forked from M-Labs/nalgebra
Merge pull request #613 from russellb23/dev
QR factorizatio nwith column pivoting
This commit is contained in:
Sébastien Crozet
GitHub
commit
06f92ad1e3
333
src/linalg/col_piv_qr.rs
Normal file
333
src/linalg/col_piv_qr.rs
Normal file
@ -0,0 +1,333 @@
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use num::Zero;
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#[cfg(feature = "serde-serialize")]
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use serde::{Deserialize, Serialize};
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use crate::allocator::{Allocator, Reallocator};
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use crate::base::{DefaultAllocator, Matrix, MatrixMN, MatrixN, Unit, VectorN};
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use crate::constraint::{SameNumberOfRows, ShapeConstraint};
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use crate::dimension::{Dim, DimMin, DimMinimum, U1};
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use crate::storage::{Storage, StorageMut};
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use crate::ComplexField;
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use crate::geometry::Reflection;
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use crate::linalg::{householder, PermutationSequence};
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/// The QR decomposition (with column pivoting) of a general matrix.
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#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
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#[cfg_attr(
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feature = "serde-serialize",
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serde(bound(serialize = "DefaultAllocator: Allocator<N, R, C> +
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Allocator<N, DimMinimum<R, C>>,
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MatrixMN<N, R, C>: Serialize,
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PermutationSequence<DimMinimum<R, C>>: Serialize,
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VectorN<N, DimMinimum<R, C>>: Serialize"))
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)]
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#[cfg_attr(
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feature = "serde-serialize",
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serde(bound(deserialize = "DefaultAllocator: Allocator<N, R, C> +
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Allocator<N, DimMinimum<R, C>>,
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MatrixMN<N, R, C>: Deserialize<'de>,
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PermutationSequence<DimMinimum<R, C>>: Deserialize<'de>,
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VectorN<N, DimMinimum<R, C>>: Deserialize<'de>"))
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)]
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#[derive(Clone, Debug)]
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pub struct ColPivQR<N: ComplexField, R: DimMin<C>, C: Dim>
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where
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DefaultAllocator: Allocator<N, R, C>
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+ Allocator<N, DimMinimum<R, C>>
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+ Allocator<(usize, usize), DimMinimum<R, C>>,
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{
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col_piv_qr: MatrixMN<N, R, C>,
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p: PermutationSequence<DimMinimum<R, C>>,
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diag: VectorN<N, DimMinimum<R, C>>,
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}
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impl<N: ComplexField, R: DimMin<C>, C: Dim> Copy for ColPivQR<N, R, C>
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where
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DefaultAllocator: Allocator<N, R, C>
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+ Allocator<N, DimMinimum<R, C>>
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+ Allocator<(usize, usize), DimMinimum<R, C>>,
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MatrixMN<N, R, C>: Copy,
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PermutationSequence<DimMinimum<R, C>>: Copy,
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VectorN<N, DimMinimum<R, C>>: Copy,
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{
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}
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impl<N: ComplexField, R: DimMin<C>, C: Dim> ColPivQR<N, R, C>
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where
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DefaultAllocator: Allocator<N, R, C>
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+ Allocator<N, R>
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+ Allocator<N, DimMinimum<R, C>>
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+ Allocator<(usize, usize), DimMinimum<R, C>>,
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{
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/// Computes the ColPivQR decomposition using householder reflections.
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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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let mut diag = unsafe { MatrixMN::new_uninitialized_generic(min_nrows_ncols, U1) };
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if min_nrows_ncols.value() == 0 {
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return ColPivQR {
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col_piv_qr: matrix,
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p,
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diag,
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};
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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..).icamax_full();
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let col_piv = piv.1 + i;
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matrix.swap_columns(i, col_piv);
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p.append_permutation(i, col_piv);
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householder::clear_column_unchecked(&mut matrix, &mut diag[i], i, 0, None);
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}
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ColPivQR {
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col_piv_qr: matrix,
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p,
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diag,
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}
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}
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/// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
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#[inline]
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pub fn r(&self) -> MatrixMN<N, DimMinimum<R, C>, C>
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where
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DefaultAllocator: Allocator<N, DimMinimum<R, C>, C>,
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{
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let (nrows, ncols) = self.col_piv_qr.data.shape();
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let mut res = self
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.col_piv_qr
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.rows_generic(0, nrows.min(ncols))
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.upper_triangle();
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res.set_partial_diagonal(self.diag.iter().map(|e| N::from_real(e.modulus())));
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res
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}
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/// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
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///
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/// This is usually faster than `r` but consumes `self`.
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#[inline]
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pub fn unpack_r(self) -> MatrixMN<N, DimMinimum<R, C>, C>
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where
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DefaultAllocator: Reallocator<N, R, C, DimMinimum<R, C>, C>,
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{
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let (nrows, ncols) = self.col_piv_qr.data.shape();
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let mut res = self
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.col_piv_qr
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.resize_generic(nrows.min(ncols), ncols, N::zero());
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res.fill_lower_triangle(N::zero(), 1);
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res.set_partial_diagonal(self.diag.iter().map(|e| N::from_real(e.modulus())));
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res
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}
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/// Computes the orthogonal matrix `Q` of this decomposition.
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pub fn q(&self) -> MatrixMN<N, R, DimMinimum<R, C>>
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where
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DefaultAllocator: Allocator<N, R, DimMinimum<R, C>>,
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{
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let (nrows, ncols) = self.col_piv_qr.data.shape();
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// NOTE: we could build the identity matrix and call q_mul on it.
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// Instead we don't so that we take in account the matrix sparseness.
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let mut res = Matrix::identity_generic(nrows, nrows.min(ncols));
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let dim = self.diag.len();
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for i in (0..dim).rev() {
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let axis = self.col_piv_qr.slice_range(i.., i);
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// TODO: sometimes, the axis might have a zero magnitude.
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let refl = Reflection::new(Unit::new_unchecked(axis), N::zero());
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let mut res_rows = res.slice_range_mut(i.., i..);
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refl.reflect_with_sign(&mut res_rows, self.diag[i].signum());
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}
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res
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}
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/// Retrieves the column permutation 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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/// Unpacks this decomposition into its two matrix factors.
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pub fn unpack(
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self,
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) -> (
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MatrixMN<N, R, DimMinimum<R, C>>,
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MatrixMN<N, DimMinimum<R, C>, C>,
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PermutationSequence<DimMinimum<R, C>>,
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)
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where
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DimMinimum<R, C>: DimMin<C, Output = DimMinimum<R, C>>,
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DefaultAllocator: Allocator<N, R, DimMinimum<R, C>>
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+ Reallocator<N, R, C, DimMinimum<R, C>, C>
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+ Allocator<(usize, usize), DimMinimum<R, C>>,
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{
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(self.q(), self.r(), self.p)
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}
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#[doc(hidden)]
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pub fn col_piv_qr_internal(&self) -> &MatrixMN<N, R, C> {
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&self.col_piv_qr
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}
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/// Multiplies the provided matrix by the transpose of the `Q` matrix of this decomposition.
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pub fn q_tr_mul<R2: Dim, C2: Dim, S2>(&self, rhs: &mut Matrix<N, R2, C2, S2>)
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where
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S2: StorageMut<N, R2, C2>,
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{
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let dim = self.diag.len();
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for i in 0..dim {
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let axis = self.col_piv_qr.slice_range(i.., i);
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let refl = Reflection::new(Unit::new_unchecked(axis), N::zero());
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let mut rhs_rows = rhs.rows_range_mut(i..);
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refl.reflect_with_sign(&mut rhs_rows, self.diag[i].signum().conjugate());
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}
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}
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}
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impl<N: ComplexField, D: DimMin<D, Output = D>> ColPivQR<N, D, D>
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where
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DefaultAllocator:
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Allocator<N, D, D> + Allocator<N, D> + Allocator<(usize, usize), DimMinimum<D, D>>,
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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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/// Returns `None` if `self` is not invertible.
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pub fn solve<R2: Dim, C2: Dim, S2>(
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&self,
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b: &Matrix<N, R2, C2, S2>,
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) -> Option<MatrixMN<N, R2, C2>>
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where
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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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{
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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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} 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 garbage.
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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
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S2: StorageMut<N, R2, C2>,
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ShapeConstraint: SameNumberOfRows<R2, D>,
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{
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assert_eq!(
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self.col_piv_qr.nrows(),
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b.nrows(),
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"ColPivQR solve matrix dimension mismatch."
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);
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assert!(
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self.col_piv_qr.is_square(),
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"ColPivQR solve: unable to solve a non-square system."
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);
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self.q_tr_mul(b);
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let solved = self.solve_upper_triangular_mut(b);
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self.p.inv_permute_rows(b);
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solved
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}
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// TODO: duplicate code from the `solve` module.
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fn solve_upper_triangular_mut<R2: Dim, C2: Dim, S2>(
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&self,
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b: &mut Matrix<N, R2, C2, S2>,
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) -> bool
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where
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S2: StorageMut<N, R2, C2>,
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ShapeConstraint: SameNumberOfRows<R2, D>,
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{
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let dim = self.col_piv_qr.nrows();
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for k in 0..b.ncols() {
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let mut b = b.column_mut(k);
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for i in (0..dim).rev() {
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let coeff;
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unsafe {
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let diag = self.diag.vget_unchecked(i).modulus();
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if diag.is_zero() {
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return false;
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}
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coeff = b.vget_unchecked(i).unscale(diag);
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*b.vget_unchecked_mut(i) = coeff;
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}
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b.rows_range_mut(..i)
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.axpy(-coeff, &self.col_piv_qr.slice_range(..i, i), N::one());
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}
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}
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true
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}
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/// Computes the inverse of the decomposed matrix.
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///
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/// Returns `None` if the decomposed matrix is not invertible.
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pub fn try_inverse(&self) -> Option<MatrixN<N, D>> {
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assert!(
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self.col_piv_qr.is_square(),
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"ColPivQR inverse: unable to compute the inverse of a non-square matrix."
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);
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// TODO: is there a less naive method ?
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let (nrows, ncols) = self.col_piv_qr.data.shape();
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let mut res = MatrixN::identity_generic(nrows, ncols);
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if self.solve_mut(&mut res) {
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Some(res)
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} else {
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None
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}
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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!(
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self.col_piv_qr.is_square(),
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"ColPivQR: unable to test the invertibility of a non-square matrix."
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);
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for i in 0..self.diag.len() {
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if self.diag[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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/// Computes the determinant of the decomposed matrix.
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pub fn determinant(&self) -> N {
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let dim = self.col_piv_qr.nrows();
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assert!(
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self.col_piv_qr.is_square(),
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"ColPivQR determinant: unable to compute the determinant of a non-square matrix."
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);
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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.diag.vget_unchecked(i) };
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}
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res * self.p.determinant()
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}
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}
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@ -1,8 +1,8 @@
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use crate::storage::Storage;
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use crate::{
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Allocator, Bidiagonal, Cholesky, ComplexField, DefaultAllocator, Dim, DimDiff, DimMin,
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DimMinimum, DimSub, FullPivLU, Hessenberg, Matrix, Schur, SymmetricEigen, SymmetricTridiagonal,
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LU, QR, SVD, U1,
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Allocator, Bidiagonal, Cholesky, ColPivQR, ComplexField, DefaultAllocator, Dim, DimDiff,
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DimMin, DimMinimum, DimSub, FullPivLU, Hessenberg, Matrix, Schur, SymmetricEigen,
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SymmetricTridiagonal, LU, QR, SVD, U1,
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};
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/// # Rectangular matrix decomposition
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@ -13,8 +13,9 @@ use crate::{
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/// | Decomposition | Factors | Details |
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/// | -------------------------|---------------------|--------------|
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/// | QR | `Q * R` | `Q` is an unitary matrix, and `R` is upper-triangular. |
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/// | QR with column pivoting | `Q * R * P⁻¹` | `Q` is an unitary matrix, and `R` is upper-triangular. `P` is a permutation matrix. |
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/// | LU with partial pivoting | `P⁻¹ * L * U` | `L` is lower-triangular with a diagonal filled with `1` and `U` is upper-triangular. `P` is a permutation matrix. |
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/// | LU with full pivoting | `P⁻¹ * L * U ~ Q⁻¹` | `L` is lower-triangular with a diagonal filled with `1` and `U` is upper-triangular. `P` and `Q` are permutation matrices. |
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/// | LU with full pivoting | `P⁻¹ * L * U * Q⁻¹` | `L` is lower-triangular with a diagonal filled with `1` and `U` is upper-triangular. `P` and `Q` are permutation matrices. |
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/// | SVD | `U * Σ * Vᵀ` | `U` and `V` are two orthogonal matrices and `Σ` is a diagonal matrix containing the singular values. |
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impl<N: ComplexField, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
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/// Computes the bidiagonalization using householder reflections.
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@ -60,6 +61,18 @@ impl<N: ComplexField, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
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QR::new(self.into_owned())
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}
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/// Computes the QR decomposition (with column pivoting) of this matrix.
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pub fn col_piv_qr(self) -> ColPivQR<N, R, C>
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where
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R: DimMin<C>,
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DefaultAllocator: Allocator<N, R, C>
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+ Allocator<N, R>
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+ Allocator<N, DimMinimum<R, C>>
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+ Allocator<(usize, usize), DimMinimum<R, C>>,
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{
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ColPivQR::new(self.into_owned())
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}
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/// Computes the Singular Value Decomposition using implicit shift.
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pub fn svd(self, compute_u: bool, compute_v: bool) -> SVD<N, R, C>
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where
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@ -8,6 +8,7 @@ mod determinant;
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// TODO: this should not be needed. However, the exp uses
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// explicit float operations on `f32` and `f64`. We need to
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// get rid of these to allow exp to be used on a no-std context.
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mod col_piv_qr;
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mod decomposition;
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#[cfg(feature = "std")]
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mod exp;
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@ -31,6 +32,7 @@ mod symmetric_tridiagonal;
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pub use self::bidiagonal::*;
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pub use self::cholesky::*;
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pub use self::col_piv_qr::*;
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pub use self::convolution::*;
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#[cfg(feature = "std")]
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pub use self::exp::*;
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@ -60,8 +60,8 @@ where
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return QR { qr: matrix, diag };
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}
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for ite in 0..min_nrows_ncols.value() {
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householder::clear_column_unchecked(&mut matrix, &mut diag[ite], ite, 0, None);
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for i in 0..min_nrows_ncols.value() {
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householder::clear_column_unchecked(&mut matrix, &mut diag[i], i, 0, None);
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}
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QR { qr: matrix, diag }
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161
tests/linalg/col_piv_qr.rs
Normal file
161
tests/linalg/col_piv_qr.rs
Normal file
@ -0,0 +1,161 @@
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#[cfg_attr(rustfmt, rustfmt_skip)]
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use na::Matrix4;
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#[test]
|
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fn col_piv_qr() {
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let m = Matrix4::new(
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1.0, -1.0, 2.0, 1.0, -1.0, 3.0, -1.0, -1.0, 3.0, -5.0, 5.0, 3.0, 1.0, 2.0, 1.0, -2.0,
|
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);
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let col_piv_qr = m.col_piv_qr();
|
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assert!(relative_eq!(
|
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col_piv_qr.determinant(),
|
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0.0,
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epsilon = 1.0e-7
|
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));
|
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|
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let (q, r, p) = col_piv_qr.unpack();
|
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|
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let mut qr = q * r;
|
||||
p.inv_permute_columns(&mut qr);
|
||||
|
||||
assert!(relative_eq!(m, qr, epsilon = 1.0e-7));
|
||||
}
|
||||
|
||||
#[cfg(feature = "arbitrary")]
|
||||
mod quickcheck_tests {
|
||||
macro_rules! gen_tests(
|
||||
($module: ident, $scalar: ty) => {
|
||||
mod $module {
|
||||
use na::{DMatrix, DVector, Matrix3x5, Matrix4, Matrix4x3, Matrix5x3, Vector4};
|
||||
use std::cmp;
|
||||
#[allow(unused_imports)]
|
||||
use crate::core::helper::{RandScalar, RandComplex};
|
||||
|
||||
quickcheck! {
|
||||
fn col_piv_qr(m: DMatrix<$scalar>) -> bool {
|
||||
let m = m.map(|e| e.0);
|
||||
let col_piv_qr = m.clone().col_piv_qr();
|
||||
let (q, r, p) = col_piv_qr.unpack();
|
||||
let mut qr = &q * &r;
|
||||
p.inv_permute_columns(&mut qr);
|
||||
|
||||
println!("m: {}", m);
|
||||
println!("col_piv_qr: {}", &q * &r);
|
||||
|
||||
relative_eq!(m, &qr, epsilon = 1.0e-7) &&
|
||||
q.is_orthogonal(1.0e-7)
|
||||
}
|
||||
|
||||
fn col_piv_qr_static_5_3(m: Matrix5x3<$scalar>) -> bool {
|
||||
let m = m.map(|e| e.0);
|
||||
let col_piv_qr = m.col_piv_qr();
|
||||
let (q, r, p) = col_piv_qr.unpack();
|
||||
let mut qr = q * r;
|
||||
p.inv_permute_columns(&mut qr);
|
||||
|
||||
relative_eq!(m, qr, epsilon = 1.0e-7) &&
|
||||
q.is_orthogonal(1.0e-7)
|
||||
}
|
||||
|
||||
fn col_piv_qr_static_3_5(m: Matrix3x5<$scalar>) -> bool {
|
||||
let m = m.map(|e| e.0);
|
||||
let col_piv_qr = m.col_piv_qr();
|
||||
let (q, r, p) = col_piv_qr.unpack();
|
||||
let mut qr = q * r;
|
||||
p.inv_permute_columns(&mut qr);
|
||||
|
||||
relative_eq!(m, qr, epsilon = 1.0e-7) &&
|
||||
q.is_orthogonal(1.0e-7)
|
||||
}
|
||||
|
||||
fn col_piv_qr_static_square(m: Matrix4<$scalar>) -> bool {
|
||||
let m = m.map(|e| e.0);
|
||||
let col_piv_qr = m.col_piv_qr();
|
||||
let (q, r, p) = col_piv_qr.unpack();
|
||||
let mut qr = q * r;
|
||||
p.inv_permute_columns(&mut qr);
|
||||
|
||||
println!("{}{}{}{}", q, r, qr, m);
|
||||
|
||||
relative_eq!(m, qr, epsilon = 1.0e-7) &&
|
||||
q.is_orthogonal(1.0e-7)
|
||||
}
|
||||
|
||||
fn col_piv_qr_solve(n: usize, nb: usize) -> bool {
|
||||
if n != 0 && nb != 0 {
|
||||
let n = cmp::min(n, 50); // To avoid slowing down the test too much.
|
||||
let nb = cmp::min(nb, 50); // To avoid slowing down the test too much.
|
||||
let m = DMatrix::<$scalar>::new_random(n, n).map(|e| e.0);
|
||||
|
||||
let col_piv_qr = m.clone().col_piv_qr();
|
||||
let b1 = DVector::<$scalar>::new_random(n).map(|e| e.0);
|
||||
let b2 = DMatrix::<$scalar>::new_random(n, nb).map(|e| e.0);
|
||||
|
||||
if col_piv_qr.is_invertible() {
|
||||
let sol1 = col_piv_qr.solve(&b1).unwrap();
|
||||
let sol2 = col_piv_qr.solve(&b2).unwrap();
|
||||
|
||||
return relative_eq!(&m * sol1, b1, epsilon = 1.0e-6) &&
|
||||
relative_eq!(&m * sol2, b2, epsilon = 1.0e-6)
|
||||
}
|
||||
}
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
fn col_piv_qr_solve_static(m: Matrix4<$scalar>) -> bool {
|
||||
let m = m.map(|e| e.0);
|
||||
let col_piv_qr = m.col_piv_qr();
|
||||
let b1 = Vector4::<$scalar>::new_random().map(|e| e.0);
|
||||
let b2 = Matrix4x3::<$scalar>::new_random().map(|e| e.0);
|
||||
|
||||
if col_piv_qr.is_invertible() {
|
||||
let sol1 = col_piv_qr.solve(&b1).unwrap();
|
||||
let sol2 = col_piv_qr.solve(&b2).unwrap();
|
||||
|
||||
relative_eq!(m * sol1, b1, epsilon = 1.0e-6) &&
|
||||
relative_eq!(m * sol2, b2, epsilon = 1.0e-6)
|
||||
}
|
||||
else {
|
||||
false
|
||||
}
|
||||
}
|
||||
|
||||
fn col_piv_qr_inverse(n: usize) -> bool {
|
||||
let n = cmp::max(1, cmp::min(n, 15)); // To avoid slowing down the test too much.
|
||||
let m = DMatrix::<$scalar>::new_random(n, n).map(|e| e.0);
|
||||
|
||||
if let Some(m1) = m.clone().col_piv_qr().try_inverse() {
|
||||
let id1 = &m * &m1;
|
||||
let id2 = &m1 * &m;
|
||||
|
||||
id1.is_identity(1.0e-5) && id2.is_identity(1.0e-5)
|
||||
}
|
||||
else {
|
||||
true
|
||||
}
|
||||
}
|
||||
|
||||
fn col_piv_qr_inverse_static(m: Matrix4<$scalar>) -> bool {
|
||||
let m = m.map(|e| e.0);
|
||||
let col_piv_qr = m.col_piv_qr();
|
||||
|
||||
if let Some(m1) = col_piv_qr.try_inverse() {
|
||||
let id1 = &m * &m1;
|
||||
let id2 = &m1 * &m;
|
||||
|
||||
id1.is_identity(1.0e-5) && id2.is_identity(1.0e-5)
|
||||
}
|
||||
else {
|
||||
true
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
);
|
||||
|
||||
gen_tests!(complex, RandComplex<f64>);
|
||||
gen_tests!(f64, RandScalar<f64>);
|
||||
}
|
||||
@ -1,6 +1,7 @@
|
||||
mod balancing;
|
||||
mod bidiagonal;
|
||||
mod cholesky;
|
||||
mod col_piv_qr;
|
||||
mod convolution;
|
||||
mod eigen;
|
||||
mod exp;
|
||||
|
||||
Loading…
Reference in New Issue
Block a user