gmm_dense_matrix_functions.h

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/**@file gmm_dense_matrix_functions.h
   @author  Konstantinos Poulios <poulios.konstantinos@gmail.com>
   @date December 10, 2014.
   @brief Common matrix functions for dense matrices.
*/
#ifndef GMM_DENSE_MATRIX_FUNCTIONS_H
#define GMM_DENSE_MATRIX_FUNCTIONS_H


namespace gmm {


  /**
     Matrix square root for upper triangular matrices (from GNU Octave).
  */
  template <typename T>
  void sqrtm_utri_inplace(dense_matrix<T>& A)
  {
    typedef typename number_traits<T>::magnitude_type R;
    bool singular = false;

    // The following code is equivalent to this triple loop:
    //
    //   n = rows (A);
    //   for j = 1:n
    //     A(j,j) = sqrt (A(j,j));
    //     for i = j-1:-1:1
    //       A(i,j) /= (A(i,i) + A(j,j));
    //       k = 1:i-1;
    //   t storing a    A(k,j) -= A(k,i) * A(i,j);
    //     endfor
    //   endfor

    R tol = R(0); // default_tol(R()) * gmm::mat_maxnorm(A);

    const size_type n = mat_nrows(A);
    for (int j=0; j < int(n); j++) {
      typename dense_matrix<T>::iterator colj = A.begin() + j*n;
      if (gmm::abs(colj[j]) > tol)
        colj[j] = gmm::sqrt(colj[j]);
      else
        singular = true;

      for (int i=j-1; i >= 0; i--) {
        typename dense_matrix<T>::const_iterator coli = A.begin() + i*n;
        T colji = colj[i] = safe_divide(colj[i], (coli[i] + colj[j]));
        for (int k = 0; k < i; k++)
          colj[k] -= coli[k] * colji;
      }
    }

    if (singular)
      GMM_WARNING1("Matrix is singular, may not have a square root");
  }


  template <typename T>
  void sqrtm(const dense_matrix<std::complex<T> >& A,
             dense_matrix<std::complex<T> >& SQRTMA)
  {
    GMM_ASSERT1(gmm::mat_nrows(A) == gmm::mat_ncols(A),
                "Matrix square root requires a square matrix");
    gmm::resize(SQRTMA, gmm::mat_nrows(A), gmm::mat_ncols(A));
    dense_matrix<std::complex<T> > S(A), Q(A), TMP(A);
    #if defined(GMM_USES_LAPACK)
    schur(TMP, S, Q);
    #else
    GMM_ASSERT1(false, "Please recompile with lapack and blas librairies "
                "to use sqrtm matrix function.");
    #endif
    sqrtm_utri_inplace(S);
    gmm::mult(Q, S, TMP);
    gmm::mult(TMP, gmm::transposed(Q), SQRTMA);
  }

  template <typename T>
  void sqrtm(const dense_matrix<T>& A,
             dense_matrix<std::complex<T> >& SQRTMA)
  {
    dense_matrix<std::complex<T> > cA(mat_nrows(A), mat_ncols(A));
    gmm::copy(A, gmm::real_part(cA));
    sqrtm(cA, SQRTMA);
  }

  template <typename T>
  void sqrtm(const dense_matrix<T>& A, dense_matrix<T>& SQRTMA)
  {
    dense_matrix<std::complex<T> > cA(mat_nrows(A), mat_ncols(A));
    gmm::copy(A, gmm::real_part(cA));
    dense_matrix<std::complex<T> > cSQRTMA(cA);
    sqrtm(cA, cSQRTMA);
    gmm::resize(SQRTMA, gmm::mat_nrows(A), gmm::mat_ncols(A));
    gmm::copy(gmm::real_part(cSQRTMA), SQRTMA);
//    dense_matrix<std::complex<T1> >::const_reference
//      it = cSQRTMA.begin(), ite = cSQRTMA.end();
//    dense_matrix<std::complex<T1> >::reference
//      rit = SQRTMA.begin();
//    for (; it != ite; ++it, ++rit) *rit = it->real();
  }


  /**
   Matrix logarithm for upper triangular matrices (from GNU/Octave)
  */
  template <typename T>
  void logm_utri_inplace(dense_matrix<T>& S)
  {
    typedef typename number_traits<T>::magnitude_type R;

    size_type n = gmm::mat_nrows(S);
    GMM_ASSERT1(n == gmm::mat_ncols(S),
                "Matrix logarithm is not defined for non-square matrices");
    for (size_type i=0; i < n-1; ++i)
      if (gmm::abs(S(i+1,i)) > default_tol(T())) {
        GMM_ASSERT1(false, "An upper triangular matrix is expected");
        break;
      }
    for (size_type i=0; i < n-1; ++i)
      if (gmm::real(S(i,i)) <= -default_tol(R()) &&
          gmm::abs(gmm::imag(S(i,i))) <= default_tol(R())) {
        GMM_ASSERT1(false, "Principal matrix logarithm is not defined "
                           "for matrices with negative eigenvalues");
        break;
      }

    // Algorithm 11.9 in "Function of matrices", by N. Higham
    R theta[] = { R(0),R(0),R(1.61e-2),R(5.38e-2),R(1.13e-1),R(1.86e-1),R(2.6429608311114350e-1) };

    R scaling(1);
    size_type p(0), m(6), opt_iters(100);
    for (size_type k=0; k < opt_iters; ++k, scaling *= R(2)) {
      dense_matrix<T> auxS(S);
      for (size_type i = 0; i < n; ++i) auxS(i,i) -= R(1);
      R tau = gmm::mat_norm1(auxS);
      if (tau <= theta[6]) {
        ++p;
        size_type j1(6), j2(6);
        for (size_type j=0; j < 6; ++j)
          if (tau <= theta[j]) { j1 = j; break; }
        for (size_type j=0; j < j1; ++j)
          if (tau <= 2*theta[j]) { j2 = j; break; }
        if (j1 - j2 <= 1 || p == 2) { m = j1; break; }
      }
      sqrtm_utri_inplace(S);
      if (k == opt_iters-1)
        GMM_WARNING1 ("Maximum number of square roots exceeded; "
                      "the calculated matrix logarithm may still be accurate");
    }

    for (size_type i = 0; i < n; ++i) S(i,i) -= R(1);

    if (m > 0) {

      std::vector<R> nodes, wts;
      switch(m) {
      case 0: {
        R nodes_[] = { R(0.5) };
        R wts_[] = { R(1) };
        nodes.assign(nodes_, nodes_+m+1);
        wts.assign(wts_, wts_+m+1);
        } break;
      case 1: {
        R nodes_[] = { R(0.211324865405187),R(0.788675134594813) };
        R wts_[] = { R(0.5),R(0.5) };
        nodes.assign(nodes_, nodes_+m+1);
        wts.assign(wts_, wts_+m+1);
        } break;
      case 2: {
        R nodes_[] = { R(0.112701665379258),R(0.500000000000000),R(0.887298334620742) };
        R wts_[] = { R(0.277777777777778),R(0.444444444444444),R(0.277777777777778) };
        nodes.assign(nodes_, nodes_+m+1);
        wts.assign(wts_, wts_+m+1);
        } break;
      case 3: {
        R nodes_[] = { R(0.0694318442029737),R(0.3300094782075718),R(0.6699905217924281),R(0.9305681557970263) };
        R wts_[] = { R(0.173927422568727),R(0.326072577431273),R(0.326072577431273),R(0.173927422568727) };
        nodes.assign(nodes_, nodes_+m+1);
        wts.assign(wts_, wts_+m+1);
        } break;
      case 4: {
        R nodes_[] = { R(0.0469100770306681),R(0.2307653449471584),R(0.5000000000000000),R(0.7692346550528415),R(0.9530899229693319) };
        R wts_[] = { R(0.118463442528095),R(0.239314335249683),R(0.284444444444444),R(0.239314335249683),R(0.118463442528094) };
        nodes.assign(nodes_, nodes_+m+1);
        wts.assign(wts_, wts_+m+1);
        } break;
      case 5: {
        R nodes_[] = { R(0.0337652428984240),R(0.1693953067668678),R(0.3806904069584015),R(0.6193095930415985),R(0.8306046932331322),R(0.9662347571015761) };
        R wts_[] = { R(0.0856622461895853),R(0.1803807865240693),R(0.2339569672863452),R(0.2339569672863459),R(0.1803807865240693),R(0.0856622461895852) };
        nodes.assign(nodes_, nodes_+m+1);
        wts.assign(wts_, wts_+m+1);
        } break;
      case 6: {
        R nodes_[] = { R(0.0254460438286208),R(0.1292344072003028),R(0.2970774243113015),R(0.4999999999999999),R(0.7029225756886985),R(0.8707655927996973),R(0.9745539561713792) };
        R wts_[] = { R(0.0647424830844348),R(0.1398526957446384),R(0.1909150252525594),R(0.2089795918367343),R(0.1909150252525595),R(0.1398526957446383),R(0.0647424830844349) };
        nodes.assign(nodes_, nodes_+m+1);
        wts.assign(wts_, wts_+m+1);
        } break;
      }

      dense_matrix<T> auxS1(S), auxS2(S);
      std::vector<T> auxvec(n);
      gmm::clear(S);
      for (size_type j=0; j <= m; ++j) {
        gmm::copy(gmm::scaled(auxS1, nodes[j]), auxS2);
        gmm::add(gmm::identity_matrix(), auxS2);
        // S += wts[i] * auxS1 * inv(auxS2)
        for (size_type i=0; i < n; ++i) {
          gmm::copy(gmm::mat_row(auxS1, i), auxvec);
          gmm::lower_tri_solve(gmm::transposed(auxS2), auxvec, false);
          gmm::add(gmm::scaled(auxvec, wts[j]), gmm::mat_row(S, i));
        }
      }
    }
    gmm::scale(S, scaling);
  }

  /**
   Matrix logarithm (from GNU/Octave)
  */
  template <typename T>
  void logm(const dense_matrix<T>& A, dense_matrix<T>& LOGMA)
  {
    typedef typename number_traits<T>::magnitude_type R;
    size_type n = gmm::mat_nrows(A);
    GMM_ASSERT1(n == gmm::mat_ncols(A),
                "Matrix logarithm is not defined for non-square matrices");
    dense_matrix<T> S(A), Q(A);
    #if defined(GMM_USES_LAPACK)
    schur(A, S, Q); // A = Q * S * Q^T
    #else
    GMM_ASSERT1(false, "Please recompile with lapack and blas librairies "
                "to use logm matrix function.");
    #endif

    bool convert_to_complex(false);
    if (!is_complex(T()))
      for (size_type i=0; i < n-1; ++i)
        if (gmm::abs(S(i+1,i)) > default_tol(T())) {
          convert_to_complex = true;
          break;
        }

    gmm::resize(LOGMA, n, n);
    if (convert_to_complex) {
      dense_matrix<std::complex<R> > cS(n,n), cQ(n,n), auxmat(n,n);
      gmm::copy(gmm::real_part(S), gmm::real_part(cS));
      gmm::copy(gmm::real_part(Q), gmm::real_part(cQ));
      block2x2_reduction(cS, cQ, default_tol(R())*R(3));
      for (size_type j=0; j < n-1; ++j)
        for (size_type i=j+1; i < n; ++i)
          cS(i,j) = T(0);
      logm_utri_inplace(cS);
      gmm::mult(cQ, cS, auxmat);
      gmm::mult(auxmat, gmm::transposed(cQ), cS);
      // Remove small complex values which may have entered calculation
      gmm::copy(gmm::real_part(cS), LOGMA);
//      GMM_ASSERT1(gmm::mat_norm1(gmm::imag_part(cS)) < n*default_tol(T()),
//                  "Internal error, imag part should be zero");
    } else {
      dense_matrix<T> auxmat(n,n);
      logm_utri_inplace(S);
      gmm::mult(Q, S, auxmat);
      gmm::mult(auxmat, gmm::transposed(Q), LOGMA);
    }

  }

}

#endif