nalgebra/linalg/bidiagonal.rs
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#[cfg(feature = "serde-serialize-no-std")]
use serde::{Deserialize, Serialize};
use crate::allocator::Allocator;
use crate::base::{DefaultAllocator, Matrix, OMatrix, OVector, Unit};
use crate::dimension::{Const, Dim, DimDiff, DimMin, DimMinimum, DimSub, U1};
use simba::scalar::ComplexField;
use crate::geometry::Reflection;
use crate::linalg::householder;
use crate::num::Zero;
use std::mem::MaybeUninit;
/// The bidiagonalization of a general matrix.
#[cfg_attr(feature = "serde-serialize-no-std", derive(Serialize, Deserialize))]
#[cfg_attr(
feature = "serde-serialize-no-std",
serde(bound(serialize = "DimMinimum<R, C>: DimSub<U1>,
DefaultAllocator: Allocator<R, C> +
Allocator<DimMinimum<R, C>> +
Allocator<DimDiff<DimMinimum<R, C>, U1>>,
OMatrix<T, R, C>: Serialize,
OVector<T, DimMinimum<R, C>>: Serialize,
OVector<T, DimDiff<DimMinimum<R, C>, U1>>: Serialize"))
)]
#[cfg_attr(
feature = "serde-serialize-no-std",
serde(bound(deserialize = "DimMinimum<R, C>: DimSub<U1>,
DefaultAllocator: Allocator<R, C> +
Allocator<DimMinimum<R, C>> +
Allocator<DimDiff<DimMinimum<R, C>, U1>>,
OMatrix<T, R, C>: Deserialize<'de>,
OVector<T, DimMinimum<R, C>>: Deserialize<'de>,
OVector<T, DimDiff<DimMinimum<R, C>, U1>>: Deserialize<'de>"))
)]
#[derive(Clone, Debug)]
pub struct Bidiagonal<T: ComplexField, R: DimMin<C>, C: Dim>
where
DimMinimum<R, C>: DimSub<U1>,
DefaultAllocator:
Allocator<R, C> + Allocator<DimMinimum<R, C>> + Allocator<DimDiff<DimMinimum<R, C>, U1>>,
{
// TODO: perhaps we should pack the axes into different vectors so that axes for `v_t` are
// contiguous. This prevents some useless copies.
uv: OMatrix<T, R, C>,
/// The diagonal elements of the decomposed matrix.
diagonal: OVector<T, DimMinimum<R, C>>,
/// The off-diagonal elements of the decomposed matrix.
off_diagonal: OVector<T, DimDiff<DimMinimum<R, C>, U1>>,
upper_diagonal: bool,
}
impl<T: ComplexField, R: DimMin<C>, C: Dim> Copy for Bidiagonal<T, R, C>
where
DimMinimum<R, C>: DimSub<U1>,
DefaultAllocator:
Allocator<R, C> + Allocator<DimMinimum<R, C>> + Allocator<DimDiff<DimMinimum<R, C>, U1>>,
OMatrix<T, R, C>: Copy,
OVector<T, DimMinimum<R, C>>: Copy,
OVector<T, DimDiff<DimMinimum<R, C>, U1>>: Copy,
{
}
impl<T: ComplexField, R: DimMin<C>, C: Dim> Bidiagonal<T, R, C>
where
DimMinimum<R, C>: DimSub<U1>,
DefaultAllocator: Allocator<R, C>
+ Allocator<C>
+ Allocator<R>
+ Allocator<DimMinimum<R, C>>
+ Allocator<DimDiff<DimMinimum<R, C>, U1>>,
{
/// Computes the Bidiagonal decomposition using householder reflections.
pub fn new(mut matrix: OMatrix<T, R, C>) -> Self {
let (nrows, ncols) = matrix.shape_generic();
let min_nrows_ncols = nrows.min(ncols);
let dim = min_nrows_ncols.value();
assert!(
dim != 0,
"Cannot compute the bidiagonalization of an empty matrix."
);
let mut diagonal = Matrix::uninit(min_nrows_ncols, Const::<1>);
let mut off_diagonal = Matrix::uninit(min_nrows_ncols.sub(Const::<1>), Const::<1>);
let mut axis_packed = Matrix::zeros_generic(ncols, Const::<1>);
let mut work = Matrix::zeros_generic(nrows, Const::<1>);
let upper_diagonal = nrows.value() >= ncols.value();
if upper_diagonal {
for ite in 0..dim - 1 {
diagonal[ite] = MaybeUninit::new(householder::clear_column_unchecked(
&mut matrix,
ite,
0,
None,
));
off_diagonal[ite] = MaybeUninit::new(householder::clear_row_unchecked(
&mut matrix,
&mut axis_packed,
&mut work,
ite,
1,
));
}
diagonal[dim - 1] = MaybeUninit::new(householder::clear_column_unchecked(
&mut matrix,
dim - 1,
0,
None,
));
} else {
for ite in 0..dim - 1 {
diagonal[ite] = MaybeUninit::new(householder::clear_row_unchecked(
&mut matrix,
&mut axis_packed,
&mut work,
ite,
0,
));
off_diagonal[ite] = MaybeUninit::new(householder::clear_column_unchecked(
&mut matrix,
ite,
1,
None,
));
}
diagonal[dim - 1] = MaybeUninit::new(householder::clear_row_unchecked(
&mut matrix,
&mut axis_packed,
&mut work,
dim - 1,
0,
));
}
// Safety: diagonal and off_diagonal have been fully initialized.
let (diagonal, off_diagonal) =
unsafe { (diagonal.assume_init(), off_diagonal.assume_init()) };
Bidiagonal {
uv: matrix,
diagonal,
off_diagonal,
upper_diagonal,
}
}
/// Indicates whether this decomposition contains an upper-diagonal matrix.
#[inline]
#[must_use]
pub fn is_upper_diagonal(&self) -> bool {
self.upper_diagonal
}
#[inline]
fn axis_shift(&self) -> (usize, usize) {
if self.upper_diagonal {
(0, 1)
} else {
(1, 0)
}
}
/// Unpacks this decomposition into its three matrix factors `(U, D, V^t)`.
///
/// The decomposed matrix `M` is equal to `U * D * V^t`.
#[inline]
pub fn unpack(
self,
) -> (
OMatrix<T, R, DimMinimum<R, C>>,
OMatrix<T, DimMinimum<R, C>, DimMinimum<R, C>>,
OMatrix<T, DimMinimum<R, C>, C>,
)
where
DefaultAllocator: Allocator<DimMinimum<R, C>, DimMinimum<R, C>>
+ Allocator<R, DimMinimum<R, C>>
+ Allocator<DimMinimum<R, C>, C>,
{
// TODO: optimize by calling a reallocator.
(self.u(), self.d(), self.v_t())
}
/// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
#[inline]
#[must_use]
pub fn d(&self) -> OMatrix<T, DimMinimum<R, C>, DimMinimum<R, C>>
where
DefaultAllocator: Allocator<DimMinimum<R, C>, DimMinimum<R, C>>,
{
let (nrows, ncols) = self.uv.shape_generic();
let d = nrows.min(ncols);
let mut res = OMatrix::identity_generic(d, d);
res.set_partial_diagonal(
self.diagonal
.iter()
.map(|e| T::from_real(e.clone().modulus())),
);
let start = self.axis_shift();
res.view_mut(start, (d.value() - 1, d.value() - 1))
.set_partial_diagonal(
self.off_diagonal
.iter()
.map(|e| T::from_real(e.clone().modulus())),
);
res
}
/// Computes the orthogonal matrix `U` of this `U * D * V` decomposition.
// TODO: code duplication with householder::assemble_q.
// Except that we are returning a rectangular matrix here.
#[must_use]
pub fn u(&self) -> OMatrix<T, R, DimMinimum<R, C>>
where
DefaultAllocator: Allocator<R, DimMinimum<R, C>>,
{
let (nrows, ncols) = self.uv.shape_generic();
let mut res = Matrix::identity_generic(nrows, nrows.min(ncols));
let dim = self.diagonal.len();
let shift = self.axis_shift().0;
for i in (0..dim - shift).rev() {
let axis = self.uv.view_range(i + shift.., i);
// Sometimes, the axis might have a zero magnitude.
if axis.norm_squared().is_zero() {
continue;
}
let refl = Reflection::new(Unit::new_unchecked(axis), T::zero());
let mut res_rows = res.view_range_mut(i + shift.., i..);
let sign = if self.upper_diagonal {
self.diagonal[i].clone().signum()
} else {
self.off_diagonal[i].clone().signum()
};
refl.reflect_with_sign(&mut res_rows, sign);
}
res
}
/// Computes the orthogonal matrix `V_t` of this `U * D * V_t` decomposition.
#[must_use]
pub fn v_t(&self) -> OMatrix<T, DimMinimum<R, C>, C>
where
DefaultAllocator: Allocator<DimMinimum<R, C>, C>,
{
let (nrows, ncols) = self.uv.shape_generic();
let min_nrows_ncols = nrows.min(ncols);
let mut res = Matrix::identity_generic(min_nrows_ncols, ncols);
let mut work = Matrix::zeros_generic(min_nrows_ncols, Const::<1>);
let mut axis_packed = Matrix::zeros_generic(ncols, Const::<1>);
let shift = self.axis_shift().1;
for i in (0..min_nrows_ncols.value() - shift).rev() {
let axis = self.uv.view_range(i, i + shift..);
let mut axis_packed = axis_packed.rows_range_mut(i + shift..);
axis_packed.tr_copy_from(&axis);
// Sometimes, the axis might have a zero magnitude.
if axis_packed.norm_squared().is_zero() {
continue;
}
let refl = Reflection::new(Unit::new_unchecked(axis_packed), T::zero());
let mut res_rows = res.view_range_mut(i.., i + shift..);
let sign = if self.upper_diagonal {
self.off_diagonal[i].clone().signum()
} else {
self.diagonal[i].clone().signum()
};
refl.reflect_rows_with_sign(&mut res_rows, &mut work.rows_range_mut(i..), sign);
}
res
}
/// The diagonal part of this decomposed matrix.
#[must_use]
pub fn diagonal(&self) -> OVector<T::RealField, DimMinimum<R, C>>
where
DefaultAllocator: Allocator<DimMinimum<R, C>>,
{
self.diagonal.map(|e| e.modulus())
}
/// The off-diagonal part of this decomposed matrix.
#[must_use]
pub fn off_diagonal(&self) -> OVector<T::RealField, DimDiff<DimMinimum<R, C>, U1>>
where
DefaultAllocator: Allocator<DimDiff<DimMinimum<R, C>, U1>>,
{
self.off_diagonal.map(|e| e.modulus())
}
#[doc(hidden)]
pub fn uv_internal(&self) -> &OMatrix<T, R, C> {
&self.uv
}
}
// impl<T: ComplexField, D: DimMin<D, Output = D> + DimSub<Dyn>> Bidiagonal<T, D, D>
// where DefaultAllocator: Allocator<D, D> +
// Allocator<D> {
// /// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
// pub fn solve<R2: Dim, C2: Dim, S2>(&self, b: &Matrix<T, R2, C2, S2>) -> OMatrix<T, R2, C2>
// where S2: StorageMut<T, R2, C2>,
// ShapeConstraint: SameNumberOfRows<R2, D> {
// let mut res = b.clone_owned();
// self.solve_mut(&mut res);
// res
// }
//
// /// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
// pub fn solve_mut<R2: Dim, C2: Dim, S2>(&self, b: &mut Matrix<T, R2, C2, S2>)
// where S2: StorageMut<T, R2, C2>,
// ShapeConstraint: SameNumberOfRows<R2, D> {
//
// assert_eq!(self.uv.nrows(), b.nrows(), "Bidiagonal solve matrix dimension mismatch.");
// assert!(self.uv.is_square(), "Bidiagonal solve: unable to solve a non-square system.");
//
// self.q_tr_mul(b);
// self.solve_upper_triangular_mut(b);
// }
//
// // TODO: duplicate code from the `solve` module.
// fn solve_upper_triangular_mut<R2: Dim, C2: Dim, S2>(&self, b: &mut Matrix<T, R2, C2, S2>)
// where S2: StorageMut<T, R2, C2>,
// ShapeConstraint: SameNumberOfRows<R2, D> {
//
// let dim = self.uv.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);
// coeff = *b.vget_unchecked(i) / diag;
// *b.vget_unchecked_mut(i) = coeff;
// }
//
// b.rows_range_mut(.. i).axpy(-coeff, &self.uv.view_range(.. i, i), T::one());
// }
// }
// }
//
// /// Computes the inverse of the decomposed matrix.
// pub fn inverse(&self) -> OMatrix<T, D, D> {
// assert!(self.uv.is_square(), "Bidiagonal inverse: unable to compute the inverse of a non-square matrix.");
//
// // TODO: is there a less naive method ?
// let (nrows, ncols) = self.uv.shape_generic();
// let mut res = OMatrix::identity_generic(nrows, ncols);
// self.solve_mut(&mut res);
// res
// }
//
// // /// Computes the determinant of the decomposed matrix.
// // pub fn determinant(&self) -> T {
// // let dim = self.uv.nrows();
// // assert!(self.uv.is_square(), "Bidiagonal determinant: unable to compute the determinant of a non-square matrix.");
//
// // let mut res = T::one();
// // for i in 0 .. dim {
// // res *= unsafe { *self.diag.vget_unchecked(i) };
// // }
//
// // res self.q_determinant()
// // }
// }