numerical::svd_decomposition< T > Class Template Reference

The singular value decomposition A = U Σ Vᴴ of an m × n matrix. More...

#include <decomposition.hpp>

Collaboration diagram for numerical::svd_decomposition< T >:

Public Member Functions
	svd_decomposition (const matrix< T > &a, svd_method method=svd_method::divide_and_conquer, bool full=false)
	Decomposes a.
const vector< double > &	singular_values () const noexcept
	The singular values, descending.
const matrix< T > &	u () const &noexcept
	The left singular vectors U (columns): m × k, or m × m full.
matrix< T >	u () &&
	U, moved out of an expiring decomposition (v and the singular values stay).
const matrix< T > &	v () const &noexcept
	The right singular vectors V (columns): n × k, or n × n full.
matrix< T >	vh () const
	Vᴴ (k × n, or n × n full), so that A = U Σ Vᴴ reads u() * diag(singular_values()) * vh() (full, Σ padded with zeros to m × n).

Private Member Functions
void	factor (const matrix< T > &a, svd_method method, bool full)
	Decomposes an m × n a with m ≥ n into u_, singular_values_ and v_, U square when full.

Static Private Member Functions
static matrix< T >	left_vectors (buffer< T > q, buffer< T > left, std::size_t rows, std::size_t cols, std::size_t width)
	U from the Householder q (rows × width) and the left vectors left of R (cols × cols): Q times them, and when Q is complete (width = rows) its columns past cols after them.
static void	orthogonalise (buffer< T > &x, const buffer< T > &basis, std::size_t n, std::size_t count)
	Removes from x its components along the first count columns of the n × n basis.

Private Attributes
vector< double >	singular_values_
	Singular values, descending.
matrix< T >	u_
	m×k left singular vectors.
matrix< T >	v_
	n×k right singular vectors.

Detailed Description

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

The singular value decomposition A = U Σ Vᴴ of an m × n matrix.

Template Parameters

T	The scalar element type (double or std::complex<double>).

With k = min(m, n): the k singular values, descending; U (m × k) and V (n × k) with orthonormal columns, also where A is rank-deficient. Computed after a Householder QR of A (of Aᴴ when m < n): by default R is reduced to bidiagonal form and the bidiagonal decomposed by divide and conquer (svd_method::divide_and_conquer), the method of LAPACK's dgesdd, most of its work in matrix products, or by implicit-shift QR (svd_method::bidiagonal_qr, dgesvd's); either way each singular value to an absolute accuracy of the largest times the machine epsilon. One-sided Jacobi on R (svd_method::jacobi) gives every singular value to high relative accuracy — the small ones of a graded matrix too — at several times the cost. vh is Vᴴ, the third factor NumPy's svd returns. Full (full true, NumPy's full_matrices=True), U is m × m and V n × n: the columns past k are an orthonormal basis of the complement, the rest of the complete Householder Q of the larger side.

Definition at line 6017 of file decomposition.hpp.

Constructor & Destructor Documentation

svd_decomposition()

template<typename T >

numerical::svd_decomposition< T >::svd_decomposition ( const matrix< T > &  a, svd_method  method = svd_method::divide_and_conquer, bool  full = false  )

inlineexplicit

Decomposes a.

Parameters

[in]	a	The matrix.
[in]	method	The algorithm for R.
[in]	full	Whether U and V are square (m × m, n × n), as NumPy's full_matrices=True.

Exceptions

non_finite_error	If an entry is infinite or NaN.
convergence_error	If the iteration does not converge within its bound (not observed).

Definition at line 6029 of file decomposition.hpp.

Member Function Documentation

factor()

template<typename T >

void numerical::svd_decomposition< T >::factor ( const matrix< T > &  a, svd_method  method, bool  full  )

inlineprivate

Decomposes an m × n a with m ≥ n into u_, singular_values_ and v_, U square when full.

Definition at line 6101 of file decomposition.hpp.

left_vectors()

template<typename T >

static matrix< T > numerical::svd_decomposition< T >::left_vectors ( buffer< T >  q, buffer< T >  left, std::size_t  rows, std::size_t  cols, std::size_t  width  )

inlinestaticprivate

U from the Householder q (rows × width) and the left vectors left of R (cols × cols): Q times them, and when Q is complete (width = rows) its columns past cols after them.

Definition at line 6080 of file decomposition.hpp.

orthogonalise()

template<typename T >

static void numerical::svd_decomposition< T >::orthogonalise ( buffer< T > &  x, const buffer< T > &  basis, std::size_t  n, std::size_t  count  )

inlinestaticprivate

Removes from x its components along the first count columns of the n × n basis.

Definition at line 6227 of file decomposition.hpp.

singular_values()

template<typename T >

const vector< double > & numerical::svd_decomposition< T >::singular_values ( ) const

inlinenoexcept

The singular values, descending.

Definition at line 6044 of file decomposition.hpp.

u() [1/2]

template<typename T >

matrix< T > numerical::svd_decomposition< T >::u ( ) &&

inline

U, moved out of an expiring decomposition (v and the singular values stay).

Definition at line 6056 of file decomposition.hpp.

u() [2/2]

template<typename T >

const matrix< T > & numerical::svd_decomposition< T >::u ( ) const &

inlinenoexcept

The left singular vectors U (columns): m × k, or m × m full.

Definition at line 6050 of file decomposition.hpp.

v()

template<typename T >

const matrix< T > & numerical::svd_decomposition< T >::v ( ) const &

inlinenoexcept

The right singular vectors V (columns): n × k, or n × n full.

Definition at line 6062 of file decomposition.hpp.

vh()

template<typename T >

matrix< T > numerical::svd_decomposition< T >::vh ( ) const

inline

Vᴴ (k × n, or n × n full), so that A = U Σ Vᴴ reads u() * diag(singular_values()) * vh() (full, Σ padded with zeros to m × n).

Definition at line 6069 of file decomposition.hpp.

Member Data Documentation

singular_values_

template<typename T >

vector<double> numerical::svd_decomposition< T >::singular_values_

private

Singular values, descending.

Definition at line 6243 of file decomposition.hpp.

u_

template<typename T >

matrix<T> numerical::svd_decomposition< T >::u_

private

m×k left singular vectors.

Definition at line 6244 of file decomposition.hpp.

v_

template<typename T >

matrix<T> numerical::svd_decomposition< T >::v_

private

n×k right singular vectors.

Definition at line 6245 of file decomposition.hpp.

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

• include/numerical/decomposition.hpp