The VXL Project

Feature Requests item #523333, was opened at 2002-02-27 03:03
You can respond by visiting: 
http://sourceforge.net/tracker/?func=detail&atid=381096&aid=523333&group_id=24293

Category: Interface Improvements (example)
Group: None
Status: Open
Priority: 5
Submitted By: lapresté jean-thierry (lapreste)
Assigned to: Nobody/Anonymous (nobody)
Summary: some pbs wit vnl_svd

Initial Comment:
I have found no real bugs in svd, but I tried

 #define C_Matrix vnl_matrix < double > 

 I have a non square X and I do

  vnl_svd < double > svd(X,  1.0E-8);
  C_Matrix v_sol = svd.solve(Y);

And that doesn't worked fine v_sol being filled
with inf 

I tried 

 C_Matrix inv =  svd.pinverse();
 C_Matrix v_sol = inv*Y; 

and that worked fine

So I have investigated the two files vnl_svd.h and
vnl_svd.txx and propose some modifications that
do not change the interface and a new function

Now I can write :

  vnl_svd < double > svd(X);
  C_Matrix v_sol = svd.solve(Y,  1.0E-8);

and it works. Perhaps I missed something, however
I join my versions of the files and would like to know
how to
participate to further devlopments or corrections if I
find
any. 

   sincerely

    J.T. Lapresté

lapreste@...

------------------------------------------------------------------
------------------------------------------------------------------
here vnl_svd.h 
------------------------------------------------------------------
#ifndef vnl_svd_h_
#define vnl_svd_h_
#ifdef __GNUC__
#pragma interface
#endif

//:
//  \file
//  \brief Holds the singular value decomposition of a
vnl_matrix.
//  \author Andrew W. Fitzgibbon, Oxford IERG
//  \date   15 Jul 96
//
// \verbatim
//  Modifications
// F. Schaffalitzky, Oxford IESRG, 26 Mar 1999
//     1. The singular values are now stored as reals
(not complexes) when T is complex.
//     2. Fixed bug : for complex T, matrices have to
be conjugated as well as transposed.
//   Feb.2002 - Peter Vanroose - brief doxygen comment
placed on single line
// J.T. Laprest, Lasmea Clermont-Ferrand,  27-Feb-2002
//     1. The typename singval_t must be used in solve,
it allows to add a supplementary 
//        tolerance parameter and mahe svdsolve
consistent with pinverse 
//     2. The modified interfaces are
//        vnl_matrix<T> vnl_svd<T>::solve(vnl_matrix<T>
const& B, singval_t tol)  const;
//        vnl_vector<T> solve (vnl_vector<T> const& y,
singval_t tol=singval_t(0)) const;
//        void          solve (T const *rhs, T *lhs, 
singval_t tol=singval_t(0)) const; 
//     3. The routine 
//        vnl_matrix<T>
vnl_svd<T>::solve_preinverted(vnl_matrix<T> const& B,
singval_t tol)  const;
//        has been added 
// \endverbatim

#include <vnl/vnl_numeric_traits.h>
#include <vnl/vnl_vector.h>
#include <vnl/vnl_matrix.h>
#include <vnl/vnl_diag_matrix.h>
#include <vcl_iosfwd.h>

//: 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
enquiries 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.

export template <class T>
class vnl_svd {
public:
  //: The singular values of a matrix of complex<T> are
of type T, not complex<T>
  typedef typename vnl_numeric_traits<T>::abs_t
singval_t;

  //:
  // 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();

  vnl_svd(vnl_matrix<T> const &M, double zero_out_tol =
0.0);
 ~vnl_svd() {}

  // Data
Access---------------------------------------------------------------

  //: find weights below threshold tol, zero them out,
and update W_ and Winverse_
  void            zero_out_absolute(double tol = 1e-8);
//sqrt(machine epsilon)

  //: find weights below tol*max(w) and zero them out
  void            zero_out_relative(double tol = 1e-8);
//sqrt(machine epsilon)
  int             singularities () const { return
W_.n() - rank(); }
  int             rank () const { return rank_; }
  singval_t       well_condition () const { return
sigma_min()/sigma_max(); }

    //: Calculate determinant as product of diagonals
in W.
  singval_t       determinant_magnitude () const;
  singval_t       norm() const;

  //: Return the matrix U.
  vnl_matrix<T>      & U()       { return U_; }

  //: Return the matrix U.
  vnl_matrix<T> const& U() const { return U_; }

  //: Return the matrix U's (i,j)th entry (to avoid
svd.U()(i,j); ).
  T U(int i, int j) { return U_(i,j); }

  //: Get at DiagMatrix (q.v.) of singular values,
sorted from largest to smallest
  vnl_diag_matrix<singval_t>       & W()             {
return W_; }

  //: Get at DiagMatrix (q.v.) of singular values,
sorted from largest to smallest
  vnl_diag_matrix<singval_t> const & W() const       {
return W_; }
  vnl_diag_matrix<singval_t>       & Winverse()      {
return Winverse_; }
  vnl_diag_matrix<singval_t> const & Winverse() const {
return Winverse_; }
  singval_t                   & W(int i, int j) {
return W_(i,j); }
  singval_t                   & W(int i)        {
return W_(i,i); }
  singval_t     sigma_max() const { return W_(0,0);
}       // largest
  singval_t     sigma_min() const { return
W_(n_-1,n_-1); } // smallest

  //: Return the matrix V.
  vnl_matrix<T>      & V()       { return V_; }

  //: Return the matrix V.
  vnl_matrix<T> const& V() const { return V_; }

  //: Return the matrix V's (i,j)th entry (to avoid
svd.V()(i,j); ).
  T V(int i, int j) { return V_(i,j); }

  //:
  vnl_matrix<T> inverse () const;

  //: pseudo-inverse (for non-square matrix).
  vnl_matrix<T> pinverse () const;

  //: Calculate inverse of transpose.
  vnl_matrix<T> tinverse () const;

  //: Recompose SVD to U*W*V'
  vnl_matrix<T> recompose () const;

  //: Solve the matrix equation M X = B, returning X
  vnl_matrix<T> solve (vnl_matrix<T> const& B,
singval_t tol =  singval_t(0)) const;

  //: Solve the matrix equation M X = B.
  // Assuming that the singular values W have been
preinverted by the caller.
  vnl_matrix<T>
vnl_svd<T>::solve_preinverted(vnl_matrix<T> const& B,
singval_t tol)  const;

  //: Solve the matrix-vector system M x = y, returning
x.
  vnl_vector<T> solve (vnl_vector<T> const& y,
singval_t tol=singval_t(0)) const;
  void          solve (T const *rhs, T *lhs,  singval_t
tol=singval_t(0)) const; // min ||A*lhs - rhs||

  //: Solve the matrix-vector system M x = y.
  // Assuming that the singular values W have been
preinverted by the caller.
  void solve_preinverted(vnl_vector<T> const& rhs,
vnl_vector<T>* out) const;

  //: Return N such that M * N = 0
  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;

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

  //: Return the rightmost column of U.
  //  Does not check to see whether or not the matrix
actually was rank-deficient.
  vnl_vector<T> left_nullvector() const;

private:

  int m_, n_;              // Size of M, local cache.
  vnl_matrix<T> U_;        // Columns Ui are basis for
range of M for Wi != 0
  vnl_diag_matrix<singval_t> W_;// Singular values,
sorted in decreasing order
  vnl_diag_matrix<singval_t> Winverse_;
  vnl_matrix<T> V_;       // Columns Vi are basis for
nullspace of M for Wi = 0
  unsigned rank_;
  bool have_max_;
  singval_t max_;
  bool have_min_;
  singval_t min_;
  double last_tol_;

  // Disallow assignment.
  vnl_svd(vnl_svd<T> const &) { }
  vnl_svd<T>& operator=(vnl_svd<T> const &) { return
*this; }
};

template <class T>
inline
vnl_matrix<T> vnl_svd_inverse(vnl_matrix<T> const& m)
{
  return vnl_svd<T>(m).inverse();
}

// this aint no friend.
export template <class T>
vcl_ostream& operator<<(vcl_ostream&, vnl_svd<T> const&
svd);

#endif // vnl_svd_h_

--------------------------------------------------------------------
here vnl_svd.txx
--------------------------------------------------------------------
#ifndef vnl_svd_txx_
#define vnl_svd_txx_

//:
// \file

#include "vnl_svd.h"

#include <vcl_cstdlib.h> // vcl_abort()
#include <vcl_cassert.h>
#include <vcl_complex.h>
#include <vcl_fstream.h>
#include <vcl_algorithm.h> // min

#include <vnl/vnl_math.h>
#include <vnl/vnl_fortran_copy.h>
#include <vnl/algo/vnl_netlib.h>

#ifdef HAS_FSM_PACK
template <typename T> int
fsm_svdc_cxx(vnl_netlib_svd_proto(T));
# define vnl_linpack_svdc fsm_svdc_cxx
#else
// use C++ overloading to call the right linpack
routine from the template code :
#define macro(p, T) \
inline int vnl_linpack_svdc(vnl_netlib_svd_proto(T)) \
{ return p##svdc_(vnl_netlib_svd_params); }
macro(s, float);
macro(d, double);
macro(c, vcl_complex<float>);
macro(z, vcl_complex<double>);
#undef macro
#endif

//--------------------------------------------------------------------------------

static bool test_heavily = false;

template <class T>
vnl_svd<T>::vnl_svd(vnl_matrix<T> const& M, double
zero_out_tol):
  m_(M.rows()),
  n_(M.columns()),
  U_(m_, n_),
  W_(n_),
  Winverse_(n_),
  V_(n_, n_)
{
  assert(m_ > 0);
  assert(n_ > 0);

  {
    int n = M.rows();
    int p = M.columns();
    int mm = vcl_min(n+1,p);

    // Copy source matrix into fortran storage
    // SVD is slow, don't worry about the cost of this
transpose.
    vnl_fortran_copy<T> X(M);

    // Make workspace vectors.
    vnl_vector<T> work(n, T(0));
    vnl_vector<T> uspace(n*p, T(0));
    vnl_vector<T> vspace(p*p, T(0));
    vnl_vector<T> wspace(mm, T(0)); // complex fortran
routine actually _wants_ complex W!
    vnl_vector<T> espace(p, T(0));

    // Call Linpack SVD
    int info = 0;
    vnl_linpack_svdc((T*)X, n, n, p,
                     wspace.data_block(),
                     espace.data_block(),
                     uspace.data_block(), n,
                     vspace.data_block(), p,
                     work.data_block(),
                     21, &info);

    // Error return?
    if (info != 0) {
      // If info is non-zero, it contains the number of
singular values
      // for this the SVD algorithm failed to converge.
The condition is
      // not bogus. Even if the returned singular
values are sensible, 
      // the singular vectors can be utterly wrong.

      // It is possible the failure was due to NaNs or
infinities in the
      // matrix. Check for that now.
      M.assert_finite();

      // If we get here it might be because the scalar
type has such 
      // extreme precision that too few iterations were
performed to
      // converge to within machine precision (that is
the svdc criterion).
      // The only solution to that is to increase the
maximum number of
      // iterations in the netlib code. Diagnose the
problem here by
      // printing a warning message.
      vcl_cerr << __FILE__ ": suspicious return value
(" << info << ") from SVDC\n"
               << __FILE__ ": M is " << M.rows() << 'x'
<< M.cols() << vcl_endl;
    }

    // Copy fortran outputs into our storage
    {
      const T *d = uspace.data_block();
      for(int j = 0; j < p; ++j)
        for(int i = 0; i < n; ++i)
          U_(i,j) = *d++;
    }

    for(int j = 0; j < mm; ++j)
      W_(j, j) = vcl_abs(wspace(j)); // we get rid of
complexness here.

    for(int j = mm; j < n_; ++j)
      W_(j, j) = 0;

    {
      const T *d = vspace.data_block();
      for(int j = 0; j < p; ++j)
        for(int i = 0; i < p; ++i)
          V_(i,j) = *d++;
    }
  }

  if (test_heavily) {
    // Test that recomposed matrix == M
    typedef typename vnl_numeric_traits<T>::abs_t
abs_t;
    abs_t recomposition_residual = vcl_abs((recompose()
- M).fro_norm());
    abs_t n = vcl_abs(M.fro_norm());
    abs_t thresh = m_ * vnl_math::eps * n;
    if (recomposition_residual > thresh) {
      vcl_cerr << "vnl_svd<T>::vnl_svd<T>() -- Warning,
recomposition_residual = "
               << recomposition_residual << vcl_endl
               << "fro_norm(M) = " << n << vcl_endl
               << "eps*fro_norm(M) = " << thresh <<
vcl_endl
               << "Press return to continue\n";
      char x;
      vcl_cin.get(&x, 1, '\n');
    }
  }

  if (zero_out_tol >= 0)
    // Zero out small sv's and update rank count.
    zero_out_absolute(double(+zero_out_tol));
  else
    // negative tolerance implies relative to max elt.
    zero_out_relative(double(-zero_out_tol));
}

#if 0
// Assignment
template <class T>
vnl_svd<T>& vnl_svd<T>::operator=(vnl_svd<T> const&
that)
{
  U_ = that.U_;
  W_ = that.W_;
  Winverse_ = that.Winverse_;
  V_ = that.V_;
  rank_ = that.rank_;
  return *this;
}
#endif

template <class T>
vcl_ostream& operator<<(vcl_ostream& s, const
vnl_svd<T>& svd)
{
  s << "vnl_svd<T>:\n"
//  << "M = [\n" << M << "]\n"
    << "U = [\n" << svd.U() << "]\n"
    << "W = " << svd.W() << "\n"
    << "V = [\n" << svd.V() << "]\n"
    << "rank = " << svd.rank() << vcl_endl;
  return s;
}

//-----------------------------------------------------------------------------
// Chunky bits.

//: find weights below threshold tol, zero them out,
and update W_ and Winverse_
template <class T>
void
vnl_svd<T>::zero_out_absolute(double tol)
{
  last_tol_ = tol;
  rank_ = W_.n();
  for (unsigned k = 0; k < W_.n(); k++) {
    singval_t & weight = W_(k, k);
    if (weight <= tol) {                    // no need
for vcl_abs(weight)
      Winverse_(k,k) = 0;
      weight = 0;
      --rank_;
    } else {
      Winverse_(k,k) = singval_t(1.0)/weight;
    }
  }
}

//: find weights below tol*max(w) and zero them out
template <class T> void
vnl_svd<T>::zero_out_relative(double tol) //
sqrt(machine epsilon)
{
  zero_out_absolute(tol * vcl_abs(sigma_max()));
}

//: Calculate determinant as product of diagonals in W.
template <class T>
vnl_svd<T>::singval_t
vnl_svd<T>::determinant_magnitude() const
{
  {
    static bool warned = false;
    if (!warned && m_ != n_) {
      vcl_cerr << __FILE__ ": called
determinant_magnitude() on SVD of non-square matrix" <<
vcl_endl;
      warned = true;
    }
  }
  singval_t product = W_(0, 0);
  for (unsigned long k = 1; k < W_.columns(); k++)
    product *= W_(k, k);

  return product;
}

template <class T>
vnl_svd<T>::singval_t vnl_svd<T>::norm() const
{
  return vcl_abs(sigma_max());
}

//: Recompose SVD to U*W*V'
template <class T>
vnl_matrix<T> vnl_svd<T>::recompose() const
{
  vnl_matrix<T> W(W_.rows(),W_.columns());
  W.fill(T(0));
  for (unsigned i=0;i<rank_;i++)
    W(i,i)=W_(i,i);

  return U_*W*V_.conjugate_transpose();
}

template <class T>
vnl_matrix<T> vnl_svd<T>::inverse() const
{
  return pinverse();
}

//: Calculate pseudo-inverse.
template <class T>
vnl_matrix<T> vnl_svd<T>::pinverse()  const
{
  vnl_matrix<T>
Winverse(Winverse_.rows(),Winverse_.columns());
  Winverse.fill(T(0));
  for (unsigned i=0;i<rank_;i++)
    Winverse(i,i)=Winverse_(i,i);

  return V_ * Winverse * U_.conjugate_transpose();
}

//: Calculate inverse of transpose.
template <class T>
vnl_matrix<T> vnl_svd<T>::tinverse()  const
{
  vnl_matrix<T>
Winverse(Winverse_.rows(),Winverse_.columns());
  Winverse.fill(T(0));
  for (unsigned i=0;i<rank_;i++)
    Winverse(i,i)=Winverse_(i,i);

  return U_ * Winverse * V_.conjugate_transpose();
}

//: Solve the matrix equation M X = B, returning X
template <class T>
vnl_matrix<T> vnl_svd<T>::solve(vnl_matrix<T> const& B,
singval_t tol)  const
{
  vnl_matrix<T> x;                                     
// solution matrix
  if (U_.rows() < U_.columns()) {                      
// augment y with extra rows of
    vnl_matrix<T> yy(U_.rows(), B.columns(), T(0));    
// zeros, so that it matches
    yy.update(B);                                      
// cols of u.transpose. ???
    x = U_.conjugate_transpose() * yy;
  } else
    x = U_.conjugate_transpose() * B;
  unsigned long i, j;
  for (i = 0; i < x.rows(); i++) {                     
// multiply with diagonal 1/W
    singval_t weight = W_(i, i);
    if (weight >= tol){; //vnl_numeric_traits<T>::zero)
      weight = singval_t(1) / weight;
    } else {
      weight = singval_t(0);
    }
    for (j = 0; j < x.columns(); j++)
      x(i, j) *= weight;
  }
  x = V_ * x;                                          
// premultiply with v.
  return x;
}

//: Solve the matrix equation M X = B, returning X
// Assuming that the singular values W have been
preinverted by the caller.
template <class T>
vnl_matrix<T>
vnl_svd<T>::solve_preinverted(vnl_matrix<T> const& B,
singval_t tol)  const
{
  vnl_matrix<T> x;                                     
// solution matrix
  if (U_.rows() < U_.columns()) {                      
// augment y with extra rows of
    vnl_matrix<T> yy(U_.rows(), B.columns(), T(0));    
// zeros, so that it matches
    yy.update(B);                                      
// cols of u.transpose. ???
    x = U_.conjugate_transpose() * yy;
  } else
    x = U_.conjugate_transpose() * B;
  unsigned long i, j;
  for (i = 0; i < x.rows(); i++) {                     
// multiply with diagonal W assumed inverted
    singval_t weight = W_(i, i);
    for (j = 0; j < x.columns(); j++)
      x(i, j) *= weight;
  }
  x = V_ * x;                                          
// premultiply with v.
  return x;
}

//: Solve the matrix-vector system M x = y, returning
x.
template <class T>
vnl_vector<T> vnl_svd<T>::solve(vnl_vector<T> const& y,
singval_t tol)  const
{
  // fsm sanity check :
  if (y.size() != U_.rows()) {
    vcl_cerr << __FILE__ << ": size of rhs is
incompatible with no. of rows in U_\n"
             << "y =" << y << "\n"
             << "m_=" << m_ << "\n"
             << "n_=" << n_ << "\n"
             << "U_=\n" << U_
             << "V_=\n" << V_
             << "W_=\n" << W_;
  }

  vnl_vector<T> x(V_.rows());                   //
Solution matrix.
  if (U_.rows() < U_.columns()) {               //
Augment y with extra rows of
    vnl_vector<T> yy(U_.rows(), T(0));          //
zeros, so that it matches
    if (yy.size()<y.size()) { // fsm
      vcl_cerr << "yy=" << yy << vcl_endl
               << "y =" << y  << vcl_endl;
      // the update() call on the next line will
abort...
    }
    yy.update(y);                               // cols
of u.transpose.
    x = U_.conjugate_transpose() * yy;
  }
  else
    x = U_.conjugate_transpose() * y;

  for (unsigned i = 0; i < x.size(); i++) {        //
multiply with diagonal 1/W
    singval_t weight = W_(i, i), zero_(0);
    if (weight <= tol)
      x[i] /= weight;
    else
      x[i] = zero_;
  }
  return V_ * x;                                //
premultiply with v.
}
template <class T> // FIXME. this should implement the
above, not the other way round.
void vnl_svd<T>::solve(T const *y, T *x,  singval_t
tol) const {
  solve(vnl_vector<T>(y, m_), tol).copy_out(x);
}

//: Solve the matrix-vector system M x = y.
// Assume that the singular values W have been
preinverted by the caller.
template <class T>
void vnl_svd<T>::solve_preinverted(vnl_vector<T> const&
y, vnl_vector<T>* x_out)  const
{
  vnl_vector<T> x;              // solution matrix
  if (U_.rows() < U_.columns()) {               //
augment y with extra rows of
    vcl_cout << "vnl_svd<T>::solve_preinverted() --
Augmenting y\n";
    vnl_vector<T> yy(U_.rows(), T(0));     // zeros, so
that it match
    yy.update(y);                               // cols
of u.transpose. ??
    x = U_.conjugate_transpose() * yy;
  } else
    x = U_.conjugate_transpose() * y;
  for (unsigned i = 0; i < x.size(); i++)  // multiply
with diagonal W, assumed inverted
    x[i] *= W_(i, i);

  *x_out = V_ * x;                                     
// premultiply with v.
}

//-----------------------------------------------------------------------------
//: Return N s.t. M * N = 0
template <class T>
vnl_matrix <T> vnl_svd<T>::nullspace()  const
{
  int k = rank();
  if (k == n_)
    vcl_cerr << "vnl_svd<T>::nullspace() -- Matrix is
full rank." << last_tol_ << vcl_endl;
  return nullspace(n_-k);
}

//-----------------------------------------------------------------------------
//: Return N s.t. M * N = 0
template <class T>
vnl_matrix <T> vnl_svd<T>::nullspace(int
required_nullspace_dimension)  const
{
  return V_.extract(V_.rows(),
required_nullspace_dimension, 0, n_ -
required_nullspace_dimension);
}

//-----------------------------------------------------------------------------
//: Return N s.t. M' * N = 0
template <class T>
vnl_matrix <T> vnl_svd<T>::left_nullspace()  const
{
  int k = rank();
  if (k == n_)
    vcl_cerr << "vnl_svd<T>::left_nullspace() -- Matrix
is full rank." << last_tol_ << vcl_endl;
  return U_.extract(U_.rows(), n_-k, 0, k);
}

//: Implementation to be done yet; currently returns
left_nullspace(). - PVR.
template <class T>
vnl_matrix<T> vnl_svd<T>::left_nullspace(int
/*required_nullspace_dimension*/) const
{
  return left_nullspace();
}

//-----------------------------------------------------------------------------
//: 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.
template <class T>
vnl_vector <T> vnl_svd<T>::nullvector()  const
{
  vnl_vector<T> ret(n_);
  for(int i = 0; i < n_; ++i)
    ret(i) = V_(i, n_-1);
  return ret;
}

//-----------------------------------------------------------------------------
//: Return the rightmost column of U.
//  Does not check to see whether or not the matrix
actually was rank-deficient.
template <class T>
vnl_vector <T> vnl_svd<T>::left_nullvector()  const
{
  vnl_vector<T> ret(m_);
  int col = vcl_min(m_, n_) - 1;
  for(int i = 0; i < m_; ++i)
    ret(i) = U_(i, col);
  return ret;
}

//--------------------------------------------------------------------------------

#undef VNL_SVD_INSTANTIATE
#define VNL_SVD_INSTANTIATE(T) \
template class vnl_svd<T >; \
template vcl_ostream& operator<<(vcl_ostream &,
vnl_svd<T > const &)

#endif // vnl_svd_txx_

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