vpMatrix_qr.cpp

/****************************************************************************
 *
 * ViSP, open source Visual Servoing Platform software.
 * Copyright (C) 2005 - 2023 by Inria. All rights reserved.
 *
 * This software is free software; you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation; either version 2 of the License, or
 * (at your option) any later version.
 * See the file LICENSE.txt at the root directory of this source
 * distribution for additional information about the GNU GPL.
 *
 * For using ViSP with software that can not be combined with the GNU
 * GPL, please contact Inria about acquiring a ViSP Professional
 * Edition License.
 *
 * See https://visp.inria.fr for more information.
 *
 * This software was developed at:
 * Inria Rennes - Bretagne Atlantique
 * Campus Universitaire de Beaulieu
 * 35042 Rennes Cedex
 * France
 *
 * If you have questions regarding the use of this file, please contact
 * Inria at visp@inria.fr
 *
 * This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
 * WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
 *
 * Description:
 * Matrix QR decomposition.
 *
 * Authors:
 * Filip Novotny
 *
*****************************************************************************/
 
#include <algorithm> // for (std::min) and (std::max)
#include <cmath>     // For std::abs() on iOS
#include <cstdlib>   // For std::abs() on iOS
#include <visp3/core/vpConfig.h>
 
#include <visp3/core/vpColVector.h>
#include <visp3/core/vpMath.h>
#include <visp3/core/vpMatrix.h>
 
// Exception
#include <visp3/core/vpException.h>
#include <visp3/core/vpMatrixException.h>
 
// Debug trace
#include <visp3/core/vpDebug.h>
 
#ifdef VISP_HAVE_LAPACK
#ifdef VISP_HAVE_GSL
#if !(GSL_MAJOR_VERSION >= 2 && GSL_MINOR_VERSION >= 2)
// Needed for GSL_VERSION < 2.2 where gsl_linalg_tri_*_invert() not present
#include <gsl/gsl_blas.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_vector.h>
#endif
#include <gsl/gsl_linalg.h>
#include <gsl/gsl_permutation.h>
#endif
#ifdef VISP_HAVE_MKL
#include <mkl.h>
typedef MKL_INT integer;
 
integer allocate_work(double **work)
{
  integer dimWork = (integer)((*work)[0]);
  delete[] * work;
  *work = new double[dimWork];
  return (integer)dimWork;
}
#elif !defined(VISP_HAVE_GSL)
#ifdef VISP_HAVE_LAPACK_BUILT_IN
typedef long int integer;
#else
typedef int integer;
#endif
extern "C" int dgeqrf_(integer *m, integer *n, double *a, integer *lda, double *tau, double *work, integer *lwork,
                       integer *info);
extern "C" int dormqr_(char *side, char *trans, integer *m, integer *n, integer *k, double *a, integer *lda,
                       double *tau, double *c__, integer *ldc, double *work, integer *lwork, integer *info);
extern "C" int dorgqr_(integer *, integer *, integer *, double *, integer *, double *, double *, integer *, integer *);
extern "C" int dtrtri_(char *uplo, char *diag, integer *n, double *a, integer *lda, integer *info);
extern "C" int dgeqp3_(integer *m, integer *n, double *a, integer *lda, integer *p, double *tau, double *work,
                       integer *lwork, integer *info);
 
int allocate_work(double **work);
 
int allocate_work(double **work)
{
  unsigned int dimWork = (unsigned int)((*work)[0]);
  delete[] * work;
  *work = new double[dimWork];
  return (int)dimWork;
}
#endif
#endif
 
#if defined(VISP_HAVE_GSL)
#ifndef DOXYGEN_SHOULD_SKIP_THIS
void display_gsl(gsl_matrix *M)
{
  // display
  for (unsigned int i = 0; i < M->size1; ++i) {
    unsigned int k = i * M->tda;
    for (unsigned int j = 0; j < M->size2; ++j) {
      std::cout << M->data[k + j] << " ";
    }
    std::cout << std::endl;
  }
}
#endif
#endif
 
#if defined(VISP_HAVE_LAPACK)
vpMatrix vpMatrix::inverseByQRLapack() const
{
#if defined(VISP_HAVE_GSL)
  {
    vpMatrix inv;
    inv.resize(colNum, rowNum, false);
    gsl_matrix *gsl_A, *gsl_Q, *gsl_R, gsl_inv;
    gsl_vector *gsl_tau;
 
    gsl_A = gsl_matrix_alloc(rowNum, colNum);
    gsl_Q = gsl_matrix_alloc(rowNum, rowNum); // M by M
    gsl_R = gsl_matrix_alloc(rowNum, colNum); // M by N
    gsl_tau = gsl_vector_alloc(std::min<unsigned int>(rowNum, colNum));
 
    gsl_inv.size1 = inv.rowNum;
    gsl_inv.size2 = inv.colNum;
    gsl_inv.tda = gsl_inv.size2;
    gsl_inv.data = inv.data;
    gsl_inv.owner = 0;
    gsl_inv.block = 0;
 
    // copy input matrix since gsl_linalg_QR_decomp() is destructive
    unsigned int Atda = static_cast<unsigned int>(gsl_A->tda);
    size_t len = sizeof(double) * colNum;
    for (unsigned int i = 0; i < rowNum; ++i) {
      unsigned int k = i * Atda;
      memcpy(&gsl_A->data[k], (*this)[i], len);
    }
    gsl_linalg_QR_decomp(gsl_A, gsl_tau);
    gsl_linalg_QR_unpack(gsl_A, gsl_tau, gsl_Q, gsl_R);
#if (GSL_MAJOR_VERSION >= 2 && GSL_MINOR_VERSION >= 2)
    gsl_linalg_tri_upper_invert(gsl_R);
#else
    {
      gsl_matrix_view m;
      gsl_vector_view v;
      for (unsigned int i = 0; i < rowNum; ++i) {
        double *Tii = gsl_matrix_ptr(gsl_R, i, i);
        *Tii = 1.0 / (*Tii);
        double aii = -(*Tii);
        if (i > 0) {
          m = gsl_matrix_submatrix(gsl_R, 0, 0, i, i);
          v = gsl_matrix_subcolumn(gsl_R, i, 0, i);
          gsl_blas_dtrmv(CblasUpper, CblasNoTrans, CblasNonUnit, &m.matrix, &v.vector);
          gsl_blas_dscal(aii, &v.vector);
        }
      }
    }
#endif
    gsl_matrix_transpose(gsl_Q);
 
    gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, gsl_R, gsl_Q, 0, &gsl_inv);
    unsigned int gsl_inv_tda = static_cast<unsigned int>(gsl_inv.tda);
    size_t inv_len = sizeof(double) * inv.colNum;
    for (unsigned int i = 0; i < inv.rowNum; ++i) {
      unsigned int k = i * gsl_inv_tda;
      memcpy(inv[i], &gsl_inv.data[k], inv_len);
    }
 
    gsl_matrix_free(gsl_A);
    gsl_matrix_free(gsl_Q);
    gsl_matrix_free(gsl_R);
    gsl_vector_free(gsl_tau);
 
    return inv;
  }
#else
  {
    if (rowNum != colNum) {
      throw(vpMatrixException(vpMatrixException::matrixError, "Cannot inverse a non-square matrix (%ux%u) by QR",
                              rowNum, colNum));
    }
 
    integer rowNum_ = (integer)this->getRows();
    integer colNum_ = (integer)this->getCols();
    integer lda = (integer)rowNum_; // lda is the number of rows because we don't use a submatrix
    integer dimTau = std::min<integer>(rowNum_, colNum_);
    integer dimWork = -1;
    double *tau = new double[dimTau];
    double *work = new double[1];
    integer info;
    vpMatrix C;
    vpMatrix A = *this;
 
    try {
      // 1) Extract householder reflections (useful to compute Q) and R
      dgeqrf_(&rowNum_, // The number of rows of the matrix A.  M >= 0.
              &colNum_, // The number of columns of the matrix A.  N >= 0.
              A.data,   /*On entry, the M-by-N matrix A.
                                            On exit, the elements on and above the diagonal of
                                         the array   contain the min(M,N)-by-N upper trapezoidal
                                         matrix R (R is   upper triangular if m >= n); the
                                         elements below the diagonal,   with the array TAU,
                                         represent the orthogonal matrix Q as a   product of
                                         min(m,n) elementary reflectors.
                                          */
              &lda,     // The leading dimension of the array A.  LDA >= max(1,M).
              tau,      /*Dimension (min(M,N))
                                        The scalar factors of the elementary reflectors
                                      */
              work,     // Internal working array. dimension (MAX(1,LWORK))
              &dimWork, // The dimension of the array WORK.  LWORK >= max(1,N).
              &info     // status
      );
 
      if (info != 0) {
        std::cout << "dgeqrf_:Preparation:" << -info << "th element had an illegal value" << std::endl;
        throw vpMatrixException::badValue;
      }
      dimWork = allocate_work(&work);
 
      dgeqrf_(&rowNum_, // The number of rows of the matrix A.  M >= 0.
              &colNum_, // The number of columns of the matrix A.  N >= 0.
              A.data,   /*On entry, the M-by-N matrix A.
                                            On exit, the elements on and above the diagonal of
                                         the array   contain the min(M,N)-by-N upper trapezoidal
                                         matrix R (R is   upper triangular if m >= n); the
                                         elements below the diagonal,   with the array TAU,
                                         represent the orthogonal matrix Q as a   product of
                                         min(m,n) elementary reflectors.
                                          */
              &lda,     // The leading dimension of the array A.  LDA >= max(1,M).
              tau,      /*Dimension (min(M,N))
                                        The scalar factors of the elementary reflectors
                                      */
              work,     // Internal working array. dimension (MAX(1,LWORK))
              &dimWork, // The dimension of the array WORK.  LWORK >= max(1,N).
              &info     // status
      );
 
      if (info != 0) {
        std::cout << "dgeqrf_:" << -info << "th element had an illegal value" << std::endl;
        throw vpMatrixException::badValue;
      }
 
      // A now contains the R matrix in its upper triangular (in lapack
      // convention)
      C = A;
 
      // 2) Invert R
      dtrtri_((char *)"U", (char *)"N", &dimTau, C.data, &lda, &info);
      if (info != 0) {
        if (info < 0)
          std::cout << "dtrtri_:" << -info << "th element had an illegal value" << std::endl;
        else if (info > 0) {
          std::cout << "dtrtri_:R(" << info << "," << info << ")"
            << " is exactly zero.  The triangular matrix is singular "
            "and its inverse can not be computed."
            << std::endl;
          std::cout << "R=" << std::endl << C << std::endl;
        }
        throw vpMatrixException::badValue;
      }
 
      // 3) Zero-fill R^-1
      // the matrix is upper triangular for lapack but lower triangular for visp
      // we fill it with zeros above the diagonal (where we don't need the
      // values)
      for (unsigned int i = 0; i < C.getRows(); ++i)
        for (unsigned int j = 0; j < C.getRows(); ++j)
          if (j > i)
            C[i][j] = 0.;
 
      dimWork = -1;
      integer ldc = lda;
 
      // 4) Transpose Q and left-multiply it by R^-1
      // get R^-1*tQ
      // C contains R^-1
      // A contains Q
      dormqr_((char *)"R", (char *)"T", &rowNum_, &colNum_, &dimTau, A.data, &lda, tau, C.data, &ldc, work, &dimWork,
              &info);
      if (info != 0) {
        std::cout << "dormqr_:Preparation" << -info << "th element had an illegal value" << std::endl;
        throw vpMatrixException::badValue;
      }
      dimWork = allocate_work(&work);
 
      dormqr_((char *)"R", (char *)"T", &rowNum_, &colNum_, &dimTau, A.data, &lda, tau, C.data, &ldc, work, &dimWork,
              &info);
 
      if (info != 0) {
        std::cout << "dormqr_:" << -info << "th element had an illegal value" << std::endl;
        throw vpMatrixException::badValue;
      }
      delete[] tau;
      delete[] work;
    }
    catch (vpMatrixException &) {
      delete[] tau;
      delete[] work;
      throw;
    }
 
    return C;
  }
#endif
}
#endif
 
vpMatrix vpMatrix::inverseByQR() const
{
#if defined(VISP_HAVE_LAPACK)
  return inverseByQRLapack();
#else
  throw(vpException(vpException::fatalError, "Cannot inverse matrix by QR. Install Lapack 3rd party"));
#endif
}
 
unsigned int vpMatrix::qr(vpMatrix &Q, vpMatrix &R, bool full, bool squareR, double tol) const
{
#if defined(VISP_HAVE_LAPACK)
#if defined(VISP_HAVE_GSL)
  unsigned int m = rowNum;         // also rows of Q
  unsigned int n = colNum;         // also columns of R
  unsigned int r = std::min<unsigned int>(n, m); // a priori non-null rows of R = rank of R
  unsigned int q = r;              // columns of Q and rows of R
  unsigned int na = n;             // columns of A
 
  // cannot be full decomposition if m < n
  if (full && m > n) {
    q = m;  // Q is square
    na = m; // A is square
  }
 
  // prepare matrices and deal with r = 0
  Q.resize(m, q, false);
  if (squareR)
    R.resize(r, r, false);
  else
    R.resize(r, n, false);
  if (r == 0)
    return 0;
 
  gsl_matrix *gsl_A, *gsl_Q, *gsl_R, gsl_Qfull;
  gsl_vector *gsl_tau;
 
  gsl_A = gsl_matrix_alloc(rowNum, colNum);
  gsl_R = gsl_matrix_alloc(rowNum, colNum); // M by N
  gsl_tau = gsl_vector_alloc(std::min<unsigned int>(rowNum, colNum));
 
  // copy input matrix since gsl_linalg_QR_decomp() is destructive
  unsigned int Atda = static_cast<unsigned int>(gsl_A->tda);
  size_t len = sizeof(double) * colNum;
  for (unsigned int i = 0; i < rowNum; ++i) {
    unsigned int k = i * Atda;
    memcpy(&gsl_A->data[k], (*this)[i], len);
    //    for (unsigned int j = 0; j < colNum; ++j)
    //      gsl_A->data[k + j] = (*this)[i][j];
  }
 
  gsl_linalg_QR_decomp(gsl_A, gsl_tau);
 
  if (full) {
    // Q is M by M as expected by gsl_linalg_QR_unpack()
    // for performance purpose dont allocate gsl_Q, but rather use gsl_Qfull = Q
    gsl_Qfull.size1 = Q.rowNum;
    gsl_Qfull.size2 = Q.colNum;
    gsl_Qfull.tda = gsl_Qfull.size2;
    gsl_Qfull.data = Q.data;
    gsl_Qfull.owner = 0;
    gsl_Qfull.block = 0;
 
    gsl_linalg_QR_unpack(gsl_A, gsl_tau, &gsl_Qfull, gsl_R);
    //    std::cout << "gsl_Qfull:\n "; display_gsl(&gsl_Qfull);
    //    std::cout << "gsl_R:\n "; display_gsl(gsl_R);
  }
  else {
    gsl_Q = gsl_matrix_alloc(rowNum, rowNum); // M by M
 
    gsl_linalg_QR_unpack(gsl_A, gsl_tau, gsl_Q, gsl_R);
    //    std::cout << "gsl_Q:\n "; display_gsl(gsl_Q);
    //    std::cout << "gsl_R:\n "; display_gsl(gsl_R);
 
    unsigned int Qtda = static_cast<unsigned int>(gsl_Q->tda);
    size_t len = sizeof(double) * Q.colNum;
    for (unsigned int i = 0; i < Q.rowNum; ++i) {
      unsigned int k = i * Qtda;
      memcpy(Q[i], &gsl_Q->data[k], len);
      //      for(unsigned int j = 0; j < Q.colNum; ++j) {
      //        Q[i][j] = gsl_Q->data[k + j];
      //      }
    }
    gsl_matrix_free(gsl_Q);
  }
 
  // copy useful part of R and update rank
  na = std::min<unsigned int>(m, n);
  unsigned int Rtda = static_cast<unsigned int>(gsl_R->tda);
  size_t Rlen = sizeof(double) * R.colNum;
  for (unsigned int i = 0; i < na; ++i) {
    unsigned int k = i * Rtda;
    memcpy(R[i], &gsl_R->data[k], Rlen);
    //      for(unsigned int j = i; j < na; ++j)
    //        R[i][j] = gsl_R->data[k + j];
    if (std::abs(gsl_R->data[k + i]) < tol)
      r--;
  }
 
  gsl_matrix_free(gsl_A);
  gsl_matrix_free(gsl_R);
  gsl_vector_free(gsl_tau);
 
  return r;
#else
  integer m = (integer)rowNum; // also rows of Q
  integer n = (integer)colNum; // also columns of R
  integer r = std::min<integer>(n, m);  // a priori non-null rows of R = rank of R
  integer q = r;               // columns of Q and rows of R
  integer na = n;              // columns of A
 
  // cannot be full decomposition if m < n
  if (full && m > n) {
    q = m;  // Q is square
    na = m; // A is square
  }
 
  // prepare matrices and deal with r = 0
  Q.resize(static_cast<unsigned int>(m), static_cast<unsigned int>(q));
  if (squareR)
    R.resize(static_cast<unsigned int>(r), static_cast<unsigned int>(r));
  else
    R.resize(static_cast<unsigned int>(r), static_cast<unsigned int>(n));
  if (r == 0)
    return 0;
 
  integer dimWork = -1;
  double *qrdata = new double[m * na];
  double *tau = new double[std::min<integer>(m, q)];
  double *work = new double[1];
  integer info;
 
  // copy this to qrdata in Lapack convention
  for (integer i = 0; i < m; ++i) {
    for (integer j = 0; j < n; ++j)
      qrdata[i + m * j] = data[j + n * i];
    for (integer j = n; j < na; ++j)
      qrdata[i + m * j] = 0;
  }
 
  //   work = new double[1];
  // 1) Extract householder reflections (useful to compute Q) and R
  dgeqrf_(&m,  // The number of rows of the matrix A.  M >= 0.
          &na, // The number of columns of the matrix A.  N >= 0.
          qrdata, &m, tau,
          work,     // Internal working array. dimension (MAX(1,LWORK))
          &dimWork, // The dimension of the array WORK.  LWORK >= max(1,N).
          &info     // status
  );
 
  if (info != 0) {
    std::cout << "dgeqrf_:Preparation:" << -info << "th element had an illegal value" << std::endl;
    delete[] qrdata;
    delete[] work;
    delete[] tau;
    throw vpMatrixException::badValue;
  }
  dimWork = allocate_work(&work);
 
  dgeqrf_(&m,  // The number of rows of the matrix A.  M >= 0.
          &na, // The number of columns of the matrix A.  N >= 0.
          qrdata,
          &m, // The leading dimension of the array A.  LDA >= max(1,M).
          tau,
          work,     // Internal working array. dimension (MAX(1,LWORK))
          &dimWork, // The dimension of the array WORK.  LWORK >= max(1,N).
          &info     // status
  );
 
  if (info != 0) {
    std::cout << "dgeqrf_:" << -info << "th element had an illegal value" << std::endl;
    delete[] qrdata;
    delete[] work;
    delete[] tau;
    throw vpMatrixException::badValue;
  }
 
  // data now contains the R matrix in its upper triangular (in lapack convention)
 
  // copy useful part of R from Q and update rank
  na = std::min<integer>(m, n);
  if (squareR) {
    for (int i = 0; i < na; ++i) {
      for (int j = i; j < na; ++j)
        R[i][j] = qrdata[i + m * j];
      if (std::abs(qrdata[i + m * i]) < tol)
        r--;
    }
  }
  else {
    for (int i = 0; i < na; ++i) {
      for (int j = i; j < n; ++j)
        R[i][j] = qrdata[i + m * j];
      if (std::abs(qrdata[i + m * i]) < tol)
        r--;
    }
  }
 
  // extract Q
  dorgqr_(&m, // The number of rows of the matrix Q. M >= 0.
          &q, // The number of columns of the matrix Q. M >= N >= 0.
          &q, qrdata,
          &m, // The leading dimension of the array A.  LDA >= max(1,M).
          tau,
          work,     // Internal working array. dimension (MAX(1,LWORK))
          &dimWork, // The dimension of the array WORK.  LWORK >= max(1,N).
          &info     // status
  );
 
  // write qrdata into Q
  for (int i = 0; i < m; ++i)
    for (int j = 0; j < q; ++j)
      Q[i][j] = qrdata[i + m * j];
 
  delete[] qrdata;
  delete[] work;
  delete[] tau;
  return (unsigned int)r;
#endif // defined(VISP_HAVE_GSL)
#else
  (void)Q;
  (void)R;
  (void)full;
  (void)squareR;
  (void)tol;
  throw(vpException(vpException::fatalError, "Cannot perform QR decomposition. Install Lapack 3rd party"));
#endif
}
 
unsigned int vpMatrix::qrPivot(vpMatrix &Q, vpMatrix &R, vpMatrix &P, bool full, bool squareR, double tol) const
{
#if defined(VISP_HAVE_LAPACK)
#if defined(VISP_HAVE_GSL)
  unsigned int m = rowNum;         // also rows of Q
  unsigned int n = colNum;         // also columns of R
  unsigned int r = std::min<unsigned int>(n, m); // a priori non-null rows of R = rank of R
  unsigned int q = r;              // columns of Q and rows of R
  unsigned int na = n;             // columns of A
 
  // cannot be full decomposition if m < n
  if (full && m > n) {
    q = m;  // Q is square
    na = m; // A is square
  }
 
  // prepare Q and deal with r = 0
  Q.resize(m, q, false);
  if (r == 0) {
    if (squareR) {
      R.resize(0, 0, false);
      P.resize(0, n, false);
    }
    else {
      R.resize(r, n, false);
      P.resize(n, n);
    }
    return 0;
  }
 
  gsl_matrix gsl_A, *gsl_Q, *gsl_R, gsl_Qfull;
  gsl_vector *gsl_tau;
  gsl_permutation *gsl_p;
  gsl_vector *gsl_norm;
  int gsl_signum;
 
  gsl_A.size1 = rowNum;
  gsl_A.size2 = colNum;
  gsl_A.tda = gsl_A.size2;
  gsl_A.data = this->data;
  gsl_A.owner = 0;
  gsl_A.block = 0;
 
  gsl_R = gsl_matrix_alloc(rowNum, colNum); // M by N
  gsl_tau = gsl_vector_alloc(std::min<unsigned int>(rowNum, colNum));
  gsl_norm = gsl_vector_alloc(colNum);
  gsl_p = gsl_permutation_alloc(n);
 
  if (full) {
    // Q is M by M as expected by gsl_linalg_QR_unpack()
    // for performance purpose dont allocate gsl_Q, but rather use gsl_Qfull = Q
    gsl_Qfull.size1 = Q.rowNum;
    gsl_Qfull.size2 = Q.colNum;
    gsl_Qfull.tda = gsl_Qfull.size2;
    gsl_Qfull.data = Q.data;
    gsl_Qfull.owner = 0;
    gsl_Qfull.block = 0;
 
    gsl_linalg_QRPT_decomp2(&gsl_A, &gsl_Qfull, gsl_R, gsl_tau, gsl_p, &gsl_signum, gsl_norm);
    //    std::cout << "gsl_Qfull:\n "; display_gsl(&gsl_Qfull);
    //    std::cout << "gsl_R:\n "; display_gsl(gsl_R);
  }
  else {
    gsl_Q = gsl_matrix_alloc(rowNum, rowNum); // M by M
 
    gsl_linalg_QRPT_decomp2(&gsl_A, gsl_Q, gsl_R, gsl_tau, gsl_p, &gsl_signum, gsl_norm);
    //    std::cout << "gsl_Q:\n "; display_gsl(gsl_Q);
    //    std::cout << "gsl_R:\n "; display_gsl(gsl_R);
 
    unsigned int Qtda = static_cast<unsigned int>(gsl_Q->tda);
    size_t len = sizeof(double) * Q.colNum;
    for (unsigned int i = 0; i < Q.rowNum; ++i) {
      unsigned int k = i * Qtda;
      memcpy(Q[i], &gsl_Q->data[k], len);
      //      for(unsigned int j = 0; j < Q.colNum; ++j) {
      //        Q[i][j] = gsl_Q->data[k + j];
      //      }
    }
    gsl_matrix_free(gsl_Q);
  }
 
  // update rank
  na = std::min<unsigned int>(m, n);
  unsigned int Rtda = static_cast<unsigned int>(gsl_R->tda);
  for (unsigned int i = 0; i < na; ++i) {
    unsigned int k = i * Rtda;
    if (std::abs(gsl_R->data[k + i]) < tol)
      r--;
  }
 
  if (squareR) {
    R.resize(r, r, false); // R r x r
    P.resize(r, n);
    for (unsigned int i = 0; i < r; ++i)
      P[i][gsl_p->data[i]] = 1;
  }
  else {
    R.resize(na, n, false); // R is min(m,n) x n of rank r
    P.resize(n, n);
    for (unsigned int i = 0; i < n; ++i)
      P[i][gsl_p->data[i]] = 1;
  }
 
  // copy useful part of R
  size_t Rlen = sizeof(double) * R.colNum;
  for (unsigned int i = 0; i < na; ++i) {
    unsigned int k = i * Rtda;
    memcpy(R[i], &gsl_R->data[k], Rlen);
  }
 
  gsl_matrix_free(gsl_R);
  gsl_vector_free(gsl_tau);
  gsl_vector_free(gsl_norm);
  gsl_permutation_free(gsl_p);
 
  return r;
#else
  integer m = (integer)rowNum; // also rows of Q
  integer n = (integer)colNum; // also columns of R
  integer r = std::min<integer>(n, m);  // a priori non-null rows of R = rank of R
  integer q = r;               // columns of Q and rows of R
  integer na = n;              // columns of A
 
  // cannot be full decomposition if m < n
  // cannot be full decomposition if m < n
  if (full && m > n) {
    q = m;  // Q is square
    na = m; // A is square
  }
 
  // prepare Q and deal with r = 0
  Q.resize(static_cast<unsigned int>(m), static_cast<unsigned int>(q));
  if (r == 0) {
    if (squareR) {
      R.resize(0, 0);
      P.resize(0, static_cast<unsigned int>(n));
    }
    else {
      R.resize(static_cast<unsigned int>(r), static_cast<unsigned int>(n));
      P.resize(static_cast<unsigned int>(n), static_cast<unsigned int>(n));
    }
    return 0;
  }
 
  integer dimWork = -1;
  integer min_q_m = std::min<integer>(q, m);
  double *qrdata = new double[m * na];
  double *tau = new double[min_q_m];
  double *work = new double[1];
  integer *p = new integer[na];
  for (int i = 0; i < na; ++i)
    p[i] = 0;
 
  integer info;
 
  // copy this to qrdata in Lapack convention
  for (integer i = 0; i < m; ++i) {
    for (integer j = 0; j < n; ++j)
      qrdata[i + m * j] = data[j + n * i];
    for (integer j = n; j < na; ++j)
      qrdata[i + m * j] = 0;
  }
 
  // 1) Extract householder reflections (useful to compute Q) and R
  dgeqp3_(&m,     // The number of rows of the matrix A.  M >= 0.
          &na,    // The number of columns of the matrix A.  N >= 0.
          qrdata, /*On entry, the M-by-N matrix A.        */
          &m,     // The leading dimension of the array A.  LDA >= max(1,M).
          p,      // Dimension N
          tau,    /*Dimension (min(M,N))        */
          work,   // Internal working array. dimension (3*N)
 
          &dimWork,
          &info // status
  );
 
  if (info != 0) {
    std::cout << "dgeqp3_:Preparation:" << -info << "th element had an illegal value" << std::endl;
    delete[] qrdata;
    delete[] work;
    delete[] tau;
    delete[] p;
    throw vpMatrixException::badValue;
  }
 
  dimWork = allocate_work(&work);
 
  dgeqp3_(&m,     // The number of rows of the matrix A.  M >= 0.
          &na,    // The number of columns of the matrix A.  N >= 0.
          qrdata, /*On entry, the M-by-N matrix A.        */
          &m,     // The leading dimension of the array A.  LDA >= max(1,M).
          p,      // Dimension N
          tau,    /*Dimension (min(M,N))        */
          work,   // Internal working array. dimension (3*N)
 
          &dimWork,
          &info // status
  );
 
  if (info != 0) {
    std::cout << "dgeqp3_:" << -info << " th element had an illegal value" << std::endl;
    delete[] qrdata;
    delete[] work;
    delete[] tau;
    delete[] p;
    throw vpMatrixException::badValue;
  }
 
  // data now contains the R matrix in its upper triangular (in lapack convention)
  // get rank of R in r
  na = std::min<integer>(n, m);
  for (int i = 0; i < na; ++i)
    if (std::abs(qrdata[i + m * i]) < tol)
      r--;
 
  // write R
  if (squareR) // R r x r
  {
    R.resize(static_cast<unsigned int>(r), static_cast<unsigned int>(r));
    for (int i = 0; i < r; ++i)
      for (int j = i; j < r; ++j)
        R[i][j] = qrdata[i + m * j];
 
    // write P
    P.resize(static_cast<unsigned int>(r), static_cast<unsigned int>(n));
    for (int i = 0; i < r; ++i)
      P[i][p[i] - 1] = 1;
  }
  else // R is min(m,n) x n of rank r
  {
    R.resize(static_cast<unsigned int>(na), static_cast<unsigned int>(n));
    for (int i = 0; i < na; ++i)
      for (int j = i; j < n; ++j)
        R[i][j] = qrdata[i + m * j];
    // write P
    P.resize(static_cast<unsigned int>(n), static_cast<unsigned int>(n));
    for (int i = 0; i < n; ++i)
      P[i][p[i] - 1] = 1;
  }
 
  // extract Q
  dorgqr_(&m, // The number of rows of the matrix Q. M >= 0.
          &q, // The number of columns of the matrix Q. M >= N >= 0.
          &q, qrdata,
          &m, // The leading dimension of the array A.  LDA >= max(1,M).
          tau,
          work,     // Internal working array. dimension (MAX(1,LWORK))
          &dimWork, // The dimension of the array WORK.  LWORK >= max(1,N).
          &info     // status
  );
 
  // write qrdata into Q
  for (int i = 0; i < m; ++i)
    for (int j = 0; j < q; ++j)
      Q[i][j] = qrdata[i + m * j];
 
  delete[] qrdata;
  delete[] work;
  delete[] tau;
  delete[] p;
  return (unsigned int)r;
#endif
#else
  (void)Q;
  (void)R;
  (void)P;
  (void)full;
  (void)squareR;
  (void)tol;
  throw(vpException(vpException::fatalError, "Cannot perform QR decomposition. Install Lapack 3rd party"));
#endif
}
 
vpMatrix vpMatrix::inverseTriangular(bool upper) const
{
  if ((rowNum != colNum) || (rowNum == 0)) {
    throw(vpException(vpException::dimensionError, "Cannot inverse a triangular matrix (%d, %d) that is not square",
                      rowNum, colNum));
  }
#if defined(VISP_HAVE_LAPACK)
#if defined(VISP_HAVE_GSL)
  vpMatrix inv;
  inv.resize(colNum, rowNum, false);
  gsl_matrix gsl_inv;
 
  gsl_inv.size1 = inv.rowNum;
  gsl_inv.size2 = inv.colNum;
  gsl_inv.tda = gsl_inv.size2;
  gsl_inv.data = inv.data;
  gsl_inv.owner = 0;
  gsl_inv.block = 0;
 
  unsigned int tda = static_cast<unsigned int>(gsl_inv.tda);
  size_t len = sizeof(double) * inv.colNum;
  for (unsigned int i = 0; i < rowNum; ++i) {
    unsigned int k = i * tda;
    memcpy(&gsl_inv.data[k], (*this)[i], len);
  }
 
  if (upper) {
#if (GSL_MAJOR_VERSION >= 2 && GSL_MINOR_VERSION >= 2)
    gsl_linalg_tri_upper_invert(&gsl_inv);
#else
    {
      gsl_matrix_view m;
      gsl_vector_view v;
      for (unsigned int i = 0; i < rowNum; ++i) {
        double *Tii = gsl_matrix_ptr(&gsl_inv, i, i);
        *Tii = 1.0 / *Tii;
        double aii = -(*Tii);
        if (i > 0) {
          m = gsl_matrix_submatrix(&gsl_inv, 0, 0, i, i);
          v = gsl_matrix_subcolumn(&gsl_inv, i, 0, i);
          gsl_blas_dtrmv(CblasUpper, CblasNoTrans, CblasNonUnit, &m.matrix, &v.vector);
          gsl_blas_dscal(aii, &v.vector);
        }
      }
    }
#endif
  }
  else {
#if (GSL_MAJOR_VERSION >= 2 && GSL_MINOR_VERSION >= 2)
    gsl_linalg_tri_lower_invert(&gsl_inv);
#else
    {
      gsl_matrix_view m;
      gsl_vector_view v;
      for (unsigned int i = 0; i < rowNum; ++i) {
        size_t j = rowNum - i - 1;
        double *Tjj = gsl_matrix_ptr(&gsl_inv, j, j);
        *Tjj = 1.0 / (*Tjj);
        double ajj = -(*Tjj);
        if (j < rowNum - 1) {
          m = gsl_matrix_submatrix(&gsl_inv, j + 1, j + 1, rowNum - j - 1, rowNum - j - 1);
          v = gsl_matrix_subcolumn(&gsl_inv, j, j + 1, rowNum - j - 1);
          gsl_blas_dtrmv(CblasLower, CblasNoTrans, CblasNonUnit, &m.matrix, &v.vector);
          gsl_blas_dscal(ajj, &v.vector);
        }
      }
    }
#endif
  }
 
  return inv;
#else
  integer n = (integer)rowNum; // lda is the number of rows because we don't use a submatrix
 
  vpMatrix R = *this;
  integer info;
 
  // we just use the other (upper / lower) method from Lapack
  if (rowNum > 1 && upper) // upper triangular
    dtrtri_((char *)"L", (char *)"N", &n, R.data, &n, &info);
  else
    dtrtri_((char *)"U", (char *)"N", &n, R.data, &n, &info);
 
  if (info != 0) {
    if (info < 0)
      std::cout << "dtrtri_:" << -info << "th element had an illegal value" << std::endl;
    else if (info > 0) {
      std::cout << "dtrtri_:R(" << info << "," << info << ")"
        << " is exactly zero.  The triangular matrix is singular "
        "and its inverse can not be computed."
        << std::endl;
      throw vpMatrixException::rankDeficient;
    }
    throw vpMatrixException::badValue;
  }
  return R;
#endif
#else
  (void)upper;
  throw(vpException(vpException::fatalError, "Cannot perform triangular inverse. Install Lapack 3rd party"));
#endif
}
 
void vpMatrix::solveByQR(const vpColVector &b, vpColVector &x) const
{
  vpMatrix Q, R, P;
  unsigned int r = t().qrPivot(Q, R, P, false, true);
  x = Q.extract(0, 0, colNum, r) * R.inverseTriangular().t() * P * b;
}
 
vpColVector vpMatrix::solveByQR(const vpColVector &b) const
{
  vpColVector x(colNum);
  solveByQR(b, x);
  return x;
}