numerical::qr_decomposition< T > Class Template Reference

A QR factorisation A = Q R of an m × n matrix, as NumPy's qr, reduced by default. More...

#include <decomposition.hpp>

Public Member Functions
	qr_decomposition (const matrix< T > &a, qr_mode mode=qr_mode::reduced)
	Factorises a.
	qr_decomposition (matrix< T > &&a, qr_mode mode=qr_mode::reduced)
	Factorises a in its own storage, an expiring matrix's elements taken over: no copy of them.
matrix< T >	q () const &
	The factor Q (orthonormal columns): m × k, k = min(m, n), or m × m complete.
matrix< T >	q () &&
	Q, moved out of an expiring decomposition (r stays).
matrix< T >	r () const &
	The upper-trapezoidal factor R: k × n, k = min(m, n), or m × n complete.
matrix< T >	r () &&
	R, moved out of an expiring decomposition (q stays).
vector< T >	solve (const vector< T > &b) const
	The least-squares solution of A x ≈ b (x = R⁻¹ Qᴴ b) for a full-rank A.

Private Attributes
std::size_t	rows_
	The number of rows (m).
std::size_t	cols_
	The number of columns (n).
std::size_t	depth_
	min(m, n).
std::size_t	width_
	Q's columns and R's rows: k, or m complete.
buffer< T >	q_
	The factor Q, row-major, m × width_.
buffer< T >	r_
	The upper-trapezoidal factor R, row-major, width_ × n.

Detailed Description

template<typename T>
class numerical::qr_decomposition< T >

A QR factorisation A = Q R of an m × n matrix, as NumPy's qr, reduced by default.

Template Parameters

T	The scalar element type (a field: double, std::complex<double>, …).

With k = min(m, n), Q is m × k with orthonormal columns (Qᴴ Q = I) and R is k × n upper trapezoidal (triangular when m ≥ n) with a real, non-negative diagonal; complete (qr_mode::complete), Q is m × m and R m × n, its rows past k zero. Computed by Householder reflections, so Q stays orthonormal to rounding however ill-conditioned A is, and a rank-deficient A factorises too (with zeros on R's diagonal). solve returns the least-squares solution of A x ≈ b for a full-column-rank A (m ≥ n); for a rank-deficient or wide one, numerical::lstsq gives the minimum-norm solution.

Definition at line 2554 of file decomposition.hpp.

Constructor & Destructor Documentation

qr_decomposition() [1/2]

template<typename T >

numerical::qr_decomposition< T >::qr_decomposition ( const matrix< T > &  a, qr_mode  mode = qr_mode::reduced  )

inlineexplicit

Factorises a.

Parameters

[in]	a	The matrix, any shape.
[in]	mode	qr_mode::reduced (Q m × k) or qr_mode::complete (Q m × m).

Exceptions

non_finite_error

If an entry is infinite or NaN.

Definition at line 2563 of file decomposition.hpp.

qr_decomposition() [2/2]

template<typename T >

numerical::qr_decomposition< T >::qr_decomposition ( matrix< T > &&  a, qr_mode  mode = qr_mode::reduced  )

inlineexplicit

Factorises a in its own storage, an expiring matrix's elements taken over: no copy of them.

Parameters

[in]	a	The matrix, any shape.
[in]	mode	qr_mode::reduced (Q m × k) or qr_mode::complete (Q m × m).

Exceptions

non_finite_error

If an entry is infinite or NaN.

Definition at line 2575 of file decomposition.hpp.

Member Function Documentation

q() [1/2]

template<typename T >

matrix< T > numerical::qr_decomposition< T >::q ( ) &&

inline

Q, moved out of an expiring decomposition (r stays).

Definition at line 2594 of file decomposition.hpp.

q() [2/2]

template<typename T >

matrix< T > numerical::qr_decomposition< T >::q ( ) const &

inline

The factor Q (orthonormal columns): m × k, k = min(m, n), or m × m complete.

Definition at line 2588 of file decomposition.hpp.

r() [1/2]

template<typename T >

matrix< T > numerical::qr_decomposition< T >::r ( ) &&

inline

R, moved out of an expiring decomposition (q stays).

Definition at line 2606 of file decomposition.hpp.

r() [2/2]

template<typename T >

matrix< T > numerical::qr_decomposition< T >::r ( ) const &

inline

The upper-trapezoidal factor R: k × n, k = min(m, n), or m × n complete.

Definition at line 2600 of file decomposition.hpp.

solve()

template<typename T >

vector< T > numerical::qr_decomposition< T >::solve ( const vector< T > &  b ) const

inline

The least-squares solution of A x ≈ b (x = R⁻¹ Qᴴ b) for a full-rank A.

Parameters

[in]

b

The right-hand side (size m).

Returns

The solution x (size n).

Exceptions

dimension_error	If b's size differs from the row count, or A is wide (m < n: its system is underdetermined, and numerical::lstsq gives its minimum-norm solution).
singular_matrix_error	If A is rank-deficient: a diagonal entry of R is under detail::rank_tolerance (numerical::lstsq solves such a system).

Definition at line 2620 of file decomposition.hpp.

Member Data Documentation

cols_

template<typename T >

std::size_t numerical::qr_decomposition< T >::cols_

private

The number of columns (n).

Definition at line 2658 of file decomposition.hpp.

depth_

template<typename T >

std::size_t numerical::qr_decomposition< T >::depth_

private

min(m, n).

Definition at line 2659 of file decomposition.hpp.

q_

template<typename T >

buffer<T> numerical::qr_decomposition< T >::q_

private

The factor Q, row-major, m × width_.

Definition at line 2661 of file decomposition.hpp.

r_

template<typename T >

buffer<T> numerical::qr_decomposition< T >::r_

private

The upper-trapezoidal factor R, row-major, width_ × n.

Definition at line 2662 of file decomposition.hpp.

rows_

template<typename T >

std::size_t numerical::qr_decomposition< T >::rows_

private

The number of rows (m).

Definition at line 2657 of file decomposition.hpp.

width_

template<typename T >

std::size_t numerical::qr_decomposition< T >::width_

private

Q's columns and R's rows: k, or m complete.

Definition at line 2660 of file decomposition.hpp.

---

The documentation for this class was generated from the following file:

• include/numerical/decomposition.hpp