vnl_svd< T > Class Template Reference

#include <vnl_svd.h>

Inheritance diagram for vnl_svd< T >:

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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 62 of file vnl_svd.h.


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>.

Public Member Functions

 vnl_svd (vnl_matrix< T > const &M, double zero_out_tol=0.0)
 Construct an vnl_svd<T> object from $m \times n$ matrix $M$.
 ~vnl_svd ()
void zero_out_absolute (double tol=1e-8)
 find weights below threshold tol, zero them out, and update W_ and Winverse_.
void zero_out_relative (double tol=1e-8)
 find weights below tol*max(w) and zero them out.
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.
singval_t norm () const
vnl_matrix< T > & U ()
 Return the matrix U.
vnl_matrix< T > const & U () const
 Return the matrix U.
U (int i, int j) const
 Return the matrix U's (i,j)th entry (to avoid svd.U()(i,j); ).
vnl_diag_matrix< singval_t > & W ()
 Get at DiagMatrix (q.v.
vnl_diag_matrix< singval_t >
const & 
W () const
 Get at DiagMatrix (q.v.
vnl_diag_matrix< singval_t > & Winverse ()
vnl_diag_matrix< singval_t >
const & 
Winverse () const
singval_tW (int i, int j)
singval_tW (int i)
singval_t sigma_max () const
singval_t sigma_min () const
vnl_matrix< T > & V ()
 Return the matrix V.
vnl_matrix< T > const & V () const
 Return the matrix V.
V (int i, int j) const
 Return the matrix V's (i,j)th entry (to avoid svd.V()(i,j); ).
vnl_matrix< T > inverse () const
vnl_matrix< T > pinverse (unsigned int rank=~0u) const
 pseudo-inverse (for non-square matrix) of desired rank.
vnl_matrix< T > tinverse (unsigned int rank=~0u) const
 Calculate inverse of transpose, using desired rank.
vnl_matrix< T > recompose (unsigned int rank=~0u) const
 Recompose SVD to U*W*V', using desired rank.
vnl_matrix< T > solve (vnl_matrix< T > const &B) const
 Solve the matrix equation M X = B, returning X.
vnl_vector< T > solve (vnl_vector< T > const &y) const
 Solve the matrix-vector system M x = y, returning x.
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.
vnl_matrix< T > nullspace () const
 Return N such that M * N = 0.
vnl_matrix< T > left_nullspace () const
 Return N such that M' * N = 0.
vnl_matrix< T > nullspace (int required_nullspace_dimension) const
 Return N such that M * N = 0.
vnl_matrix< T > left_nullspace (int required_nullspace_dimension) const
 Implementation to be done yet; currently returns left_nullspace(). - PVR.
vnl_vector< T > nullvector () const
 Return the rightmost column of V.
vnl_vector< T > left_nullvector () const
 Return the rightmost column of U.
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_tW_
vnl_diag_matrix< singval_tWinverse_
vnl_matrix< T > V_
unsigned rank_
bool have_max_
singval_t max_
bool have_min_
singval_t min_
double last_tol_
bool valid_

Member Typedef Documentation

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 66 of file vnl_svd.h.


Constructor & Destructor Documentation

template<class T>
vnl_svd< T >::vnl_svd ( vnl_matrix< T > const &  M,
double  zero_out_tol = 0.0 
) [inline]

Construct an 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 35 of file vnl_svd.txx.

template<class T>
vnl_svd< T >::~vnl_svd (  )  [inline]

Definition at line 84 of file vnl_svd.h.

template<class T>
vnl_svd< T >::vnl_svd ( vnl_svd< T > const &   )  [inline, private]

Definition at line 193 of file vnl_svd.h.


Member Function Documentation

template<class T>
void vnl_svd< T >::zero_out_absolute ( double  tol = 1e-8  )  [inline]

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

Definition at line 179 of file vnl_svd.txx.

template<class T>
void vnl_svd< T >::zero_out_relative ( double  tol = 1e-8  )  [inline]

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

Definition at line 199 of file vnl_svd.txx.

template<class T>
int vnl_svd< T >::singularities (  )  const [inline]

Definition at line 93 of file vnl_svd.h.

template<class T>
unsigned int vnl_svd< T >::rank (  )  const [inline]

Definition at line 94 of file vnl_svd.h.

template<class T>
singval_t vnl_svd< T >::well_condition (  )  const [inline]

Definition at line 95 of file vnl_svd.h.

template<class T>
vnl_svd< T >::singval_t vnl_svd< T >::determinant_magnitude (  )  const [inline]

Calculate determinant as product of diagonals in W.

Definition at line 209 of file vnl_svd.txx.

template<class T>
vnl_svd< T >::singval_t vnl_svd< T >::norm (  )  const [inline]

Definition at line 222 of file vnl_svd.txx.

template<class T>
vnl_matrix<T>& vnl_svd< T >::U (  )  [inline]

Return the matrix U.

Definition at line 102 of file vnl_svd.h.

template<class T>
vnl_matrix<T> const& vnl_svd< T >::U (  )  const [inline]

Return the matrix U.

Definition at line 105 of file vnl_svd.h.

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 108 of file vnl_svd.h.

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 111 of file vnl_svd.h.

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 114 of file vnl_svd.h.

template<class T>
vnl_diag_matrix<singval_t>& vnl_svd< T >::Winverse (  )  [inline]

Definition at line 115 of file vnl_svd.h.

template<class T>
vnl_diag_matrix<singval_t> const& vnl_svd< T >::Winverse (  )  const [inline]

Definition at line 116 of file vnl_svd.h.

template<class T>
singval_t& vnl_svd< T >::W ( int  i,
int  j 
) [inline]

Definition at line 117 of file vnl_svd.h.

template<class T>
singval_t& vnl_svd< T >::W ( int  i  )  [inline]

Definition at line 118 of file vnl_svd.h.

template<class T>
singval_t vnl_svd< T >::sigma_max (  )  const [inline]

Definition at line 119 of file vnl_svd.h.

template<class T>
singval_t vnl_svd< T >::sigma_min (  )  const [inline]

Definition at line 120 of file vnl_svd.h.

template<class T>
vnl_matrix<T>& vnl_svd< T >::V (  )  [inline]

Return the matrix V.

Definition at line 123 of file vnl_svd.h.

template<class T>
vnl_matrix<T> const& vnl_svd< T >::V (  )  const [inline]

Return the matrix V.

Definition at line 126 of file vnl_svd.h.

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 129 of file vnl_svd.h.

template<class T>
vnl_matrix<T> vnl_svd< T >::inverse (  )  const [inline]

Definition at line 132 of file vnl_svd.h.

template<class T>
vnl_matrix< T > vnl_svd< T >::pinverse ( unsigned int  rank = ~0u  )  const [inline]

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

Calculate pseudo-inverse.

Definition at line 243 of file vnl_svd.txx.

template<class T>
vnl_matrix< T > vnl_svd< T >::tinverse ( unsigned int  rank = ~0u  )  const [inline]

Calculate inverse of transpose, using desired rank.

Calculate (pseudo-)inverse of transpose.

Definition at line 257 of file vnl_svd.txx.

template<class T>
vnl_matrix< T > vnl_svd< T >::recompose ( unsigned int  rank = ~0u  )  const [inline]

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

Recompose SVD to U*W*V'.

Definition at line 229 of file vnl_svd.txx.

template<class T>
vnl_matrix< T > vnl_svd< T >::solve ( vnl_matrix< T > const &  B  )  const [inline]

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

Definition at line 271 of file vnl_svd.txx.

template<class T>
vnl_vector< T > vnl_svd< T >::solve ( vnl_vector< T > const &  y  )  const [inline]

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

Definition at line 293 of file vnl_svd.txx.

template<class T>
void vnl_svd< T >::solve ( T const *  rhs,
T *  lhs 
) const [inline]

Definition at line 332 of file vnl_svd.txx.

template<class T>
void vnl_svd< T >::solve_preinverted ( vnl_vector< T > const &  y,
vnl_vector< T > *  x_out 
) const [inline]

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 340 of file vnl_svd.txx.

template<class T>
vnl_matrix< T > vnl_svd< T >::nullspace (  )  const [inline]

Return N such that M * N = 0.

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

Definition at line 359 of file vnl_svd.txx.

template<class T>
vnl_matrix< T > vnl_svd< T >::left_nullspace (  )  const [inline]

Return N such that M' * N = 0.

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

Definition at line 378 of file vnl_svd.txx.

template<class T>
vnl_matrix< T > vnl_svd< T >::nullspace ( int  required_nullspace_dimension  )  const [inline]

Return N such that M * N = 0.

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

Definition at line 370 of file vnl_svd.txx.

template<class T>
vnl_matrix< T > vnl_svd< T >::left_nullspace ( int  required_nullspace_dimension  )  const [inline]

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

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

Todo:

Definition at line 388 of file vnl_svd.txx.

template<class T>
vnl_vector< T > vnl_svd< T >::nullvector (  )  const [inline]

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.

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 399 of file vnl_svd.txx.

template<class T>
vnl_vector< T > vnl_svd< T >::left_nullvector (  )  const [inline]

Return the rightmost column of U.

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

Definition at line 411 of file vnl_svd.txx.

template<class T>
bool vnl_svd< T >::valid (  )  const [inline]

Definition at line 175 of file vnl_svd.h.

template<class T>
vnl_svd<T>& vnl_svd< T >::operator= ( vnl_svd< T > const &   )  [inline, private]

Definition at line 194 of file vnl_svd.h.


Member Data Documentation

template<class T>
int vnl_svd< T >::m_ [private]

Definition at line 179 of file vnl_svd.h.

template<class T>
int vnl_svd< T >::n_ [private]

Definition at line 179 of file vnl_svd.h.

template<class T>
vnl_matrix<T> vnl_svd< T >::U_ [private]

Definition at line 180 of file vnl_svd.h.

template<class T>
vnl_diag_matrix<singval_t> vnl_svd< T >::W_ [private]

Definition at line 181 of file vnl_svd.h.

template<class T>
vnl_diag_matrix<singval_t> vnl_svd< T >::Winverse_ [private]

Definition at line 182 of file vnl_svd.h.

template<class T>
vnl_matrix<T> vnl_svd< T >::V_ [private]

Definition at line 183 of file vnl_svd.h.

template<class T>
unsigned vnl_svd< T >::rank_ [private]

Definition at line 184 of file vnl_svd.h.

template<class T>
bool vnl_svd< T >::have_max_ [private]

Definition at line 185 of file vnl_svd.h.

template<class T>
singval_t vnl_svd< T >::max_ [private]

Definition at line 186 of file vnl_svd.h.

template<class T>
bool vnl_svd< T >::have_min_ [private]

Definition at line 187 of file vnl_svd.h.

template<class T>
singval_t vnl_svd< T >::min_ [private]

Definition at line 188 of file vnl_svd.h.

template<class T>
double vnl_svd< T >::last_tol_ [private]

Definition at line 189 of file vnl_svd.h.

template<class T>
bool vnl_svd< T >::valid_ [private]

Definition at line 190 of file vnl_svd.h.


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

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