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Change the QR implementation, inspired by LAPACK

This commit is contained in:
Jannik Schürg 2020-05-09 16:36:57 +02:00
parent b0ccf13c16
commit 085061a184
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2 changed files with 271 additions and 221 deletions
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240
src/linalg/qr.rs
View File

@ -3,15 +3,12 @@ use num::Zero;
use serde::{Deserialize, Serialize};
use crate::allocator::{Allocator, Reallocator};
use crate::base::{DefaultAllocator, Matrix, MatrixMN, MatrixN, Unit, VectorN};
use crate::base::{DefaultAllocator, Matrix, MatrixMN, MatrixN, VectorN};
use crate::constraint::{SameNumberOfRows, ShapeConstraint};
use crate::dimension::{Dim, DimMin, DimMinimum, U1};
use crate::storage::{Storage, StorageMut};
use simba::scalar::ComplexField;
use crate::geometry::Reflection;
use crate::linalg::householder;
/// The QR decomposition of a general matrix.
#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
#[cfg_attr(
@ -34,7 +31,7 @@ where
DefaultAllocator: Allocator<N, R, C> + Allocator<N, DimMinimum<R, C>>,
{
qr: MatrixMN<N, R, C>,
diag: VectorN<N, DimMinimum<R, C>>,
tau: VectorN<N, DimMinimum<R, C>>,
}
impl<N: ComplexField, R: DimMin<C>, C: Dim> Copy for QR<N, R, C>
@ -47,30 +44,64 @@ where
impl<N: ComplexField, R: DimMin<C>, C: Dim> QR<N, R, C>
where
DefaultAllocator: Allocator<N, R, C> + Allocator<N, R> + Allocator<N, DimMinimum<R, C>>,
DefaultAllocator:
Allocator<N, R, C> + Allocator<N, R> + Allocator<N, C> + Allocator<N, DimMinimum<R, C>>,
{
/// Computes the QR decomposition using householder reflections.
/// Computes the QR 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 diag = unsafe { MatrixMN::new_uninitialized_generic(min_nrows_ncols, U1) };
let mut tau = unsafe { MatrixMN::new_uninitialized_generic(min_nrows_ncols, U1) };
if min_nrows_ncols.value() == 0 {
return QR {
qr: matrix,
diag: diag,
return QR { qr: matrix, tau };
}
let mut work = unsafe { MatrixMN::new_uninitialized_generic(ncols, U1) };
for icol in 0..min_nrows_ncols.value() {
let (mut left, mut right) = matrix.columns_range_pair_mut(icol, icol + 1..);
let mut axis = left.rows_range_mut(icol..);
// Compute the scaled Housholder vector
let (beta, tau_i) = {
let alpha = unsafe { *axis.vget_unchecked(0) };
let xnorm = axis.rows_range(1..).norm();
if xnorm.is_zero() && alpha.imaginary().is_zero() {
(alpha, N::zero())
} else {
let a_r = alpha.real();
let a_i = alpha.imaginary();
// FIXME: Use LAPACK's ?LAPY3 once we have F::max
let reflection_norm = (a_r * a_r + a_i * a_i + xnorm * xnorm).sqrt();
// FIXME: Use reflection_norm.copysign(a_r)
let beta = -reflection_norm.abs() * a_r.signum();
// FIXME: Check for tiny beta and recompute, cf. LAPACK's ?LARFG
let tau = (N::from_real(beta) - alpha).unscale(beta);
let tmp = alpha - N::from_real(beta);
axis.rows_range_mut(1..).apply(|x| x / tmp);
(N::from_real(beta), tau)
}
};
unsafe {
*tau.vget_unchecked_mut(icol) = tau_i;
}
if !tau_i.is_zero() {
// apply the Householder reflection to the remaining columns
unsafe {
*axis.vget_unchecked_mut(0) = N::one();
}
let mut work = work.rows_range_mut(icol + 1..);
work.gemv_ad(N::one(), &right.rows_range(icol..), &axis, N::zero());
right
.rows_range_mut(icol..)
.gerc(-tau_i.conjugate(), &axis, &work, N::one());
}
unsafe {
*axis.vget_unchecked_mut(0) = beta;
}
}
for ite in 0..min_nrows_ncols.value() {
householder::clear_column_unchecked(&mut matrix, &mut diag[ite], ite, 0, None);
}
QR {
qr: matrix,
diag: diag,
}
QR { qr: matrix, tau }
}
/// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
@ -80,9 +111,7 @@ where
DefaultAllocator: Allocator<N, DimMinimum<R, C>, C>,
{
let (nrows, ncols) = self.qr.data.shape();
let mut res = self.qr.rows_generic(0, nrows.min(ncols)).upper_triangle();
res.set_partial_diagonal(self.diag.iter().map(|e| N::from_real(e.modulus())));
res
self.qr.rows_generic(0, nrows.min(ncols)).upper_triangle()
}
/// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
@ -96,32 +125,59 @@ where
let (nrows, ncols) = self.qr.data.shape();
let mut res = self.qr.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
}
/// Compute the first `ncols` columns of `Q`.
///
/// Panics if `ncols` is bigger than the number of rows of the original matrix.
pub fn q_columns<K: Dim>(&self, ncols: K) -> MatrixMN<N, R, K>
where
DefaultAllocator: Allocator<N, R, K> + Allocator<N, K>,
{
let (q_nrows, q_ncols) = self.qr.data.shape();
assert!(
ncols.value() <= q_nrows.value(),
"k must be less than the number of columns"
);
let mut a = MatrixMN::<N, R, K>::identity_generic(q_nrows, ncols);
let mut work = unsafe { VectorN::<N, K>::new_uninitialized_generic(ncols, U1) };
let k = q_nrows.min(q_ncols);
let ncols_to_copy = k.value().min(q_ncols.value());
a.slice_range_mut(.., ..ncols_to_copy)
.copy_from(&self.qr.slice_range(.., ..ncols_to_copy));
for i in (0..k.value()).rev() {
let tau_i = unsafe { *self.tau.vget_unchecked(i) };
if i < q_ncols.value() {
unsafe {
*a.get_unchecked_mut((i, i)) = N::one();
}
let (left, mut right) = a.columns_range_pair_mut(i, i + 1..);
let axis = left.rows_range(i..);
let mut work = work.rows_range_mut(i + 1..);
work.gemv_ad(N::one(), &right.rows_range(i..), &axis, N::zero());
right
.rows_range_mut(i..)
.gerc(-tau_i, &axis, &work, N::one());
}
if i < q_nrows.value() {
a.slice_range_mut(i + 1.., i).apply(|x| x * (-tau_i));
}
unsafe {
*a.get_unchecked_mut((i, i)) = N::one() - tau_i;
}
a.slice_range_mut(0..i, i).fill(N::zero());
}
a
}
/// 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>>,
DefaultAllocator: Allocator<N, R, DimMinimum<R, C>> + Allocator<N, DimMinimum<R, C>>,
{
let (nrows, ncols) = self.qr.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.qr.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
self.q_columns::<DimMinimum<R, C>>(nrows.min(ncols))
}
/// Unpacks this decomposition into its two matrix factors.
@ -133,8 +189,9 @@ where
)
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>,
DefaultAllocator: Allocator<N, R, DimMinimum<R, C>>
+ Allocator<N, DimMinimum<R, C>>
+ Reallocator<N, R, C, DimMinimum<R, C>, C>,
{
(self.q(), self.unpack_r())
}
@ -145,19 +202,27 @@ where
}
/// 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?
pub fn q_tr_mul<R2: Dim, C2: Dim, S2>(&mut self, rhs: &mut Matrix<N, R2, C2, S2>)
where
S2: StorageMut<N, R2, C2>,
ShapeConstraint: SameNumberOfRows<R2, R>,
DefaultAllocator: Allocator<N, C2>,
{
let dim = self.diag.len();
for i in 0..dim {
let axis = self.qr.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());
let ncols = rhs.data.shape().1;
let mut work = unsafe { VectorN::<N, C2>::new_uninitialized_generic(ncols, U1) };
for i in 0..self.tau.len() {
let mut axis = self.qr.slice_range_mut(i.., i);
let temp = unsafe { *axis.vget_unchecked(0) };
unsafe {
*axis.vget_unchecked_mut(0) = N::one();
}
let tau_i = unsafe { *self.tau.vget_unchecked(i) };
work.gemv_ad(N::one(), &rhs.rows_range(i..), &axis, N::zero());
rhs.rows_range_mut(i..)
.gerc(-tau_i.conjugate(), &axis, &work, N::one());
unsafe {
*axis.vget_unchecked_mut(0) = temp;
}
}
}
}
@ -170,13 +235,13 @@ where
///
/// Returns `None` if `self` is not invertible.
pub fn solve<R2: Dim, C2: Dim, S2>(
&self,
&mut self,
b: &Matrix<N, R2, C2, S2>,
) -> Option<MatrixMN<N, R2, C2>>
where
S2: Storage<N, R2, C2>,
ShapeConstraint: SameNumberOfRows<R2, D>,
DefaultAllocator: Allocator<N, R2, C2>,
DefaultAllocator: Allocator<N, R2, C2> + Allocator<N, C2>,
{
let mut res = b.clone_owned();
@ -191,9 +256,10 @@ where
///
/// 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
pub fn solve_mut<R2: Dim, C2: Dim, S2>(&mut self, b: &mut Matrix<N, R2, C2, S2>) -> bool
where
S2: StorageMut<N, R2, C2>,
DefaultAllocator: Allocator<N, C2>,
ShapeConstraint: SameNumberOfRows<R2, D>,
{
assert_eq!(
@ -207,48 +273,13 @@ where
);
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.qr.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.qr.slice_range(..i, i), N::one());
}
}
true
self.qr.solve_upper_triangular_mut(b)
}
/// 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>> {
pub fn try_inverse(&mut self) -> Option<MatrixN<N, D>> {
assert!(
self.qr.is_square(),
"QR inverse: unable to compute the inverse of a non-square matrix."
@ -271,33 +302,14 @@ where
self.qr.is_square(),
"QR: 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;
(0..self.qr.ncols()).all(|i| unsafe { !self.qr.get_unchecked((i, i)).is_zero() })
}
}
true
}
// /// Computes the determinant of the decomposed matrix.
// pub fn determinant(&self) -> N {
// let dim = self.qr.nrows();
// assert!(self.qr.is_square(), "QR 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>>,
DefaultAllocator:
Allocator<N, R, C> + Allocator<N, R> + Allocator<N, C> + Allocator<N, DimMinimum<R, C>>,
{
/// Computes the QR decomposition of this matrix.
pub fn qr(self) -> QR<N, R, C> {

92
tests/linalg/qr.rs
View File

@ -1,7 +1,49 @@
#![cfg(feature = "arbitrary")]
use na::{Matrix2, Matrix4x2, U3, U4};
#[test]
fn simple_qr() {
#[rustfmt::skip]
let a = Matrix4x2::new(
-0.8943285241224914 , 0.12787800716234649,
-0.37320804072796987, 0.21338804264385058,
0. , -0.2456767687354977 ,
0.2456767687354977 , 0. );
let qr = a.qr();
// the reference values were generated by converting the input
// to the form `m * 2 ^ e` for integers m and e. This was then used to
// obtain the QR decomposition without rounding errors. The result was
// converted back to f64.
#[rustfmt::skip]
let r_ref = Matrix2::new(
0.99973237689865724, -0.19405501632841561,
0. , -0.2908383860381578);
assert_relative_eq!(qr.r(), r_ref);
macro_rules! gen_tests(
#[rustfmt::skip]
let q_ref = Matrix4x2::new(
-0.89456793116659196, 0.15719172406996297,
-0.3733079465583837 , -0.48461884587835711,
0. , 0.8447191998351451,
0.24574253511487697, -0.1639658791740342);
assert_relative_eq!(qr.q(), q_ref);
}
#[test]
fn q_columns() {
let a = Matrix4x2::new(0., 1., 3., 3., 1., 1., 2., 1.);
let qr = a.qr();
assert!(qr.q_columns(U4).is_orthogonal(1.0e-15));
}
#[test]
#[should_panic]
fn q_columns_panic() {
Matrix2::<f64>::zeros().qr().q_columns(U3);
}
#[cfg(feature = "arbitrary")]
mod quickcheck_test {
macro_rules! gen_tests(
($module: ident, $scalar: ty) => {
mod $module {
use na::{DMatrix, DVector, Matrix3x5, Matrix4, Matrix4x3, Matrix5x3, Vector4};
@ -16,11 +58,8 @@ macro_rules! gen_tests(
let q = qr.q();
let r = qr.r();
println!("m: {}", m);
println!("qr: {}", &q * &r);
relative_eq!(m, &q * r, epsilon = 1.0e-7) &&
q.is_orthogonal(1.0e-7)
relative_eq!(m, &q * r, epsilon = 1.0e-9) &&
q.is_orthogonal(1.0e-15)
}
fn qr_static_5_3(m: Matrix5x3<$scalar>) -> bool {
@ -29,8 +68,8 @@ macro_rules! gen_tests(
let q = qr.q();
let r = qr.r();
relative_eq!(m, q * r, epsilon = 1.0e-7) &&
q.is_orthogonal(1.0e-7)
relative_eq!(m, q * r, epsilon = 1.0e-8) &&
q.is_orthogonal(1.0e-15)
}
fn qr_static_3_5(m: Matrix3x5<$scalar>) -> bool {
@ -39,8 +78,8 @@ macro_rules! gen_tests(
let q = qr.q();
let r = qr.r();
relative_eq!(m, q * r, epsilon = 1.0e-7) &&
q.is_orthogonal(1.0e-7)
relative_eq!(m, q * r, epsilon = 1.0e-9) &&
q.is_orthogonal(1.0e-15)
}
fn qr_static_square(m: Matrix4<$scalar>) -> bool {
@ -49,10 +88,8 @@ macro_rules! gen_tests(
let q = qr.q();
let r = qr.r();
println!("{}{}{}{}", q, r, q * r, m);
relative_eq!(m, q * r, epsilon = 1.0e-7) &&
q.is_orthogonal(1.0e-7)
relative_eq!(m, q * r, epsilon = 1.0e-9) &&
q.is_orthogonal(1.0e-15)
}
fn qr_solve(n: usize, nb: usize) -> bool {
@ -61,7 +98,7 @@ macro_rules! gen_tests(
let nb = cmp::min(nb, 50); // To avoid slowing down the test too much.
let m = DMatrix::<$scalar>::new_random(n, n).map(|e| e.0);
let qr = m.clone().qr();
let mut qr = m.clone().qr();
let b1 = DVector::<$scalar>::new_random(n).map(|e| e.0);
let b2 = DMatrix::<$scalar>::new_random(n, nb).map(|e| e.0);
@ -69,8 +106,8 @@ macro_rules! gen_tests(
let sol1 = qr.solve(&b1).unwrap();
let sol2 = qr.solve(&b2).unwrap();
return relative_eq!(&m * sol1, b1, epsilon = 1.0e-6) &&
relative_eq!(&m * sol2, b2, epsilon = 1.0e-6)
return relative_eq!(&m * sol1, b1, epsilon = 1.0e-8) &&
relative_eq!(&m * sol2, b2, epsilon = 1.0e-8)
}
}
@ -79,7 +116,7 @@ macro_rules! gen_tests(
fn qr_solve_static(m: Matrix4<$scalar>) -> bool {
let m = m.map(|e| e.0);
let qr = m.qr();
let mut qr = m.qr();
let b1 = Vector4::<$scalar>::new_random().map(|e| e.0);
let b2 = Matrix4x3::<$scalar>::new_random().map(|e| e.0);
@ -87,8 +124,8 @@ macro_rules! gen_tests(
let sol1 = qr.solve(&b1).unwrap();
let sol2 = qr.solve(&b2).unwrap();
relative_eq!(m * sol1, b1, epsilon = 1.0e-6) &&
relative_eq!(m * sol2, b2, epsilon = 1.0e-6)
relative_eq!(m * sol1, b1, epsilon = 1.0e-8) &&
relative_eq!(m * sol2, b2, epsilon = 1.0e-8)
}
else {
false
@ -103,7 +140,7 @@ macro_rules! gen_tests(
let id1 = &m * &m1;
let id2 = &m1 * &m;
id1.is_identity(1.0e-5) && id2.is_identity(1.0e-5)
id1.is_identity(1.0e-7) && id2.is_identity(1.0e-7)
}
else {
true
@ -112,13 +149,13 @@ macro_rules! gen_tests(
fn qr_inverse_static(m: Matrix4<$scalar>) -> bool {
let m = m.map(|e| e.0);
let qr = m.qr();
let mut qr = m.qr();
if let Some(m1) = qr.try_inverse() {
let id1 = &m * &m1;
let id2 = &m1 * &m;
id1.is_identity(1.0e-5) && id2.is_identity(1.0e-5)
id1.is_identity(1.0e-7) && id2.is_identity(1.0e-7)
}
else {
true
@ -127,7 +164,8 @@ macro_rules! gen_tests(
}
}
}
);
);
gen_tests!(complex, RandComplex<f64>);
gen_tests!(f64, RandScalar<f64>);
gen_tests!(complex, RandComplex<f64>);
gen_tests!(f64, RandScalar<f64>);
}