Holds the singular value decomposition of a vnl_matrix. More...

#include <vnl_algo_fwd.h>

Inheritance diagram for vnl_svd< T >:

Public Types
typedef vnl_numeric_traits< T >::abs_t	singval_t
	The singular values of a matrix of complex<T> are of type T, not complex<T>. More...

Public Member Functions
	vnl_svd (vnl_matrix< T > const &M, double zero_out_tol=0.0)
	Construct a vnl_svd<T> object from $m \times n$ matrix . More...

virtual	~vnl_svd ()=default

void	zero_out_absolute (double tol=1e-8)
	find weights below threshold tol, zero them out, and update W_ and Winverse_. More...

void	zero_out_relative (double tol=1e-8)
	find weights below tol*max(w) and zero them out. More...

int	singularities () const

unsigned int	rank () const

singval_t	well_condition () const

singval_t	determinant_magnitude () const
	Calculate determinant as product of diagonals in W. More...

singval_t	norm () const

vnl_matrix< T > &	U ()
	Return the matrix U. More...

vnl_matrix< T > const &	U () const
	Return the matrix U. More...

T	U (int i, int j) const
	Return the matrix U's (i,j)th entry (to avoid svd.U()(i,j); ). More...

vnl_diag_matrix< singval_t > &	W ()
	Get at DiagMatrix (q.v.) of singular values, sorted from largest to smallest. More...

vnl_diag_matrix< singval_t > const &	W () const
	Get at DiagMatrix (q.v.) of singular values, sorted from largest to smallest. More...

vnl_diag_matrix< singval_t > &	Winverse ()

vnl_diag_matrix< singval_t > const &	Winverse () const

singval_t &	W (int i, int j)

singval_t &	W (int i)

singval_t	sigma_max () const

singval_t	sigma_min () const

vnl_matrix< T > &	V ()
	Return the matrix V. More...

vnl_matrix< T > const &	V () const
	Return the matrix V. More...

T	V (int i, int j) const
	Return the matrix V's (i,j)th entry (to avoid svd.V()(i,j); ). More...

vnl_matrix< T >	inverse () const

vnl_matrix< T >	pinverse (unsigned int rank=~0u) const
	pseudo-inverse (for non-square matrix) of desired rank. More...

vnl_matrix< T >	tinverse (unsigned int rank=~0u) const
	Calculate inverse of transpose, using desired rank. More...

vnl_matrix< T >	recompose (unsigned int rank=~0u) const
	Recompose SVD to UWV', using desired rank. More...

vnl_matrix< T >	solve (vnl_matrix< T > const &B) const
	Solve the matrix equation M X = B, returning X. More...

vnl_vector< T >	solve (vnl_vector< T > const &y) const
	Solve the matrix-vector system M x = y, returning x. More...

void	solve (T const rhs, T lhs) const

void	solve_preinverted (vnl_vector< T > const &rhs, vnl_vector< T > *out) const
	Solve the matrix-vector system M x = y. More...

vnl_matrix< T >	nullspace () const
	Return N such that M * N = 0. More...

vnl_matrix< T >	left_nullspace () const
	Return N such that M' * N = 0. More...

vnl_matrix< T >	nullspace (int required_nullspace_dimension) const
	Return N such that M * N = 0. More...

vnl_matrix< T >	left_nullspace (int required_nullspace_dimension) const
	Implementation to be done yet; currently returns left_nullspace(). - PVR. More...

vnl_vector< T >	nullvector () const
	Return the rightmost column of V. More...

vnl_vector< T >	left_nullvector () const
	Return the rightmost column of U. More...

bool	valid () const

Private Member Functions
	vnl_svd (vnl_svd< T > const &)

vnl_svd< T > &	operator= (vnl_svd< T > const &)

Private Attributes
int	m_

int	n_

vnl_matrix< T >	U_

vnl_diag_matrix< singval_t >	W_

vnl_diag_matrix< singval_t >	Winverse_

vnl_matrix< T >	V_

unsigned	rank_

bool	have_max_

singval_t	max_

bool	have_min_

singval_t	min_

double	last_tol_

bool	valid_

Detailed Description

template<class T>
class vnl_svd< T >

Holds the singular value decomposition of a vnl_matrix.

The class holds three matrices U, W, V such that the original matrix $M = U W V^\top$ . The DiagMatrix W stores the singular values in decreasing order. The columns of U which correspond to the nonzero singular values form a basis for range of M, while the columns of V corresponding to the zero singular values are the nullspace.

The SVD is computed at construction time, and inquiries may then be made of the SVD. In particular, this allows easy access to multiple right-hand-side solves without the bother of putting all the RHS's into a Matrix.

This class is supplied even though there is an existing vnl_matrix method for several reasons:

It is more convenient to use as it manages all the storage for the U,S,V matrices, allowing repeated queries of the same SVD results.

It avoids namespace clutter in the Matrix class. While svd() is a perfectly reasonable method for a Matrix, there are many other decompositions that might be of interest, and adding them all would make for a very large Matrix class.

It demonstrates the holder model of compute class, implementing an algorithm on an object without adding a member that may not be of general interest. A similar pattern can be used for other decompositions which are not defined as members of the library Matrix class.

It extends readily to n-ary operations, such as generalized eigensystems, which cannot be members of just one matrix.

Definition at line 7 of file vnl_algo_fwd.h.

Member Typedef Documentation

◆ singval_t

template<class T>

typedef vnl_numeric_traits<T>::abs_t vnl_svd< T >::singval_t

The singular values of a matrix of complex<T> are of type T, not complex<T>.

Definition at line 67 of file vnl_svd.h.

Constructor & Destructor Documentation

◆ vnl_svd() [1/2]

template<class T>

vnl_svd< T >::vnl_svd	(	vnl_matrix< T > const &	M,
		double	zero_out_tol = `0.0`
	)

Construct a vnl_svd<T> object from $m \times n$ matrix $M$ .

The vnl_svd<T> object contains matrices $U$ , $W$ , $V$ such that $U W V^\top = M$ .

Uses linpack routine DSVDC to calculate an `‘economy-size’' SVD where the returned $U$ is the same size as $M$ , while $W$ and $V$ are both $n \times n$ . This is efficient for large rectangular solves where $m > n$ , typical in least squares.

The optional argument zero_out_tol is used to mark the zero singular values: If nonnegative, any s.v. smaller than zero_out_tol in absolute value is set to zero. If zero_out_tol is negative, the zeroing is relative to |zero_out_tol| * sigma_max();

Definition at line 38 of file vnl_svd.hxx.

◆ ~vnl_svd()

template<class T>

virtual vnl_svd< T >::~vnl_svd ( )

virtualdefault

◆ vnl_svd() [2/2]

template<class T>

vnl_svd< T >::vnl_svd ( vnl_svd< T > const & )

inlineprivate

Definition at line 194 of file vnl_svd.h.

Member Function Documentation

◆ determinant_magnitude()

template<class T >

vnl_svd< T >::singval_t vnl_svd< T >::determinant_magnitude ( ) const

Calculate determinant as product of diagonals in W.

Definition at line 213 of file vnl_svd.hxx.

◆ inverse()

template<class T>

vnl_matrix<T> vnl_svd< T >::inverse ( ) const

inline

Definition at line 133 of file vnl_svd.h.

◆ left_nullspace() [1/2]

template<class T >

vnl_matrix< T > vnl_svd< T >::left_nullspace ( ) const

Return N such that M' * N = 0.

Return N s.t. M' * N = 0.

Definition at line 384 of file vnl_svd.hxx.

◆ left_nullspace() [2/2]

template<class T >

vnl_matrix< T > vnl_svd< T >::left_nullspace ( int required_nullspace_dimension ) const

Implementation to be done yet; currently returns left_nullspace(). - PVR.

Todo:: Implementation to be done yet; currently returns left_nullspace().

- PVr.

Definition at line 395 of file vnl_svd.hxx.

◆ left_nullvector()

template<class T >

vnl_vector< T > vnl_svd< T >::left_nullvector ( ) const

Return the rightmost column of U.

Does not check to see whether or not the matrix actually was rank-deficient.

Definition at line 418 of file vnl_svd.hxx.

◆ norm()

template<class T >

vnl_svd< T >::singval_t vnl_svd< T >::norm ( ) const

Definition at line 226 of file vnl_svd.hxx.

◆ nullspace() [1/2]

template<class T >

vnl_matrix< T > vnl_svd< T >::nullspace ( ) const

Return N such that M * N = 0.

Return N s.t. M * N = 0.

Definition at line 365 of file vnl_svd.hxx.

◆ nullspace() [2/2]

template<class T >

vnl_matrix< T > vnl_svd< T >::nullspace ( int required_nullspace_dimension ) const

Return N such that M * N = 0.

Return N s.t. M * N = 0.

Definition at line 376 of file vnl_svd.hxx.

◆ nullvector()

template<class T >

vnl_vector< T > vnl_svd< T >::nullvector ( ) const

Return the rightmost column of V.

Does not check to see whether or not the matrix actually was rank-deficient - the caller is assumed to have examined W and decided that to his or her satisfaction.

Definition at line 406 of file vnl_svd.hxx.

◆ operator=()

template<class T>

vnl_svd<T>& vnl_svd< T >::operator= ( vnl_svd< T > const & )

inlineprivate

Definition at line 195 of file vnl_svd.h.

◆ pinverse()

template<class T >

vnl_matrix< T > vnl_svd< T >::pinverse ( unsigned int rank = ~0u ) const

pseudo-inverse (for non-square matrix) of desired rank.

Calculate pseudo-inverse.

Definition at line 247 of file vnl_svd.hxx.

◆ rank()

template<class T>

unsigned int vnl_svd< T >::rank ( ) const

inline

Definition at line 95 of file vnl_svd.h.

◆ recompose()

template<class T >

vnl_matrix< T > vnl_svd< T >::recompose ( unsigned int rank = ~0u ) const

Recompose SVD to U*W*V', using desired rank.

Recompose SVD to U*W*V'.

Definition at line 233 of file vnl_svd.hxx.

◆ sigma_max()

template<class T>

singval_t vnl_svd< T >::sigma_max ( ) const

inline

Definition at line 120 of file vnl_svd.h.

◆ sigma_min()

template<class T>

singval_t vnl_svd< T >::sigma_min ( ) const

inline

Definition at line 121 of file vnl_svd.h.

◆ singularities()

template<class T>

int vnl_svd< T >::singularities ( ) const

inline

Definition at line 94 of file vnl_svd.h.

◆ solve() [1/3]

template<class T>

vnl_matrix< T > vnl_svd< T >::solve ( vnl_matrix< T > const & B ) const

Solve the matrix equation M X = B, returning X.

Definition at line 275 of file vnl_svd.hxx.

◆ solve() [2/3]

template<class T>

vnl_vector< T > vnl_svd< T >::solve ( vnl_vector< T > const & y ) const

Solve the matrix-vector system M x = y, returning x.

Definition at line 298 of file vnl_svd.hxx.

◆ solve() [3/3]

template<class T>

void vnl_svd< T >::solve	(	T const *	rhs,
		T *	lhs
	)		const

Definition at line 337 of file vnl_svd.hxx.

◆ solve_preinverted()

template<class T>

void vnl_svd< T >::solve_preinverted	(	vnl_vector< T > const &	y,
		vnl_vector< T > *	x_out
	)		const

Solve the matrix-vector system M x = y.

Assuming that the singular values W have been preinverted by the caller.

Assume that the singular values W have been preinverted by the caller.

Definition at line 345 of file vnl_svd.hxx.

◆ tinverse()

template<class T >

vnl_matrix< T > vnl_svd< T >::tinverse ( unsigned int rank = ~0u ) const

Calculate inverse of transpose, using desired rank.

Calculate (pseudo-)inverse of transpose.

Definition at line 261 of file vnl_svd.hxx.

◆ U() [1/3]

template<class T>

vnl_matrix<T>& vnl_svd< T >::U ( )

inline

Return the matrix U.

Definition at line 103 of file vnl_svd.h.

◆ U() [2/3]

template<class T>

vnl_matrix<T> const& vnl_svd< T >::U ( ) const

inline

Return the matrix U.

Definition at line 106 of file vnl_svd.h.

◆ U() [3/3]

template<class T>

T vnl_svd< T >::U	(	int	i,
		int	j
	)		const

inline

Return the matrix U's (i,j)th entry (to avoid svd.U()(i,j); ).

Definition at line 109 of file vnl_svd.h.

◆ V() [1/3]

template<class T>

vnl_matrix<T>& vnl_svd< T >::V ( )

inline

Return the matrix V.

Definition at line 124 of file vnl_svd.h.

◆ V() [2/3]

template<class T>

vnl_matrix<T> const& vnl_svd< T >::V ( ) const

inline

Return the matrix V.

Definition at line 127 of file vnl_svd.h.

◆ V() [3/3]

template<class T>

T vnl_svd< T >::V	(	int	i,
		int	j
	)		const

inline

Return the matrix V's (i,j)th entry (to avoid svd.V()(i,j); ).

Definition at line 130 of file vnl_svd.h.

◆ valid()

template<class T>

bool vnl_svd< T >::valid ( ) const

inline

Definition at line 176 of file vnl_svd.h.

◆ W() [1/4]

template<class T>

vnl_diag_matrix<singval_t>& vnl_svd< T >::W ( )

inline

Get at DiagMatrix (q.v.) of singular values, sorted from largest to smallest.

Definition at line 112 of file vnl_svd.h.

◆ W() [2/4]

template<class T>

vnl_diag_matrix<singval_t> const& vnl_svd< T >::W ( ) const

inline

Get at DiagMatrix (q.v.) of singular values, sorted from largest to smallest.

Definition at line 115 of file vnl_svd.h.

◆ W() [3/4]

template<class T>

singval_t& vnl_svd< T >::W	(	int	i,
		int	j
	)

inline

Definition at line 118 of file vnl_svd.h.

◆ W() [4/4]

template<class T>

singval_t& vnl_svd< T >::W ( int i )

inline

Definition at line 119 of file vnl_svd.h.

◆ well_condition()

template<class T>

singval_t vnl_svd< T >::well_condition ( ) const

inline

Definition at line 96 of file vnl_svd.h.

◆ Winverse() [1/2]

template<class T>

vnl_diag_matrix<singval_t>& vnl_svd< T >::Winverse ( )

inline

Definition at line 116 of file vnl_svd.h.

◆ Winverse() [2/2]

template<class T>

vnl_diag_matrix<singval_t> const& vnl_svd< T >::Winverse ( ) const

inline

Definition at line 117 of file vnl_svd.h.

◆ zero_out_absolute()

template<class T >

void vnl_svd< T >::zero_out_absolute ( double tol = 1e-8 )

find weights below threshold tol, zero them out, and update W_ and Winverse_.

Definition at line 182 of file vnl_svd.hxx.

◆ zero_out_relative()

template<class T >

void vnl_svd< T >::zero_out_relative ( double tol = 1e-8 )

find weights below tol*max(w) and zero them out.

Definition at line 203 of file vnl_svd.hxx.

Member Data Documentation

◆ have_max_

template<class T>

bool vnl_svd< T >::have_max_

private

Definition at line 186 of file vnl_svd.h.

◆ have_min_

template<class T>

bool vnl_svd< T >::have_min_

private

Definition at line 188 of file vnl_svd.h.

◆ last_tol_

template<class T>

double vnl_svd< T >::last_tol_

private

Definition at line 190 of file vnl_svd.h.

◆ m_

template<class T>

int vnl_svd< T >::m_

private

Definition at line 180 of file vnl_svd.h.

◆ max_

template<class T>

singval_t vnl_svd< T >::max_

private

Definition at line 187 of file vnl_svd.h.

◆ min_

template<class T>

singval_t vnl_svd< T >::min_

private

Definition at line 189 of file vnl_svd.h.

◆ n_

template<class T>

int vnl_svd< T >::n_

private

Definition at line 180 of file vnl_svd.h.

◆ rank_

template<class T>

unsigned vnl_svd< T >::rank_

private

Definition at line 185 of file vnl_svd.h.

◆ U_

template<class T>

vnl_matrix<T> vnl_svd< T >::U_

private

Definition at line 181 of file vnl_svd.h.

◆ V_

template<class T>

vnl_matrix<T> vnl_svd< T >::V_

private

Definition at line 184 of file vnl_svd.h.

◆ valid_

template<class T>

bool vnl_svd< T >::valid_

private

Definition at line 191 of file vnl_svd.h.

◆ W_

template<class T>

vnl_diag_matrix<singval_t> vnl_svd< T >::W_

private

Definition at line 182 of file vnl_svd.h.

◆ Winverse_

template<class T>

vnl_diag_matrix<singval_t> vnl_svd< T >::Winverse_

private

Definition at line 183 of file vnl_svd.h.

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

core/vnl/algo/vnl_algo_fwd.h
core/vnl/algo/vnl_svd.h
core/vnl/algo/vnl_svd.hxx

Public Types

Public Member Functions

Private Member Functions

Private Attributes

Detailed Description

template<class T> class vnl_svd< T >

Member Typedef Documentation

◆ singval_t

Constructor & Destructor Documentation

◆ vnl_svd() [1/2]

◆ ~vnl_svd()

◆ vnl_svd() [2/2]

Member Function Documentation

◆ determinant_magnitude()

◆ inverse()

◆ left_nullspace() [1/2]

◆ left_nullspace() [2/2]

◆ left_nullvector()

◆ norm()

◆ nullspace() [1/2]

◆ nullspace() [2/2]

◆ nullvector()

◆ operator=()

◆ pinverse()

◆ rank()

◆ recompose()

◆ sigma_max()

◆ sigma_min()

◆ singularities()

◆ solve() [1/3]

◆ solve() [2/3]

◆ solve() [3/3]

◆ solve_preinverted()

◆ tinverse()

◆ U() [1/3]

◆ U() [2/3]

◆ U() [3/3]

◆ V() [1/3]

◆ V() [2/3]

◆ V() [3/3]

◆ valid()

◆ W() [1/4]

◆ W() [2/4]

◆ W() [3/4]

◆ W() [4/4]

◆ well_condition()

◆ Winverse() [1/2]

◆ Winverse() [2/2]

◆ zero_out_absolute()

◆ zero_out_relative()

Member Data Documentation

◆ have_max_

◆ have_min_

◆ last_tol_

◆ m_

◆ max_

◆ min_

◆ n_

◆ rank_

◆ U_

◆ V_

◆ valid_

◆ W_

◆ Winverse_

template<class T>
class vnl_svd< T >