forked from M-Labs/nalgebra
303 lines
10 KiB
Rust
303 lines
10 KiB
Rust
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use std::mem;
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use alga::general::{Field, Real};
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use core::{Scalar, Matrix, MatrixN, MatrixMN, DefaultAllocator};
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use dimension::{Dim, DimMin, DimMinimum};
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use storage::{Storage, StorageMut};
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use allocator::{Allocator, Reallocator};
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use constraint::{ShapeConstraint, SameNumberOfRows};
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use linalg::PermutationSequence;
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/// LU decomposition with partial (row) pivoting.
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pub struct LU<N: Real, R: DimMin<C>, C: Dim>
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where DefaultAllocator: Allocator<N, R, C> +
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Allocator<(usize, usize), DimMinimum<R, C>> {
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lu: MatrixMN<N, R, C>,
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p: PermutationSequence<DimMinimum<R, C>>
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}
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/// Performs a LU decomposition to overwrite `out` with the inverse of `matrix`.
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///
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/// If `matrix` is not invertible, `false` is returned and `out` may contain invalid data.
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pub fn try_invert_to<N: Real, D: Dim, S>(mut matrix: MatrixN<N, D>,
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out: &mut Matrix<N, D, D, S>)
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-> bool
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where S: StorageMut<N, D, D>,
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DefaultAllocator: Allocator<N, D, D> {
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assert!(matrix.is_square(), "LU inversion: unable to invert a rectangular matrix.");
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let dim = matrix.nrows();
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out.fill_with_identity();
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for i in 0 .. dim {
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let piv = matrix.slice_range(i .., i).iamax() + i;
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let diag = matrix[(piv, i)];
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if diag.is_zero() {
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return false;
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}
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if piv != i {
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out.swap_rows(i, piv);
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matrix.columns_range_mut(.. i).swap_rows(i, piv);
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gauss_step_swap(&mut matrix, diag, i, piv);
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}
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else {
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gauss_step(&mut matrix, diag, i);
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}
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}
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matrix.solve_lower_triangular_with_diag_mut(out, N::one());
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matrix.solve_upper_triangular_mut(out)
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}
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impl<N: Real, R: DimMin<C>, C: Dim> LU<N, R, C>
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where DefaultAllocator: Allocator<N, R, C> +
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Allocator<(usize, usize), DimMinimum<R, C>> {
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/// Computes the LU decomposition with partial (row) pivoting of `matrix`.
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pub fn new(mut matrix: MatrixMN<N, R, C>) -> Self {
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let (nrows, ncols) = matrix.data.shape();
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let min_nrows_ncols = nrows.min(ncols);
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let mut p = PermutationSequence::identity_generic(min_nrows_ncols);
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if min_nrows_ncols.value() == 0 {
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return LU { lu: matrix, p: p };
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}
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for i in 0 .. min_nrows_ncols.value() {
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let piv = matrix.slice_range(i .., i).iamax() + i;
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let diag = matrix[(piv, i)];
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if diag.is_zero() {
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// No non-zero entries on this column.
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continue;
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}
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if piv != i {
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p.append_permutation(i, piv);
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matrix.columns_range_mut(.. i).swap_rows(i, piv);
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gauss_step_swap(&mut matrix, diag, i, piv);
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}
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else {
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gauss_step(&mut matrix, diag, i);
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}
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}
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LU { lu: matrix, p: p }
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}
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#[doc(hidden)]
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pub fn lu_internal(&self) -> &MatrixMN<N, R, C> {
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&self.lu
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}
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/// The lower triangular matrix of this decomposition.
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#[inline]
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pub fn l(&self) -> MatrixMN<N, R, DimMinimum<R, C>>
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where DefaultAllocator: Allocator<N, R, DimMinimum<R, C>> {
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let (nrows, ncols) = self.lu.data.shape();
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let mut m = self.lu.columns_generic(0, nrows.min(ncols)).into_owned();
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m.fill_upper_triangle(N::zero(), 1);
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m.fill_diagonal(N::one());
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m
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}
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/// The lower triangular matrix of this decomposition.
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fn l_unpack_with_p(self) -> (MatrixMN<N, R, DimMinimum<R, C>>,
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PermutationSequence<DimMinimum<R, C>>)
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where DefaultAllocator: Reallocator<N, R, C, R, DimMinimum<R, C>> {
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let (nrows, ncols) = self.lu.data.shape();
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let mut m = self.lu.resize_generic(nrows, nrows.min(ncols), N::zero());
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m.fill_upper_triangle(N::zero(), 1);
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m.fill_diagonal(N::one());
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(m, self.p)
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}
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/// The lower triangular matrix of this decomposition.
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#[inline]
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pub fn l_unpack(self) -> MatrixMN<N, R, DimMinimum<R, C>>
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where DefaultAllocator: Reallocator<N, R, C, R, DimMinimum<R, C>> {
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let (nrows, ncols) = self.lu.data.shape();
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let mut m = self.lu.resize_generic(nrows, nrows.min(ncols), N::zero());
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m.fill_upper_triangle(N::zero(), 1);
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m.fill_diagonal(N::one());
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m
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}
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/// The upper triangular matrix of this decomposition.
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#[inline]
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pub fn u(&self) -> MatrixMN<N, DimMinimum<R, C>, C>
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where DefaultAllocator: Allocator<N, DimMinimum<R, C>, C> {
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let (nrows, ncols) = self.lu.data.shape();
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self.lu.rows_generic(0, nrows.min(ncols)).upper_triangle()
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}
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/// The row permutation matrix of this decomposition.
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#[inline]
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pub fn p(&self) -> &PermutationSequence<DimMinimum<R, C>> {
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&self.p
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}
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/// The two matrix of this decomposition and the permutation matrix: `(P, L, U)`.
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#[inline]
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pub fn unpack(self) -> (PermutationSequence<DimMinimum<R, C>>,
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MatrixMN<N, R, DimMinimum<R, C>>,
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MatrixMN<N, DimMinimum<R, C>, C>)
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where DefaultAllocator: Allocator<N, R, DimMinimum<R, C>> +
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Allocator<N, DimMinimum<R, C>, C> +
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Reallocator<N, R, C, R, DimMinimum<R, C>> {
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// Use reallocation for either l or u.
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let u = self.u();
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let (l, p) = self.l_unpack_with_p();
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(p, l, u)
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}
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}
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impl<N: Real, D: DimMin<D, Output = D>> LU<N, D, D>
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where DefaultAllocator: Allocator<N, D, D> +
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Allocator<(usize, usize), D> {
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/// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
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///
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/// Returns `None` if `self` is not invertible.
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pub fn solve<R2: Dim, C2: Dim, S2>(&self, b: &Matrix<N, R2, C2, S2>) -> Option<MatrixMN<N, R2, C2>>
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where S2: StorageMut<N, R2, C2>,
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ShapeConstraint: SameNumberOfRows<R2, D>,
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DefaultAllocator: Allocator<N, R2, C2> {
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let mut res = b.clone_owned();
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if self.solve_mut(&mut res) {
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Some(res)
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}
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else {
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None
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}
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}
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/// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
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///
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/// If the decomposed matrix is not invertible, this returns `false` and its input `b` is
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/// overwritten with meaningless informations.
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pub fn solve_mut<R2: Dim, C2: Dim, S2>(&self, b: &mut Matrix<N, R2, C2, S2>) -> bool
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where S2: StorageMut<N, R2, C2>,
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ShapeConstraint: SameNumberOfRows<R2, D> {
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assert_eq!(self.lu.nrows(), b.nrows(), "LU solve matrix dimension mismatch.");
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assert!(self.lu.is_square(), "LU solve: unable to solve a non-square system.");
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self.p.permute_rows(b);
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self.lu.solve_lower_triangular_with_diag_mut(b, N::one());
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self.lu.solve_upper_triangular_mut(b)
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}
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/// Computes the inverse of the decomposed matrix.
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///
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/// Returnrs `None` if the matrix is not invertible.
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pub fn try_inverse(&self) -> Option<MatrixN<N, D>> {
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assert!(self.lu.is_square(), "LU inverse: unable to compute the inverse of a non-square matrix.");
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let (nrows, ncols) = self.lu.data.shape();
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let mut res = MatrixN::identity_generic(nrows, ncols);
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if self.try_inverse_to(&mut res) {
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Some(res)
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}
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else {
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None
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}
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}
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/// Computes the inverse of the decomposed matrix and outputs the result to `out`.
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///
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/// If the decomposed matrix is not invertible, this returns `false` and `out` may contain
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/// meaninless informations.
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pub fn try_inverse_to<S2: StorageMut<N, D, D>>(&self, out: &mut Matrix<N, D, D, S2>) -> bool {
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assert!(self.lu.is_square(), "LU inverse: unable to compute the inverse of a non-square matrix.");
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assert!(self.lu.shape() == out.shape(), "LU inverse: mismatched output shape.");
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out.fill_with_identity();
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self.solve_mut(out)
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}
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/// Computes the determinant of the decomposed matrix.
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pub fn determinant(&self) -> N {
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let dim = self.lu.nrows();
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assert!(self.lu.is_square(), "LU determinant: unable to compute the determinant of a non-square matrix.");
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let mut res = N::one();
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for i in 0 .. dim {
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res *= unsafe { *self.lu.get_unchecked(i, i) };
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}
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res * self.p.determinant()
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}
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/// Indicates if the decomposed matrix is invertible.
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pub fn is_invertible(&self) -> bool {
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assert!(self.lu.is_square(), "QR: unable to test the invertibility of a non-square matrix.");
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for i in 0 .. self.lu.nrows() {
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if self.lu[(i, i)].is_zero() {
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return false;
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}
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}
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true
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}
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}
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#[doc(hidden)]
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/// Executes one step of gaussian elimination on the i-th row and column of `matrix`. The diagonal
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/// element `matrix[(i, i)]` is provided as argument.
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pub fn gauss_step<N, R: Dim, C: Dim, S>(matrix: &mut Matrix<N, R, C, S>, diag: N, i: usize)
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where N: Scalar + Field,
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S: StorageMut<N, R, C> {
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let mut submat = matrix.slice_range_mut(i .., i ..);
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let inv_diag = N::one() / diag;
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let (mut coeffs, mut submat) = submat.columns_range_pair_mut(0, 1 ..);
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let mut coeffs = coeffs.rows_range_mut(1 ..);
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coeffs *= inv_diag;
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let (pivot_row, mut down) = submat.rows_range_pair_mut(0, 1 ..);
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for k in 0 .. pivot_row.ncols() {
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down.column_mut(k).axpy(-pivot_row[k], &coeffs, N::one());
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}
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}
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#[doc(hidden)]
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/// Swaps the rows `i` with the row `piv` and executes one step of gaussian elimination on the i-th
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/// row and column of `matrix`. The diagonal element `matrix[(i, i)]` is provided as argument.
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pub fn gauss_step_swap<N, R: Dim, C: Dim, S>(matrix: &mut Matrix<N, R, C, S>, diag: N, i: usize, piv: usize)
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where N: Scalar + Field,
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S: StorageMut<N, R, C> {
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let piv = piv - i;
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let mut submat = matrix.slice_range_mut(i .., i ..);
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let inv_diag = N::one() / diag;
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let (mut coeffs, mut submat) = submat.columns_range_pair_mut(0, 1 ..);
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coeffs.swap((0, 0), (piv, 0));
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let mut coeffs = coeffs.rows_range_mut(1 ..);
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coeffs *= inv_diag;
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let (mut pivot_row, mut down) = submat.rows_range_pair_mut(0, 1 ..);
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for k in 0 .. pivot_row.ncols() {
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mem::swap(&mut pivot_row[k], &mut down[(piv - 1, k)]);
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down.column_mut(k).axpy(-pivot_row[k], &coeffs, N::one());
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}
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}
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