QR factorization with column pivoting
This commit is contained in:
russellb23
Crozet Sébastien
parent
f46d1b4abb
commit
f8c0195f0f
|
|
@ -0,0 +1,328 @@
|
|||
#[cfg(feature = "serde-serialize")]
|
||||
use serde::{Deserialize, Serialize};
|
||||
use num::Zero;
|
||||
|
||||
use alga::general::ComplexField;
|
||||
use crate::allocator::{Allocator, Reallocator};
|
||||
use crate::base::{DefaultAllocator, Matrix, MatrixMN, MatrixN, Unit, VectorN};
|
||||
use crate::constraint::{SameNumberOfRows, ShapeConstraint};
|
||||
use crate::dimension::{Dim, DimMin, DimMinimum, U1};
|
||||
use crate::storage::{Storage, StorageMut};
|
||||
|
||||
use crate::geometry::Reflection;
|
||||
use crate::linalg::householder;
|
||||
|
||||
//=============================================================================
|
||||
use alga::general::RealField;
|
||||
use crate::linalg::PermutationSequence;
|
||||
use crate::base::VecStorage;
|
||||
use crate::base::Dynamic;
|
||||
use crate::base::DVector;
|
||||
//=============================================================================
|
||||
|
||||
/// The QRP decomposition of a general matrix.
|
||||
#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
|
||||
#[cfg_attr(
|
||||
feature = "serde-serialize",
|
||||
serde(bound(
|
||||
serialize = "DefaultAllocator: Allocator<N, R, C> +
|
||||
Allocator<N, DimMinimum<R, C>>,
|
||||
MatrixMN<N, R, C>: Serialize,
|
||||
PermutationSequence<DimMinimum<R, C>>: Serialize,
|
||||
VectorN<N, DimMinimum<R, C>>: Serialize"
|
||||
))
|
||||
)]
|
||||
#[cfg_attr(
|
||||
feature = "serde-serialize",
|
||||
serde(bound(
|
||||
deserialize = "DefaultAllocator: Allocator<N, R, C> +
|
||||
Allocator<N, DimMinimum<R, C>>,
|
||||
MatrixMN<N, R, C>: Deserialize<'de>,
|
||||
PermutationSequence<DimMinimum<R, C>>: Deserialize<'de>,
|
||||
VectorN<N, DimMinimum<R, C>>: Deserialize<'de>"
|
||||
))
|
||||
)]
|
||||
#[derive(Clone, Debug)]
|
||||
pub struct QRP<N: ComplexField, R: DimMin<C>, C: Dim>
|
||||
where DefaultAllocator: Allocator<N, R, C> + Allocator<N, DimMinimum<R, C>> +
|
||||
Allocator<(usize, usize), DimMinimum<R, C>>,
|
||||
{
|
||||
qrp: MatrixMN<N, R, C>,
|
||||
p: PermutationSequence<DimMinimum<R, C>>,
|
||||
diag: VectorN<N, DimMinimum<R, C>>,
|
||||
}
|
||||
|
||||
impl<N: ComplexField, R: DimMin<C>, C: Dim> Copy for QRP<N, R, C>
|
||||
where
|
||||
DefaultAllocator: Allocator<N, R, C> + Allocator<N, DimMinimum<R, C>> +
|
||||
Allocator<(usize, usize), DimMinimum<R, C>>,
|
||||
MatrixMN<N, R, C>: Copy,
|
||||
PermutationSequence<DimMinimum<R, C>>: Copy,
|
||||
VectorN<N, DimMinimum<R, C>>: Copy,
|
||||
{}
|
||||
|
||||
impl<N: ComplexField, R: DimMin<C>, C: Dim> QRP<N, R, C>
|
||||
where DefaultAllocator: Allocator<N, R, C> + Allocator<N, R> + Allocator<N, DimMinimum<R, C>> + Allocator<(usize, usize), DimMinimum<R, C>>
|
||||
{
|
||||
/// Computes the QRP decomposition using householder reflections.
|
||||
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 diag = unsafe { MatrixMN::new_uninitialized_generic(min_nrows_ncols, U1) };
|
||||
|
||||
if min_nrows_ncols.value() == 0 {
|
||||
return QRP {
|
||||
qrp: matrix,
|
||||
p: p,
|
||||
diag: diag,
|
||||
};
|
||||
}
|
||||
|
||||
for ite in 0..min_nrows_ncols.value() {
|
||||
let piv = matrix.slice_range(ite.., ite..).icamax_full();
|
||||
let col_piv = piv.1 + ite;
|
||||
matrix.swap_columns(ite, col_piv);
|
||||
p.append_permutation(ite, col_piv);
|
||||
|
||||
householder::clear_column_unchecked(&mut matrix, &mut diag[ite], ite, 0, None);
|
||||
}
|
||||
|
||||
QRP {
|
||||
qrp: matrix,
|
||||
p: p,
|
||||
diag: diag,
|
||||
}
|
||||
}
|
||||
|
||||
/// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
|
||||
#[inline]
|
||||
pub fn r(&self) -> MatrixMN<N, DimMinimum<R, C>, C>
|
||||
where
|
||||
DefaultAllocator: Allocator<N, DimMinimum<R, C>, C>,
|
||||
{
|
||||
let (nrows, ncols) = self.qrp.data.shape();
|
||||
let mut res = self.qrp.rows_generic(0, nrows.min(ncols)).upper_triangle();
|
||||
res.set_partial_diagonal(self.diag.iter().map(|e| N::from_real(e.modulus())));
|
||||
res
|
||||
}
|
||||
|
||||
/// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
|
||||
///
|
||||
/// This is usually faster than `r` but consumes `self`.
|
||||
#[inline]
|
||||
pub fn unpack_r(self) -> MatrixMN<N, DimMinimum<R, C>, C>
|
||||
where
|
||||
DefaultAllocator: Reallocator<N, R, C, DimMinimum<R, C>, C>,
|
||||
{
|
||||
let (nrows, ncols) = self.qrp.data.shape();
|
||||
let mut res = self.qrp.resize_generic(nrows.min(ncols), ncols, N::zero());
|
||||
res.fill_lower_triangle(N::zero(), 1);
|
||||
res.set_partial_diagonal(self.diag.iter().map(|e| N::from_real(e.modulus())));
|
||||
res
|
||||
}
|
||||
|
||||
/// Computes the orthogonal matrix `Q` of this decomposition.
|
||||
pub fn q(&self) -> MatrixMN<N, R, DimMinimum<R, C>>
|
||||
where DefaultAllocator: Allocator<N, R, DimMinimum<R, C>> {
|
||||
let (nrows, ncols) = self.qrp.data.shape();
|
||||
|
||||
// NOTE: we could build the identity matrix and call q_mul on it.
|
||||
// Instead we don't so that we take in account the matrix sparseness.
|
||||
let mut res = Matrix::identity_generic(nrows, nrows.min(ncols));
|
||||
let dim = self.diag.len();
|
||||
|
||||
for i in (0..dim).rev() {
|
||||
let axis = self.qrp.slice_range(i.., i);
|
||||
// FIXME: sometimes, the axis might have a zero magnitude.
|
||||
let refl = Reflection::new(Unit::new_unchecked(axis), N::zero());
|
||||
|
||||
let mut res_rows = res.slice_range_mut(i.., i..);
|
||||
refl.reflect_with_sign(&mut res_rows, self.diag[i].signum());
|
||||
}
|
||||
|
||||
res
|
||||
}
|
||||
/// The column permutations of this decomposition.
|
||||
#[inline]
|
||||
pub fn p(&self) -> &PermutationSequence<DimMinimum<R, C>> {
|
||||
&self.p
|
||||
}
|
||||
|
||||
/// Unpacks this decomposition into its two matrix factors.
|
||||
pub fn unpack(
|
||||
self,
|
||||
) -> (
|
||||
MatrixMN<N, R, DimMinimum<R, C>>,
|
||||
MatrixMN<N, DimMinimum<R, C>, C>,
|
||||
)
|
||||
where
|
||||
DimMinimum<R, C>: DimMin<C, Output = DimMinimum<R, C>>,
|
||||
DefaultAllocator:
|
||||
Allocator<N, R, DimMinimum<R, C>> + Reallocator<N, R, C, DimMinimum<R, C>, C>,
|
||||
{
|
||||
(self.q(), self.unpack_r())
|
||||
}
|
||||
|
||||
#[doc(hidden)]
|
||||
pub fn qrp_internal(&self) -> &MatrixMN<N, R, C> {
|
||||
&self.qrp
|
||||
}
|
||||
|
||||
/// Multiplies the provided matrix by the transpose of the `Q` matrix of this decomposition.
|
||||
pub fn q_tr_mul<R2: Dim, C2: Dim, S2>(&self, rhs: &mut Matrix<N, R2, C2, S2>)
|
||||
// FIXME: do we need a static constraint on the number of rows of rhs?
|
||||
where S2: StorageMut<N, R2, C2> {
|
||||
let dim = self.diag.len();
|
||||
|
||||
for i in 0..dim {
|
||||
let axis = self.qrp.slice_range(i.., i);
|
||||
let refl = Reflection::new(Unit::new_unchecked(axis), N::zero());
|
||||
|
||||
let mut rhs_rows = rhs.rows_range_mut(i..);
|
||||
refl.reflect_with_sign(&mut rhs_rows, self.diag[i].signum().conjugate());
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
impl<N: ComplexField, D: DimMin<D, Output = D>> QRP<N, D, D>
|
||||
where DefaultAllocator: Allocator<N, D, D> + Allocator<N, D> +
|
||||
Allocator<(usize, usize), DimMinimum<D, D>>
|
||||
{
|
||||
/// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
|
||||
///
|
||||
/// Returns `None` if `self` 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
|
||||
/// overwritten with garbage.
|
||||
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.qrp.nrows(),
|
||||
b.nrows(),
|
||||
"QRP solve matrix dimension mismatch."
|
||||
);
|
||||
assert!(
|
||||
self.qrp.is_square(),
|
||||
"QRP solve: unable to solve a non-square system."
|
||||
);
|
||||
|
||||
self.q_tr_mul(b);
|
||||
self.solve_upper_triangular_mut(b)
|
||||
}
|
||||
|
||||
// FIXME: duplicate code from the `solve` module.
|
||||
fn solve_upper_triangular_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>,
|
||||
{
|
||||
let dim = self.qrp.nrows();
|
||||
|
||||
for k in 0..b.ncols() {
|
||||
let mut b = b.column_mut(k);
|
||||
for i in (0..dim).rev() {
|
||||
let coeff;
|
||||
|
||||
unsafe {
|
||||
let diag = self.diag.vget_unchecked(i).modulus();
|
||||
|
||||
if diag.is_zero() {
|
||||
return false;
|
||||
}
|
||||
|
||||
coeff = b.vget_unchecked(i).unscale(diag);
|
||||
*b.vget_unchecked_mut(i) = coeff;
|
||||
}
|
||||
|
||||
b.rows_range_mut(..i)
|
||||
.axpy(-coeff, &self.qrp.slice_range(..i, i), N::one());
|
||||
}
|
||||
}
|
||||
|
||||
true
|
||||
}
|
||||
|
||||
/// 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.qrp.is_square(),
|
||||
"QRP inverse: unable to compute the inverse of a non-square matrix."
|
||||
);
|
||||
|
||||
// FIXME: is there a less naive method ?
|
||||
let (nrows, ncols) = self.qrp.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.qrp.is_square(),
|
||||
"QRP: unable to test the invertibility of a non-square matrix."
|
||||
);
|
||||
|
||||
for i in 0..self.diag.len() {
|
||||
if self.diag[i].is_zero() {
|
||||
return false;
|
||||
}
|
||||
}
|
||||
|
||||
true
|
||||
}
|
||||
|
||||
// /// Computes the determinant of the decomposed matrix.
|
||||
// pub fn determinant(&self) -> N {
|
||||
// let dim = self.qrp.nrows();
|
||||
// assert!(self.qrp.is_square(), "QRP determinant: unable to compute the determinant of a non-square matrix.");
|
||||
|
||||
// let mut res = N::one();
|
||||
// for i in 0 .. dim {
|
||||
// res *= unsafe { *self.diag.vget_unchecked(i) };
|
||||
// }
|
||||
|
||||
// res self.q_determinant()
|
||||
// }
|
||||
}
|
||||
|
||||
impl<N: ComplexField, R: DimMin<C>, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S>
|
||||
where DefaultAllocator: Allocator<N, R, C> + Allocator<N, R> + Allocator<N, DimMinimum<R, C>> + Allocator<(usize, usize), DimMinimum<R, C>>
|
||||
{
|
||||
/// Computes the QRP decomposition of this matrix.
|
||||
pub fn qrp(self) -> QRP<N, R, C> {
|
||||
QRP::new(self.into_owned())
|
||||
}
|
||||
}
|
||||
Loading…
Reference in New Issue