vnl_qr.hxx

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// This is core/vnl/algo/vnl_qr.hxx
#ifndef vnl_qr_hxx_
#define vnl_qr_hxx_
//:
// \file
// \author Andrew W. Fitzgibbon, Oxford RRG
// \date   08 Dec 1996

#include <iostream>
#include <complex>
#include "vnl_qr.h"
#include <cassert>
#ifdef _MSC_VER
#  include <vcl_msvc_warnings.h>
#endif
#include <vnl/vnl_math.h>
#include <vnl/vnl_complex.h>  // vnl_math::squared_magnitude()
#include <vnl/vnl_matlab_print.h>
#include <vnl/vnl_complex_traits.h>
#include <vnl/algo/vnl_netlib.h> // dqrdc_(), dqrsl_()

// use C++ overloading to call the right linpack routine from the template code:
#ifndef DOXYGEN_SHOULD_SKIP_THIS
#define macro(p, T) \
inline void vnl_linpack_qrdc(vnl_netlib_qrdc_proto(T)) \
{ v3p_netlib_##p##qrdc_(vnl_netlib_qrdc_params); } \
inline void vnl_linpack_qrsl(vnl_netlib_qrsl_proto(T)) \
{ v3p_netlib_##p##qrsl_(vnl_netlib_qrsl_params); }
macro(s, float);
macro(d, double);
macro(c, std::complex<float>);
macro(z, std::complex<double>);
#undef macro
#endif

template <class T>
vnl_qr<T>::vnl_qr(vnl_matrix<T> const& M):
  qrdc_out_(M.columns(), M.rows()),
  qraux_(M.columns()),
  jpvt_(M.rows()),
  Q_(nullptr),
  R_(nullptr)
{
  assert(! M.empty());

  // Fill transposed O/P matrix
  long c = M.columns();
  long r = M.rows();
  for (int i = 0; i < r; ++i)
    for (int j = 0; j < c; ++j)
      qrdc_out_(j,i) = M(i,j);

  long do_pivot = 0; // Enable[!=0]/disable[==0] pivoting.
  jpvt_.fill(0); // Allow all columns to be pivoted if pivoting is enabled.

  vnl_vector<T> work(M.rows());
  vnl_linpack_qrdc(qrdc_out_.data_block(), // On output, UT is R, below diag is mangled Q
                   &r, &r, &c,
                   qraux_.data_block(), // Further information required to demangle Q
                   jpvt_.data_block(),
                   work.data_block(),
                   &do_pivot);
}

template <class T>
vnl_qr<T>::~vnl_qr()
{
  delete Q_;
  delete R_;
}

//: Return the determinant of M.  This is computed from M = Q R as follows:
// |M| = |Q| |R|
// |R| is the product of the diagonal elements.
// |Q| is (-1)^n as it is a product of Householder reflections.
// So det = -prod(-r_ii).
template <class T>
T vnl_qr<T>::determinant() const
{
  int m = std::min((int)qrdc_out_.columns(), (int)qrdc_out_.rows());
  T det = qrdc_out_(0,0);

  for (int i = 1; i < m; ++i)
    det *= -qrdc_out_(i,i);

  return det;
}

//: Unpack and return unitary part Q.
template <class T>
vnl_matrix<T> const& vnl_qr<T>::Q() const
{
  int m = qrdc_out_.columns(); // column-major storage
  int n = qrdc_out_.rows();

  bool verbose = false;

  if (!Q_) {
    Q_ = new vnl_matrix<T>(m,m);
    // extract Q.
    if (verbose) {
      std::cerr << __FILE__ ": vnl_qr<T>::Q()\n"
               << " m,n = " << m << ", " << n << '\n'
               << " qr0 = [" << qrdc_out_ << "];\n"
               << " aux = [" << qraux_ << "];\n";
    }

    Q_->set_identity();
    vnl_matrix<T>& matrQ = *Q_;

    vnl_vector<T> v(m, T(0));
    vnl_vector<T> w(m, T(0));

    // Golub and vanLoan, p199.  backward accumulation of householder matrices
    // Householder vector k is [zeros(1,k-1) qraux_[k] qrdc_out_[k,:]]
    typedef typename vnl_numeric_traits<T>::abs_t abs_t;
    for (int k = n-1; k >= 0; --k) {
      if (k >= m) continue;
      // Make housevec v, and accumulate norm at the same time.
      v[k] = qraux_[k];
      abs_t sq = vnl_math::squared_magnitude(v[k]);
      for (int j = k+1; j < m; ++j) {
        v[j] = qrdc_out_(k,j);
        sq += vnl_math::squared_magnitude(v[j]);
      }
      if (verbose) vnl_matlab_print(std::cerr, v, "v");
#ifndef DOXYGEN_SHOULD_SKIP_THIS
# define c vnl_complex_traits<T>::conjugate
#endif
      // Premultiply emerging Q by house(v), noting that v[0..k-1] == 0.
      // Q_new = (1 - (2/v'*v) v v')Q
      // or Q -= (2/v'*v) v (v'Q)
      if (sq > abs_t(0)) {
        abs_t scale = abs_t(2)/sq;
        // w = (2/v'*v) v' Q
        for (int i = k; i < m; ++i) {
          w[i] = T(0);
          for (int j = k; j < m; ++j)
            w[i] += scale * c(v[j]) * matrQ(j, i);
        }
        if (verbose) vnl_matlab_print(std::cerr, w, "w");

        // Q -= v w
        for (int i = k; i < m; ++i)
          for (int j = k; j < m; ++j)
            matrQ(i,j) -= (v[i]) * (w[j]);
      }
#undef c
    }
  }
  return *Q_;
}

//: Unpack and return R.
template <class T>
vnl_matrix<T> const& vnl_qr<T>::R() const
{
  if (!R_) {
    int m = qrdc_out_.columns(); // column-major storage
    int n = qrdc_out_.rows();
    R_ = new vnl_matrix<T>(m,n);
    vnl_matrix<T> & Rmatr = *R_;

    for (int i = 0; i < m; ++i)
      for (int j = 0; j < n; ++j)
        if (i > j)
          Rmatr(i, j) = T(0);
        else
          Rmatr(i, j) = qrdc_out_(j,i);
  }

  return *R_;
}

template <class T>
vnl_matrix<T> vnl_qr<T>::recompose() const
{
  return Q() * R();
}

// JOB: ABCDE decimal
// A     B     C     D              E
// ---   ---   ---   ---            ---
// Qb    Q'b   x     norm(A*x - b)  A*x


//: Solve equation M x = b for x using the computed decomposition.
template <class T>
vnl_vector<T> vnl_qr<T>::solve(const vnl_vector<T>& b) const
{
  long n = qrdc_out_.columns();
  long p = qrdc_out_.rows();
  const T* b_data = b.data_block();
  vnl_vector<T> Qt_B(n);
  vnl_vector<T> x(p);

  // see comment above
  long JOB = 100;

  long info = 0;
  vnl_linpack_qrsl(qrdc_out_.data_block(),
                   &n, &n, &p,
                   qraux_.data_block(),
                   b_data, (T*)nullptr, Qt_B.data_block(),
                   x.data_block(),
                   (T*)nullptr/*residual*/,
                   (T*)nullptr/*Ax*/,
                   &JOB,
                   &info);

  if (info > 0)
    std::cerr << __FILE__ ": vnl_qr<T>::solve() : matrix is rank-deficient by "
             << info << '\n';

  return x;
}

//: Return residual vector d of M x = b -> d = Q'b
template <class T>
vnl_vector<T> vnl_qr<T>::QtB(const vnl_vector<T>& b) const
{
  long n = qrdc_out_.columns();
  long p = qrdc_out_.rows();
  const T* b_data = b.data_block();
  vnl_vector<T> Qt_B(n);

  // see comment above
  long JOB = 1000;

  long info = 0;
  vnl_linpack_qrsl(qrdc_out_.data_block(),
                   &n, &n, &p,
                   qraux_.data_block(),
                   b_data,
                   (T*)nullptr,               // A: Qb
                   Qt_B.data_block(),   // B: Q'b
                   (T*)nullptr,               // C: x
                   (T*)nullptr,               // D: residual
                   (T*)nullptr,               // E: Ax
                   &JOB,
                   &info);

  if (info > 0)
    std::cerr << __FILE__ ": vnl_qr<T>::QtB() -- matrix is rank-deficient by "
             << info << '\n';

  return Qt_B;
}

template <class T>
vnl_matrix<T> vnl_qr<T>::inverse() const
{
  unsigned int r = qrdc_out_.columns();
  assert(r > 0 && r == qrdc_out_.rows());
  vnl_matrix<T> inv(r,r);

  // Use solve() to compute the inverse matrix, using (00..010..00) as rhs
  vnl_vector<T> rhs(r,T(0));
  for (unsigned int i=0; i<r; ++i)
  {
    rhs(i) = T(1);
    vnl_vector<T> col = this->solve(rhs); // returns i-th column of inverse
    inv.set_column(i,col);
    rhs(i) = T(0);
  }
  return inv;
}

template <class T>
vnl_matrix<T> vnl_qr<T>::tinverse() const
{
  unsigned int r = qrdc_out_.columns();
  assert(r > 0 && r == qrdc_out_.rows());
  vnl_matrix<T> tinv(r,r);

  // Use solve() to compute the inverse matrix, using (00..010..00) as rhs
  vnl_vector<T> rhs(r,T(0));
  for (unsigned int i=0; i<r; ++i)
  {
    rhs(i) = T(1);
    vnl_vector<T> col = this->solve(rhs); // returns i-th column of inverse
    tinv.set_row(i,col);
    rhs(i) = T(0);
  }
  return tinv;
}

template <class T>
vnl_matrix<T> vnl_qr<T>::solve(vnl_matrix<T> const& rhs) const
{
  assert(rhs.rows() == qrdc_out_.columns()); // column-major storage
  int c = qrdc_out_.rows();
  int n = rhs.columns();
  vnl_matrix<T> result(c,n);

  for (int i=0; i<n; ++i)
  {
    vnl_vector<T> b = rhs.get_column(i);
    vnl_vector<T> col = this->solve(b); // returns i-th column of result
    result.set_column(i,col);
  }
  return result;
}

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

#define VNL_QR_INSTANTIATE(T) \
 template class VNL_ALGO_EXPORT vnl_qr<T >; \
 /*template VNL_EXPORT T vnl_qr_determinant(vnl_matrix<T > const&) */

#endif