant: unable to compute the determinant of a non-square matrix.");
let dim = self.lu.nrows();
let mut res = self.lu[(dim - 1, dim - 1)];
if !res.is_zero() {
for i in 0 .. dim - 1 {
res *= unsafe { *self.lu.get_unchecked(i, i) };
}
res * self.p.determinant() * self.q.determinant()
}
else {
N::zero()
}
}
}