vnl_svd_fixed< T, R, C > Class Template Reference

#include <vnl_svd_fixed.h>

List of all members.

Detailed Description

template<class T, unsigned int R, unsigned int C>
class vnl_svd_fixed< T, R, C >

Holds the singular value decomposition of a vnl_matrix_fixed.

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.

Definition at line 35 of file vnl_svd_fixed.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_fixed (vnl_matrix_fixed< T, R, C > const &M, double zero_out_tol=0.0)
 Construct an vnl_svd_fixed<T> object from $m \times n$ matrix $M$.
 ~vnl_svd_fixed ()
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_fixed< T, R, C > & U ()
 Return the matrix U.
vnl_matrix_fixed< T, R, C
> const & 		U () const
 Return the matrix U.
T U (int i, int j) const
 Return the matrix U's (i,j)th entry (to avoid svd.U()(i,j); ).
vnl_diag_matrix_fixed< singval_t,
C > & 		W ()
 Get at DiagMatrix (q.v.
vnl_diag_matrix_fixed< singval_t,
C > const & 		W () const
 Get at DiagMatrix (q.v.
vnl_diag_matrix_fixed< singval_t,
C > & 		Winverse ()
vnl_diag_matrix_fixed< singval_t,
C > 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_fixed< T, C, C > & V ()
 Return the matrix V.
vnl_matrix_fixed< T, C, C
> const & 		V () const
 Return the matrix V.
T V (int i, int j) const
 Return the matrix V's (i,j)th entry (to avoid svd.V()(i,j); ).
vnl_matrix_fixed< T, C, R > inverse () const
vnl_matrix_fixed< T, C, R > pinverse (unsigned int rank=~0u) const
 pseudo-inverse (for non-square matrix) of desired rank.
vnl_matrix_fixed< T, R, C > tinverse (unsigned int rank=~0u) const
 Calculate inverse of transpose, using desired rank.
vnl_matrix_fixed< T, R, C > 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_fixed< T, C > solve (vnl_vector_fixed< T, R > 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_fixed< T, R > const &rhs, vnl_vector_fixed< T, C > *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_fixed< T, C > nullvector () const
 Return the rightmost column of V.
vnl_vector_fixed< T, R > left_nullvector () const
 Return the rightmost column of U.
bool valid () const

Private Member Functions

 vnl_svd_fixed (vnl_svd_fixed< T, R, C > const &)
vnl_svd_fixed< T, R, C > & operator= (vnl_svd_fixed< T, R, C > const &)

Private Attributes

vnl_matrix_fixed< T, R, C > U_
vnl_diag_matrix_fixed< singval_t,
C > 		W_
vnl_diag_matrix_fixed< singval_t,
C > 		Winverse_
vnl_matrix_fixed< T, C, C > 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, unsigned int R, unsigned int C>
typedef vnl_numeric_traits<T>::abs_t vnl_svd_fixed< T, R, C >::singval_t

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

Definition at line 39 of file vnl_svd_fixed.h.

Constructor & Destructor Documentation

template<class T, unsigned int R, unsigned int C>
vnl_svd_fixed< T, R, C >::vnl_svd_fixed ( vnl_matrix_fixed< T, R, C > const &  M,
double  zero_out_tol = 0.0 
)

Construct an vnl_svd_fixed<T> object from $m \times n$ matrix $M$.

The vnl_svd_fixed<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_fixed.txx.

template<class T, unsigned int R, unsigned int C>
vnl_svd_fixed< T, R, C >::~vnl_svd_fixed (  )  [inline]

Definition at line 57 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_svd_fixed< T, R, C >::vnl_svd_fixed ( vnl_svd_fixed< T, R, C > const &   )  [inline, private]

Member Function Documentation

template<class T, unsigned int R, unsigned int C>
void vnl_svd_fixed< T, R, C >::zero_out_absolute ( double  tol = 1e-8  ) 

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

Definition at line 170 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
void vnl_svd_fixed< T, R, C >::zero_out_relative ( double  tol = 1e-8  ) 

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

Definition at line 192 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
int vnl_svd_fixed< T, R, C >::singularities (  )  const [inline]

Definition at line 66 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
unsigned int vnl_svd_fixed< T, R, C >::rank (  )  const [inline]

Definition at line 67 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
singval_t vnl_svd_fixed< T, R, C >::well_condition (  )  const [inline]

Definition at line 68 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_svd_fixed< T, R, C >::singval_t vnl_svd_fixed< T, R, C >::determinant_magnitude (  )  const

Calculate determinant as product of diagonals in W.

Definition at line 202 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
vnl_svd_fixed< T, R, C >::singval_t vnl_svd_fixed< T, R, C >::norm (  )  const

Definition at line 215 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
vnl_matrix_fixed<T,R,C>& vnl_svd_fixed< T, R, C >::U (  )  [inline]

Return the matrix U.

Definition at line 75 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_matrix_fixed<T,R,C> const& vnl_svd_fixed< T, R, C >::U (  )  const [inline]

Return the matrix U.

Definition at line 78 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
T vnl_svd_fixed< T, R, C >::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 81 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_diag_matrix_fixed<singval_t, C>& vnl_svd_fixed< T, R, C >::W (  )  [inline]

Get at DiagMatrix (q.v.

) of singular values, sorted from largest to smallest.

Definition at line 84 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_diag_matrix_fixed<singval_t, C> const& vnl_svd_fixed< T, R, C >::W (  )  const [inline]

Get at DiagMatrix (q.v.

) of singular values, sorted from largest to smallest.

Definition at line 87 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_diag_matrix_fixed<singval_t, C>& vnl_svd_fixed< T, R, C >::Winverse (  )  [inline]

Definition at line 88 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_diag_matrix_fixed<singval_t, C> const& vnl_svd_fixed< T, R, C >::Winverse (  )  const [inline]

Definition at line 89 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
singval_t& vnl_svd_fixed< T, R, C >::W ( int  i,
int  j 
) [inline]

Definition at line 90 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
singval_t& vnl_svd_fixed< T, R, C >::W ( int  i  )  [inline]

Definition at line 91 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
singval_t vnl_svd_fixed< T, R, C >::sigma_max (  )  const [inline]

Definition at line 92 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
singval_t vnl_svd_fixed< T, R, C >::sigma_min (  )  const [inline]

Definition at line 93 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_matrix_fixed<T,C,C>& vnl_svd_fixed< T, R, C >::V (  )  [inline]

Return the matrix V.

Definition at line 96 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_matrix_fixed<T,C,C> const& vnl_svd_fixed< T, R, C >::V (  )  const [inline]

Return the matrix V.

Definition at line 99 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
T vnl_svd_fixed< T, R, C >::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 102 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_matrix_fixed<T,C,R> vnl_svd_fixed< T, R, C >::inverse (  )  const [inline]

Definition at line 105 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_matrix_fixed< T, C, R > vnl_svd_fixed< T, R, C >::pinverse ( unsigned int  rank = ~0u  )  const

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

Definition at line 235 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
vnl_matrix_fixed< T, R, C > vnl_svd_fixed< T, R, C >::tinverse ( unsigned int  rank = ~0u  )  const

Calculate inverse of transpose, using desired rank.

Definition at line 248 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
vnl_matrix_fixed< T, R, C > vnl_svd_fixed< T, R, C >::recompose ( unsigned int  rank = ~0u  )  const

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

Definition at line 222 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
vnl_matrix< T > vnl_svd_fixed< T, R, C >::solve ( vnl_matrix< T > const &  B  )  const

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

Definition at line 261 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
vnl_vector_fixed< T, C > vnl_svd_fixed< T, R, C >::solve ( vnl_vector_fixed< T, R > const &  y  )  const

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

Definition at line 284 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
void vnl_svd_fixed< T, R, C >::solve ( T const *  rhs,
T *  lhs 
) const

Definition at line 300 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
void vnl_svd_fixed< T, R, C >::solve_preinverted ( vnl_vector_fixed< T, R > const &  rhs,
vnl_vector_fixed< T, C > *  out 
) const

Solve the matrix-vector system M x = y.

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

Definition at line 308 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
vnl_matrix< T > vnl_svd_fixed< T, R, C >::nullspace (  )  const

Return N such that M * N = 0.

Definition at line 321 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
vnl_matrix< T > vnl_svd_fixed< T, R, C >::left_nullspace (  )  const

Return N such that M' * N = 0.

Definition at line 340 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
vnl_matrix< T > vnl_svd_fixed< T, R, C >::nullspace ( int  required_nullspace_dimension  )  const

Return N such that M * N = 0.

Definition at line 332 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
vnl_matrix< T > vnl_svd_fixed< T, R, C >::left_nullspace ( int  required_nullspace_dimension  )  const

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

Todo:

Definition at line 350 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
vnl_vector_fixed< T, C > vnl_svd_fixed< T, R, C >::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 361 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
vnl_vector_fixed< T, R > vnl_svd_fixed< T, R, C >::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 373 of file vnl_svd_fixed.txx.

template<class T, unsigned int R, unsigned int C>
bool vnl_svd_fixed< T, R, C >::valid (  )  const [inline]

Definition at line 148 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_svd_fixed<T,R,C>& vnl_svd_fixed< T, R, C >::operator= ( vnl_svd_fixed< T, R, C > const &   )  [inline, private]

Definition at line 166 of file vnl_svd_fixed.h.

Member Data Documentation

template<class T, unsigned int R, unsigned int C>
vnl_matrix_fixed<T, R, C> vnl_svd_fixed< T, R, C >::U_ [private]

Definition at line 152 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_diag_matrix_fixed<singval_t, C> vnl_svd_fixed< T, R, C >::W_ [private]

Definition at line 153 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_diag_matrix_fixed<singval_t, C> vnl_svd_fixed< T, R, C >::Winverse_ [private]

Definition at line 154 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
vnl_matrix_fixed<T, C, C> vnl_svd_fixed< T, R, C >::V_ [private]

Definition at line 155 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
unsigned vnl_svd_fixed< T, R, C >::rank_ [private]

Definition at line 156 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
bool vnl_svd_fixed< T, R, C >::have_max_ [private]

Definition at line 157 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
singval_t vnl_svd_fixed< T, R, C >::max_ [private]

Definition at line 158 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
bool vnl_svd_fixed< T, R, C >::have_min_ [private]

Definition at line 159 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
singval_t vnl_svd_fixed< T, R, C >::min_ [private]

Definition at line 160 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
double vnl_svd_fixed< T, R, C >::last_tol_ [private]

Definition at line 161 of file vnl_svd_fixed.h.

template<class T, unsigned int R, unsigned int C>
bool vnl_svd_fixed< T, R, C >::valid_ [private]

Definition at line 162 of file vnl_svd_fixed.h.

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