linalg::QR< Mat > Struct Template Reference

Factors A = Q R using Householder reflections. More...

#include <qr.hpp>

Public Types
using	T = matrix_scalar_t< Mat >
using	ScalarT = real_type_t< T >

Public Member Functions
constexpr	QR (const Mat &A)
	Factor A using Householder reflections.
constexpr	QR (const Mat &A, const config::FactorConfig< ScalarT > &config_in)
	Factor A with an explicit solve configuration.
constexpr auto	Q () const
	Reconstruct the full orthogonal or unitary factor.
constexpr auto	R () const
	Extract the upper-trapezoidal factor.
constexpr auto	rank (ScalarT tol=1e-6) const
	Estimate rank from the diagonal of R.
constexpr auto	solve (const Vector< T, Rows > &b) const
	Solve A x = b in the least-squares sense.

Static Public Attributes
static constexpr std::size_t	Rows = mat_rows_v<Mat>
static constexpr std::size_t	Cols = mat_cols_v<Mat>

Detailed Description

template<MatrixContainer Mat>
struct linalg::QR< Mat >

Factors A = Q R using Householder reflections.

Successive reflections eliminate each column below its diagonal. The upper triangle stores R; reflector tails and coefficients reconstruct the full orthogonal or unitary Q. For wide matrices, factor A^* to obtain an LQ form.

Template Parameters

Mat	Input matrix type with at least as many rows as columns.

Definition at line 24 of file qr.hpp.

Member Typedef Documentation

ScalarT

template<MatrixContainer Mat>

using linalg::QR< Mat >::ScalarT = real_type_t<T>

Definition at line 27 of file qr.hpp.

T

template<MatrixContainer Mat>

using linalg::QR< Mat >::T = matrix_scalar_t<Mat>

Definition at line 26 of file qr.hpp.

Constructor & Destructor Documentation

QR() [1/2]

template<MatrixContainer Mat>

constexpr linalg::QR< Mat >::QR ( const Mat &  A )

inlineconstexpr

Factor A using Householder reflections.

Parameters

[in]

A

Input matrix with Rows >= Cols.

Definition at line 41 of file qr.hpp.

References linalg::mat_rows_v.

QR() [2/2]

template<MatrixContainer Mat>

constexpr linalg::QR< Mat >::QR ( const Mat &  A, const config::FactorConfig< ScalarT > &  config_in  )

inlineconstexpr

Factor A with an explicit solve configuration.

Parameters

[in]	A	Input matrix with Rows >= Cols.
[in]	config_in	Factorization settings and triangular pivot floor.

Definition at line 49 of file qr.hpp.

References linalg::mat_rows_v.

Member Function Documentation

Q()

template<MatrixContainer Mat>

constexpr auto linalg::QR< Mat >::Q ( ) const

inlineconstexpr

Reconstruct the full orthogonal or unitary factor.

Returns

Matrix<T, Rows, Rows> containing Q in A = Q R.

Definition at line 59 of file qr.hpp.

Referenced by linalg::QR< Mat >::solve().

R()

template<MatrixContainer Mat>

constexpr auto linalg::QR< Mat >::R ( ) const

inlineconstexpr

Extract the upper-trapezoidal factor.

Returns

Matrix<T, Rows, Cols> containing R in A = Q R.

Definition at line 66 of file qr.hpp.

References linalg::get_upper_right_triangular_matrix().

Referenced by linalg::QR< Mat >::rank(), and linalg::QR< Mat >::solve().

rank()

template<MatrixContainer Mat>

constexpr auto linalg::QR< Mat >::rank ( ScalarT  tol = 1e-6 ) const

inlineconstexpr

Estimate rank from the diagonal of R.

Parameters

[in]

tol

Absolute threshold for counting nonzero diagonal entries.

Returns

Number of diagonal magnitudes greater than tol (std::size_t).

Definition at line 74 of file qr.hpp.

References linalg::QR< Mat >::Cols, linalg::mat_rows_v, linalg::QR< Mat >::R(), linalg::QR< Mat >::rank(), and linalg::QR< Mat >::Rows.

Referenced by linalg::QR< Mat >::rank().

solve()

template<MatrixContainer Mat>

constexpr auto linalg::QR< Mat >::solve ( const Vector< T, Rows > &  b ) const

inlineconstexpr

Solve A x = b in the least-squares sense.

Project c = Q^* b, then back-substitute through the leading block of R.

Parameters

[in]

b

Right-hand side with Rows entries.

Returns

Tuple (Vector<T, Cols> x, NumericalEvent event).

Definition at line 91 of file qr.hpp.

References linalg::conjugate_transpose(), linalg::mat_rows_v, linalg::QR< Mat >::Q(), linalg::QR< Mat >::R(), and linalg::transpose().

Member Data Documentation

Cols

template<MatrixContainer Mat>

constexpr std::size_t linalg::QR< Mat >::Cols = mat_cols_v<Mat>

staticconstexpr

Definition at line 31 of file qr.hpp.

Referenced by linalg::QR< Mat >::rank().

Rows

template<MatrixContainer Mat>

constexpr std::size_t linalg::QR< Mat >::Rows = mat_rows_v<Mat>

staticconstexpr

Definition at line 30 of file qr.hpp.

Referenced by linalg::QR< Mat >::rank().

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

• include/akreon/linalg/spectral/qr.hpp