libgmm++-dev 4.2.1~beta1~svn4482~dfsg-2build1 / usr / include / gmm / gmm_dense_qr.h

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/**@file gmm_dense_qr.h
   @author  Caroline Lecalvez, Caroline.Lecalvez@gmm.insa-tlse.fr, Yves Renard <Yves.Renard@insa-lyon.fr>
   @date September 12, 2003.
   @brief Dense QR factorization.
*/
#ifndef GMM_DENSE_QR_H
#define GMM_DENSE_QR_H

#include "gmm_dense_Householder.h"

namespace gmm {


  /**
     QR factorization using Householder method (complex and real version).
  */
  template <typename MAT1>
  void qr_factor(const MAT1 &A_) { 
    MAT1 &A = const_cast<MAT1 &>(A_);
    typedef typename linalg_traits<MAT1>::value_type value_type;

    size_type m = mat_nrows(A), n = mat_ncols(A);
    GMM_ASSERT2(m >= n, "dimensions mismatch");

    std::vector<value_type> W(m), V(m);

    for (size_type j = 0; j < n; ++j) {
      sub_interval SUBI(j, m-j), SUBJ(j, n-j);
      V.resize(m-j); W.resize(n-j);

      for (size_type i = j; i < m; ++i) V[i-j] = A(i, j);
      house_vector(V);

      row_house_update(sub_matrix(A, SUBI, SUBJ), V, W);
      for (size_type i = j+1; i < m; ++i) A(i, j) = V[i-j];
    }
  }

  
  // QR comes from QR_factor(QR) where the upper triangular part stands for R
  // and the lower part contains the Householder reflectors.
  // A <- AQ
  template <typename MAT1, typename MAT2>
  void apply_house_right(const MAT1 &QR, const MAT2 &A_) { 
    MAT2 &A = const_cast<MAT2 &>(A_);
    typedef typename linalg_traits<MAT1>::value_type T;
    size_type m = mat_nrows(QR), n = mat_ncols(QR);
    GMM_ASSERT2(m == mat_ncols(A), "dimensions mismatch");
    if (m == 0) return;
    std::vector<T> V(m), W(mat_nrows(A));
    V[0] = T(1);
    for (size_type j = 0; j < n; ++j) {
      V.resize(m-j);
      for (size_type i = j+1; i < m; ++i) V[i-j] = QR(i, j);
      col_house_update(sub_matrix(A, sub_interval(0, mat_nrows(A)),
				  sub_interval(j, m-j)), V, W);
    }
  }

  // QR comes from QR_factor(QR) where the upper triangular part stands for R
  // and the lower part contains the Householder reflectors.
  // A <- Q*A
  template <typename MAT1, typename MAT2>
  void apply_house_left(const MAT1 &QR, const MAT2 &A_) { 
    MAT2 &A = const_cast<MAT2 &>(A_);
    typedef typename linalg_traits<MAT1>::value_type T;
    size_type m = mat_nrows(QR), n = mat_ncols(QR);
    GMM_ASSERT2(m == mat_nrows(A), "dimensions mismatch");
    if (m == 0) return;
    std::vector<T> V(m), W(mat_ncols(A));
    V[0] = T(1);
    for (size_type j = 0; j < n; ++j) {
      V.resize(m-j);
      for (size_type i = j+1; i < m; ++i) V[i-j] = QR(i, j);
      row_house_update(sub_matrix(A, sub_interval(j, m-j),
				  sub_interval(0, mat_ncols(A))), V, W);
    }
  }  

  /** Compute the QR factorization, where Q is assembled. */
  template <typename MAT1, typename MAT2, typename MAT3>
    void qr_factor(const MAT1 &A, const MAT2 &QQ, const MAT3 &RR) { 
    MAT2 &Q = const_cast<MAT2 &>(QQ); MAT3 &R = const_cast<MAT3 &>(RR); 
    typedef typename linalg_traits<MAT1>::value_type value_type;

    size_type m = mat_nrows(A), n = mat_ncols(A);
    GMM_ASSERT2(m >= n, "dimensions mismatch");
    gmm::copy(A, Q);
    
    std::vector<value_type> W(m);
    dense_matrix<value_type> VV(m, n);

    for (size_type j = 0; j < n; ++j) {
      sub_interval SUBI(j, m-j), SUBJ(j, n-j);

      for (size_type i = j; i < m; ++i) VV(i,j) = Q(i, j);
      house_vector(sub_vector(mat_col(VV,j), SUBI));

      row_house_update(sub_matrix(Q, SUBI, SUBJ),
		       sub_vector(mat_col(VV,j), SUBI), sub_vector(W, SUBJ));
    }

    gmm::copy(sub_matrix(Q, sub_interval(0, n), sub_interval(0, n)), R);
    gmm::copy(identity_matrix(), Q);
    
    for (size_type j = n-1; j != size_type(-1); --j) {
      sub_interval SUBI(j, m-j), SUBJ(j, n-j);
      row_house_update(sub_matrix(Q, SUBI, SUBJ), 
		       sub_vector(mat_col(VV,j), SUBI), sub_vector(W, SUBJ));
    }
  }

  ///@cond DOXY_SHOW_ALL_FUNCTIONS
  template <typename TA, typename TV, typename Ttol, 
	    typename MAT, typename VECT>
  void extract_eig(const MAT &A, VECT &V, Ttol tol, TA, TV) {
    size_type n = mat_nrows(A);
    if (n == 0) return;
    tol *= Ttol(2);
    Ttol tol_i = tol * gmm::abs(A(0,0)), tol_cplx = tol_i;
    for (size_type i = 0; i < n; ++i) {
      if (i < n-1) {
	tol_i = (gmm::abs(A(i,i))+gmm::abs(A(i+1,i+1)))*tol;
	tol_cplx = std::max(tol_cplx, tol_i);
      }
      if ((i < n-1) && gmm::abs(A(i+1,i)) >= tol_i) {
	TA tr = A(i,i) + A(i+1, i+1);
	TA det = A(i,i)*A(i+1, i+1) - A(i,i+1)*A(i+1, i);
	TA delta = tr*tr - TA(4) * det;
	if (delta < -tol_cplx) {
	  GMM_WARNING1("A complex eigenvalue has been detected : "
		      << std::complex<TA>(tr/TA(2), gmm::sqrt(-delta)/TA(2)));
	  V[i] = V[i+1] = tr / TA(2);
	}
	else {
	  delta = std::max(TA(0), delta);
	  V[i  ] = TA(tr + gmm::sqrt(delta))/ TA(2);
	  V[i+1] = TA(tr -  gmm::sqrt(delta))/ TA(2);
	}
	++i;
      }
      else
	V[i] = TV(A(i,i));
    }
  }

  template <typename TA, typename TV, typename Ttol, 
	    typename MAT, typename VECT>
  void extract_eig(const MAT &A, VECT &V, Ttol tol, TA, std::complex<TV>) {
    size_type n = mat_nrows(A);
    tol *= Ttol(2);
    for (size_type i = 0; i < n; ++i)
      if ((i == n-1) ||
	  gmm::abs(A(i+1,i)) < (gmm::abs(A(i,i))+gmm::abs(A(i+1,i+1)))*tol)
	V[i] = std::complex<TV>(A(i,i));
      else {
	TA tr = A(i,i) + A(i+1, i+1);
	TA det = A(i,i)*A(i+1, i+1) - A(i,i+1)*A(i+1, i);
	TA delta = tr*tr - TA(4) * det;
	if (delta < TA(0)) {
	  V[i] = std::complex<TV>(tr / TA(2), gmm::sqrt(-delta) / TA(2));
	  V[i+1] = std::complex<TV>(tr / TA(2), -gmm::sqrt(-delta)/ TA(2));
	}
	else {
	  V[i  ] = TA(tr + gmm::sqrt(delta)) / TA(2);
	  V[i+1] = TA(tr -  gmm::sqrt(delta)) / TA(2);
	}
	++i;
      }
  }

  template <typename TA, typename TV, typename Ttol,
	    typename MAT, typename VECT>
  void extract_eig(const MAT &A, VECT &V, Ttol tol, std::complex<TA>, TV) {
    typedef std::complex<TA> T;
    size_type n = mat_nrows(A);
    if (n == 0) return;
    tol *= Ttol(2);
    Ttol tol_i = tol * gmm::abs(A(0,0)), tol_cplx = tol_i;
    for (size_type i = 0; i < n; ++i) {
      if (i < n-1) {
	tol_i = (gmm::abs(A(i,i))+gmm::abs(A(i+1,i+1)))*tol;
	tol_cplx = std::max(tol_cplx, tol_i);
      }
      if ((i == n-1) || gmm::abs(A(i+1,i)) < tol_i) {
	if (gmm::abs(std::imag(A(i,i))) > tol_cplx)
	  GMM_WARNING1("A complex eigenvalue has been detected : "
		      << T(A(i,i)) << " : "  << gmm::abs(std::imag(A(i,i)))
		      / gmm::abs(std::real(A(i,i))) << " : " << tol_cplx);
	V[i] = std::real(A(i,i));
      }
      else {
	T tr = A(i,i) + A(i+1, i+1);
	T det = A(i,i)*A(i+1, i+1) - A(i,i+1)*A(i+1, i);
	T delta = tr*tr - TA(4) * det;
	T a1 = (tr + gmm::sqrt(delta)) / TA(2);
	T a2 = (tr - gmm::sqrt(delta)) / TA(2);
	if (gmm::abs(std::imag(a1)) > tol_cplx)
	  GMM_WARNING1("A complex eigenvalue has been detected : " << a1);
	if (gmm::abs(std::imag(a2)) > tol_cplx)
	  GMM_WARNING1("A complex eigenvalue has been detected : " << a2);

	V[i] = std::real(a1); V[i+1] = std::real(a2);
	++i;
      }
    }
  }

  template <typename TA, typename TV, typename Ttol,
	    typename MAT, typename VECT>
  void extract_eig(const MAT &A, VECT &V, Ttol tol,
		   std::complex<TA>, std::complex<TV>) {
    size_type n = mat_nrows(A);
    tol *= Ttol(2);
    for (size_type i = 0; i < n; ++i)
      if ((i == n-1) ||
	  gmm::abs(A(i+1,i)) < (gmm::abs(A(i,i))+gmm::abs(A(i+1,i+1)))*tol)
	V[i] = std::complex<TV>(A(i,i));
      else {
	std::complex<TA> tr = A(i,i) + A(i+1, i+1);
	std::complex<TA> det = A(i,i)*A(i+1, i+1) - A(i,i+1)*A(i+1, i);
	std::complex<TA> delta = tr*tr - TA(4) * det;
	V[i] = (tr + gmm::sqrt(delta)) / TA(2);
	V[i+1] = (tr - gmm::sqrt(delta)) / TA(2);
	++i;
      }
  }

  ///@endcond
  /**
     Compute eigenvalue vector.
  */
  template <typename MAT, typename Ttol, typename VECT> inline
  void extract_eig(const MAT &A, const VECT &V, Ttol tol) {
    extract_eig(A, const_cast<VECT&>(V), tol,
		typename linalg_traits<MAT>::value_type(),
		typename linalg_traits<VECT>::value_type());
  }

  /* ********************************************************************* */
  /*    Stop criterion for QR algorithms                                   */
  /* ********************************************************************* */

  template <typename MAT, typename Ttol>
  void qr_stop_criterion(MAT &A, size_type &p, size_type &q, Ttol tol) {
    typedef typename linalg_traits<MAT>::value_type T;
    typedef typename number_traits<T>::magnitude_type R;
    R rmin = default_min(R()) * R(2);
    size_type n = mat_nrows(A);
    if (n <= 2) { q = n; p = 0; }
    else {
      for (size_type i = 1; i < n-q; ++i)
	if (gmm::abs(A(i,i-1)) < (gmm::abs(A(i,i))+ gmm::abs(A(i-1,i-1)))*tol
	    || gmm::abs(A(i,i-1)) < rmin)
	  A(i,i-1) = T(0);
      
      while ((q < n-1 && A(n-1-q, n-2-q) == T(0)) ||
	     (q < n-2 && A(n-2-q, n-3-q) == T(0))) ++q;
      if (q >= n-2) q = n;
      p = n-q; if (p) --p; if (p) --p;
      while (p > 0 && A(p,p-1) != T(0)) --p;
    }
  }
  
  template <typename MAT, typename Ttol> inline
  void symmetric_qr_stop_criterion(const MAT &AA, size_type &p, size_type &q,
				Ttol tol) {
    typedef typename linalg_traits<MAT>::value_type T;
    typedef typename number_traits<T>::magnitude_type R;
    R rmin = default_min(R()) * R(2);
    MAT& A = const_cast<MAT&>(AA);
    size_type n = mat_nrows(A);
    if (n <= 1) { q = n; p = 0; }
    else {
      for (size_type i = 1; i < n-q; ++i)
	if (gmm::abs(A(i,i-1)) < (gmm::abs(A(i,i))+ gmm::abs(A(i-1,i-1)))*tol
	    || gmm::abs(A(i,i-1)) < rmin)
	  A(i,i-1) = T(0);
      
      while (q < n-1 && A(n-1-q, n-2-q) == T(0)) ++q;
      if (q >= n-1) q = n;
      p = n-q; if (p) --p; if (p) --p;
      while (p > 0 && A(p,p-1) != T(0)) --p;
    }
  }

  template <typename VECT1, typename VECT2, typename Ttol> inline
  void symmetric_qr_stop_criterion(const VECT1 &diag, const VECT2 &sdiag_,
				   size_type &p, size_type &q, Ttol tol) {
    typedef typename linalg_traits<VECT2>::value_type T;
    typedef typename number_traits<T>::magnitude_type R;
    R rmin = default_min(R()) * R(2);
    VECT2 &sdiag = const_cast<VECT2 &>(sdiag_);
    size_type n = vect_size(diag);
    if (n <= 1) { q = n; p = 0; return; }
    for (size_type i = 1; i < n-q; ++i)
      if (gmm::abs(sdiag[i-1]) < (gmm::abs(diag[i])+ gmm::abs(diag[i-1]))*tol
	  || gmm::abs(sdiag[i-1]) < rmin)
	sdiag[i-1] = T(0);
    while (q < n-1 && sdiag[n-2-q] == T(0)) ++q;
    if (q >= n-1) q = n;
    p = n-q; if (p) --p; if (p) --p;
    while (p > 0 && sdiag[p-1] != T(0)) --p;
  }

  /* ********************************************************************* */
  /*    2x2 blocks reduction for Schur vectors                             */
  /* ********************************************************************* */

  template <typename MATH, typename MATQ, typename Ttol>
  void block2x2_reduction(MATH &H, MATQ &Q, Ttol tol) {
    typedef typename linalg_traits<MATH>::value_type T;
    typedef typename number_traits<T>::magnitude_type R;

    size_type n = mat_nrows(H), nq = mat_nrows(Q);
    sub_interval SUBQ(0, nq), SUBL(0, 2);
    std::vector<T> v(2), w(std::max(n, nq)); v[0] = T(1);
    if (n < 2) return;
    tol *= Ttol(2);
    Ttol tol_i = tol * gmm::abs(H(0,0)), tol_cplx = tol_i;
    for (size_type i = 0; i < n-1; ++i) {
      tol_i = (gmm::abs(H(i,i))+gmm::abs(H(i+1,i+1)))*tol;
      tol_cplx = std::max(tol_cplx, tol_i);
      
      if (gmm::abs(H(i+1,i)) > tol_i) { // 2x2 block detected
	T tr = (H(i+1, i+1) - H(i,i)) / T(2);
	T delta = tr*tr + H(i,i+1)*H(i+1, i);
	
	if (is_complex(T()) || gmm::real(delta) >= R(0)) {
	  sub_interval SUBI(i, 2);
	  T theta = (tr - gmm::sqrt(delta)) / H(i+1,i);
	  R a = gmm::abs(theta);
	  v[1] = (a == R(0)) ? T(-1)
	    : gmm::conj(theta) * (R(1) - gmm::sqrt(a*a + R(1)) / a);
	  row_house_update(sub_matrix(H, SUBI), v, sub_vector(w, SUBL));
	  col_house_update(sub_matrix(H, SUBI), v, sub_vector(w, SUBL));
	  col_house_update(sub_matrix(Q, SUBQ, SUBI), v, sub_vector(w, SUBQ));
	}
	++i;
      }
    }
  }

  /* ********************************************************************* */
  /*    Basic qr algorithm.                                                */
  /* ********************************************************************* */

  #define tol_type_for_qr typename number_traits<typename \
                          linalg_traits<MAT1>::value_type>::magnitude_type
  #define default_tol_for_qr \
    (gmm::default_tol(tol_type_for_qr()) *  tol_type_for_qr(3))

  // QR method for real or complex square matrices based on QR factorisation.
  // eigval has to be a complex vector if A has complex eigeinvalues.
  // Very slow method. Use implicit_qr_method instead.
  template <typename MAT1, typename VECT, typename MAT2>
    void rudimentary_qr_algorithm(const MAT1 &A, const VECT &eigval_,
				  const MAT2 &eigvect_,
				  tol_type_for_qr tol = default_tol_for_qr,
				  bool compvect = true) {
    VECT &eigval = const_cast<VECT &>(eigval_);
    MAT2 &eigvect = const_cast<MAT2 &>(eigvect_);

    typedef typename linalg_traits<MAT1>::value_type value_type;

    size_type n = mat_nrows(A), p, q = 0, ite = 0;
    dense_matrix<value_type> Q(n, n), R(n,n), A1(n,n); 
    gmm::copy(A, A1);

    Hessenberg_reduction(A1, eigvect, compvect);
    qr_stop_criterion(A1, p, q, tol);

    while (q < n) {
      qr_factor(A1, Q, R);
      gmm::mult(R, Q, A1);
      if (compvect) { gmm::mult(eigvect, Q, R); gmm::copy(R, eigvect); }
      
      qr_stop_criterion(A1, p, q, tol);
      ++ite;
      GMM_ASSERT1(ite < n*1000, "QR algorithm failed");
    }
    if (compvect) block2x2_reduction(A1, Q, tol);
    extract_eig(A1, eigval, tol); 
  }

  template <typename MAT1, typename VECT>
    void rudimentary_qr_algorithm(const MAT1 &a, VECT &eigval,
				  tol_type_for_qr tol = default_tol_for_qr) {
    dense_matrix<typename linalg_traits<MAT1>::value_type> m(0,0);
    rudimentary_qr_algorithm(a, eigval, m, tol, false); 
  }

  /* ********************************************************************* */
  /*    Francis QR step.                                                   */
  /* ********************************************************************* */

  template <typename MAT1, typename MAT2>
    void Francis_qr_step(const MAT1& HH, const MAT2 &QQ, bool compute_Q) {
    MAT1& H = const_cast<MAT1&>(HH); MAT2& Q = const_cast<MAT2&>(QQ);
    typedef typename linalg_traits<MAT1>::value_type value_type;
    size_type n = mat_nrows(H), nq = mat_nrows(Q); 
    
    std::vector<value_type> v(3), w(std::max(n, nq));

    value_type s = H(n-2, n-2) + H(n-1, n-1);
    value_type t = H(n-2, n-2) * H(n-1, n-1) - H(n-2, n-1) * H(n-1, n-2);
    value_type x = H(0, 0) * H(0, 0) + H(0,1) * H(1, 0) - s * H(0,0) + t;
    value_type y = H(1, 0) * (H(0,0) + H(1,1) - s);
    value_type z = H(1, 0) * H(2, 1);

    sub_interval SUBQ(0, nq);

    for (size_type k = 0; k < n - 2; ++k) {
      v[0] = x; v[1] = y; v[2] = z;
      house_vector(v);
      size_type r = std::min(k+4, n), q = (k==0) ? 0 : k-1;
      sub_interval SUBI(k, 3), SUBJ(0, r), SUBK(q, n-q);
      
      row_house_update(sub_matrix(H, SUBI, SUBK),  v, sub_vector(w, SUBK));
      col_house_update(sub_matrix(H, SUBJ, SUBI),  v, sub_vector(w, SUBJ));
      
      if (compute_Q)
       	col_house_update(sub_matrix(Q, SUBQ, SUBI),  v, sub_vector(w, SUBQ));

      x = H(k+1, k); y = H(k+2, k);
      if (k < n-3) z = H(k+3, k);
    }
    sub_interval SUBI(n-2,2), SUBJ(0, n), SUBK(n-3,3), SUBL(0, 3);
    v.resize(2);
    v[0] = x; v[1] = y;
    house_vector(v);
    row_house_update(sub_matrix(H, SUBI, SUBK), v, sub_vector(w, SUBL));
    col_house_update(sub_matrix(H, SUBJ, SUBI), v, sub_vector(w, SUBJ));
    if (compute_Q)
      col_house_update(sub_matrix(Q, SUBQ, SUBI), v, sub_vector(w, SUBQ));
  }

  /* ********************************************************************* */
  /*    Wilkinson Double shift QR step (from Lapack).                      */
  /* ********************************************************************* */

  template <typename MAT1, typename MAT2, typename Ttol>
  void Wilkinson_double_shift_qr_step(const MAT1& HH, const MAT2 &QQ,
				      Ttol tol, bool exc, bool compute_Q) {
    MAT1& H = const_cast<MAT1&>(HH); MAT2& Q = const_cast<MAT2&>(QQ);
    typedef typename linalg_traits<MAT1>::value_type T;
    typedef typename number_traits<T>::magnitude_type R;

    size_type n = mat_nrows(H), nq = mat_nrows(Q), m;
    std::vector<T> v(3), w(std::max(n, nq));
    const R dat1(0.75), dat2(-0.4375);
    T h33, h44, h43h34, v1(0), v2(0), v3(0);
    
    if (exc) {                    /* Exceptional shift.                    */
      R s = gmm::abs(H(n-1, n-2)) + gmm::abs(H(n-2, n-3));
      h33 = h44 = dat1 * s;
      h43h34 = dat2*s*s;
    }
    else {                        /* Wilkinson double shift.               */
      h44 = H(n-1,n-1); h33 = H(n-2, n-2);
      h43h34 = H(n-1, n-2) * H(n-2, n-1);
    }

    /* Look for two consecutive small subdiagonal elements.                */
    /* Determine the effect of starting the double-shift QR iteration at   */
    /* row m, and see if this would make H(m-1, m-2) negligible.           */
    for (m = n-2; m != 0; --m) {
      T h11  = H(m-1, m-1), h22  = H(m, m);
      T h21  = H(m, m-1),   h12  = H(m-1, m);
      T h44s = h44 - h11,   h33s = h33 - h11;
      v1 = (h33s*h44s-h43h34) / h21 + h12;
      v2 = h22 - h11 - h33s - h44s;
      v3 = H(m+1, m);
      R s = gmm::abs(v1) + gmm::abs(v2) + gmm::abs(v3);
      v1 /= s; v2 /= s; v3 /= s;
      if (m == 1) break;
      T h00 = H(m-2, m-2);
      T h10 = H(m-1, m-2);
      R tst1 = gmm::abs(v1)*(gmm::abs(h00)+gmm::abs(h11)+gmm::abs(h22));
      if (gmm::abs(h10)*(gmm::abs(v2)+gmm::abs(v3)) <= tol * tst1) break;
    }

    /* Double shift QR step.                                               */
    sub_interval SUBQ(0, nq);
    for (size_type k = (m == 0) ? 0 : m-1; k < n-2; ++k) {
      v[0] = v1; v[1] = v2; v[2] = v3;
      house_vector(v);
      size_type r = std::min(k+4, n), q = (k==0) ? 0 : k-1;
      sub_interval SUBI(k, 3), SUBJ(0, r), SUBK(q, n-q);
      
      row_house_update(sub_matrix(H, SUBI, SUBK),  v, sub_vector(w, SUBK));
      col_house_update(sub_matrix(H, SUBJ, SUBI),  v, sub_vector(w, SUBJ));
      if (k > m-1) { H(k+1, k-1) = T(0); if (k < n-3) H(k+2, k-1) = T(0); }
      
      if (compute_Q)
       	col_house_update(sub_matrix(Q, SUBQ, SUBI),  v, sub_vector(w, SUBQ));

      v1 = H(k+1, k); v2 = H(k+2, k);
      if (k < n-3) v3 = H(k+3, k);
    }
    sub_interval SUBI(n-2,2), SUBJ(0, n), SUBK(n-3,3), SUBL(0, 3);
    v.resize(2); v[0] = v1; v[1] = v2;
    house_vector(v);
    row_house_update(sub_matrix(H, SUBI, SUBK), v, sub_vector(w, SUBL));
    col_house_update(sub_matrix(H, SUBJ, SUBI), v, sub_vector(w, SUBJ));
    if (compute_Q)
      col_house_update(sub_matrix(Q, SUBQ, SUBI), v, sub_vector(w, SUBQ));
  }

  /* ********************************************************************* */
  /*    Implicit QR algorithm.                                             */
  /* ********************************************************************* */

  // QR method for real or complex square matrices based on an
  // implicit QR factorisation. eigval has to be a complex vector
  // if A has complex eigeinvalues. complexity about 10n^3, 25n^3 if
  // eigenvectors are computed
  template <typename MAT1, typename VECT, typename MAT2>
    void implicit_qr_algorithm(const MAT1 &A, const VECT &eigval_,
			       const MAT2 &Q_, 
			       tol_type_for_qr tol = default_tol_for_qr,
			       bool compvect = true) {
    VECT &eigval = const_cast<VECT &>(eigval_);
    MAT2 &Q = const_cast<MAT2 &>(Q_);
    typedef typename linalg_traits<MAT1>::value_type value_type;

    size_type n(mat_nrows(A)), q(0), q_old, p(0), ite(0), its(0);
    dense_matrix<value_type> H(n,n);
    sub_interval SUBK(0,0);

    gmm::copy(A, H);
    Hessenberg_reduction(H, Q, compvect);
    qr_stop_criterion(H, p, q, tol);
    
    while (q < n) {
      sub_interval SUBI(p, n-p-q), SUBJ(0, mat_ncols(Q));
      if (compvect) SUBK = SUBI;
//       Francis_qr_step(sub_matrix(H, SUBI),
// 		      sub_matrix(Q, SUBJ, SUBK), compvect);
      Wilkinson_double_shift_qr_step(sub_matrix(H, SUBI), 
				     sub_matrix(Q, SUBJ, SUBK),
				     tol, (its == 10 || its == 20), compvect);
      q_old = q;
      qr_stop_criterion(H, p, q, tol*2);
      if (q != q_old) its = 0;
      ++its; ++ite;
      GMM_ASSERT1(ite < n*100, "QR algorithm failed");
    }
    if (compvect) block2x2_reduction(H, Q, tol);
    extract_eig(H, eigval, tol);
  }


  template <typename MAT1, typename VECT>
    void implicit_qr_algorithm(const MAT1 &a, VECT &eigval,
			       tol_type_for_qr tol = default_tol_for_qr) {
    dense_matrix<typename linalg_traits<MAT1>::value_type> m(1,1);
    implicit_qr_algorithm(a, eigval, m, tol, false); 
  }

  /* ********************************************************************* */
  /*    Implicit symmetric QR step with Wilkinson Shift.                   */
  /* ********************************************************************* */

  template <typename MAT1, typename MAT2> 
    void symmetric_Wilkinson_qr_step(const MAT1& MM, const MAT2 &ZZ,
				     bool compute_z) {
    MAT1& M = const_cast<MAT1&>(MM); MAT2& Z = const_cast<MAT2&>(ZZ);
    typedef typename linalg_traits<MAT1>::value_type T;
    typedef typename number_traits<T>::magnitude_type R;
    size_type n = mat_nrows(M);

    for (size_type i = 0; i < n; ++i) {
      M(i, i) = T(gmm::real(M(i, i)));
      if (i > 0) {
	T a = (M(i, i-1) + gmm::conj(M(i-1, i)))/R(2);
	M(i, i-1) = a; M(i-1, i) = gmm::conj(a);
      }
    }

    R d = gmm::real(M(n-2, n-2) - M(n-1, n-1)) / R(2);
    R e = gmm::abs_sqr(M(n-1, n-2));
    R nu = d + gmm::sgn(d)*gmm::sqrt(d*d+e);
    if (nu == R(0)) { M(n-1, n-2) = T(0); return; }
    R mu = gmm::real(M(n-1, n-1)) - e / nu;
    T x = M(0,0) - T(mu), z = M(1, 0), c, s;

    for (size_type k = 1; k < n; ++k) {
      Givens_rotation(x, z, c, s);

      if (k > 1) Apply_Givens_rotation_left(M(k-1,k-2), M(k,k-2), c, s);
      Apply_Givens_rotation_left(M(k-1,k-1), M(k,k-1), c, s);
      Apply_Givens_rotation_left(M(k-1,k  ), M(k,k  ), c, s);
      if (k < n-1) Apply_Givens_rotation_left(M(k-1,k+1), M(k,k+1), c, s);
      if (k > 1) Apply_Givens_rotation_right(M(k-2,k-1), M(k-2,k), c, s);
      Apply_Givens_rotation_right(M(k-1,k-1), M(k-1,k), c, s);
      Apply_Givens_rotation_right(M(k  ,k-1), M(k,k)  , c, s);
      if (k < n-1) Apply_Givens_rotation_right(M(k+1,k-1), M(k+1,k), c, s);

      if (compute_z) col_rot(Z, c, s, k-1, k);
      if (k < n-1) { x = M(k, k-1); z = M(k+1, k-1); }
    }

  }

  template <typename VECT1, typename VECT2, typename MAT> 
  void symmetric_Wilkinson_qr_step(const VECT1& diag_, const VECT2& sdiag_,
				   const MAT &ZZ, bool compute_z) {
    VECT1& diag = const_cast<VECT1&>(diag_);
    VECT2& sdiag = const_cast<VECT2&>(sdiag_);
    MAT& Z = const_cast<MAT&>(ZZ);
    typedef typename linalg_traits<VECT2>::value_type T;
    typedef typename number_traits<T>::magnitude_type R;

    size_type n = vect_size(diag);
    R d = (diag[n-2] - diag[n-1]) / R(2);
    R e = gmm::abs_sqr(sdiag[n-2]);
    R nu = d + gmm::sgn(d)*gmm::sqrt(d*d+e);
    if (nu == R(0)) { sdiag[n-2] = T(0); return; }
    R mu = diag[n-1] - e / nu;
    T x = diag[0] - T(mu), z = sdiag[0], c, s;

    T a01(0), a02(0);
    T a10(0), a11(diag[0]), a12(gmm::conj(sdiag[0])), a13(0);
    T a20(0), a21(sdiag[0]), a22(diag[1]), a23(gmm::conj(sdiag[1]));
    T a31(0), a32(sdiag[1]);

    for (size_type k = 1; k < n; ++k) {
      Givens_rotation(x, z, c, s);

      if (k > 1) Apply_Givens_rotation_left(a10, a20, c, s);
      Apply_Givens_rotation_left(a11, a21, c, s);
      Apply_Givens_rotation_left(a12, a22, c, s);
      if (k < n-1) Apply_Givens_rotation_left(a13, a23, c, s);

      if (k > 1) Apply_Givens_rotation_right(a01, a02, c, s);
      Apply_Givens_rotation_right(a11, a12, c, s);
      Apply_Givens_rotation_right(a21, a22, c, s);
      if (k < n-1) Apply_Givens_rotation_right(a31, a32, c, s);

      if (compute_z) col_rot(Z, c, s, k-1, k);

      diag[k-1] = gmm::real(a11);
      diag[k] = gmm::real(a22);
      if (k > 1) sdiag[k-2] = (gmm::conj(a01) + a10) / R(2);
      sdiag[k-1] = (gmm::conj(a12) + a21) / R(2);

      x = sdiag[k-1]; z = (gmm::conj(a13) + a31) / R(2);

      a01 = a12; a02 = a13;
      a10 = a21; a11 = a22; a12 = a23; a13 = T(0);
      a20 = a31; a21 = a32; a31 = T(0);

      if (k < n-1) {
	sdiag[k] = (gmm::conj(a23) + a32) / R(2);
	a22 = T(diag[k+1]); a32 = sdiag[k+1]; a23 = gmm::conj(a32);
      }
    }
  }

  /* ********************************************************************* */
  /*    Implicit QR algorithm for symmetric or hermitian matrices.         */
  /* ********************************************************************* */

  // implicit QR method for real square symmetric matrices or complex
  // hermitian matrices.
  // eigval has to be a complex vector if A has complex eigeinvalues.
  // complexity about 4n^3/3, 9n^3 if eigenvectors are computed
  template <typename MAT1, typename VECT, typename MAT2>
  void symmetric_qr_algorithm_old(const MAT1 &A, const VECT &eigval_,
			      const MAT2 &eigvect_,
			      tol_type_for_qr tol = default_tol_for_qr,
			      bool compvect = true) {
    VECT &eigval = const_cast<VECT &>(eigval_);
    MAT2 &eigvect = const_cast<MAT2 &>(eigvect_);
    typedef typename linalg_traits<MAT1>::value_type T;
    typedef typename number_traits<T>::magnitude_type R;

    if (compvect) gmm::copy(identity_matrix(), eigvect);
    size_type n = mat_nrows(A), q = 0, p, ite = 0;
    dense_matrix<T> Tri(n, n);
    gmm::copy(A, Tri);

    Householder_tridiagonalization(Tri, eigvect, compvect);
    
    symmetric_qr_stop_criterion(Tri, p, q, tol);
    
    while (q < n) {

      sub_interval SUBI(p, n-p-q), SUBJ(0, mat_ncols(eigvect)), SUBK(p, n-p-q);
      if (!compvect) SUBK = sub_interval(0,0);
      symmetric_Wilkinson_qr_step(sub_matrix(Tri, SUBI), 
				  sub_matrix(eigvect, SUBJ, SUBK), compvect);
      
      symmetric_qr_stop_criterion(Tri, p, q, tol*R(2));
      ++ite;
      GMM_ASSERT1(ite < n*100, "QR algorithm failed. Probably, your matrix"
		  " is not real symmetric or complex hermitian");
    }
    
    extract_eig(Tri, eigval, tol);
  }

  template <typename MAT1, typename VECT, typename MAT2>
  void symmetric_qr_algorithm(const MAT1 &A, const VECT &eigval_,
			      const MAT2 &eigvect_,
			      tol_type_for_qr tol = default_tol_for_qr,
			      bool compvect = true) {
    VECT &eigval = const_cast<VECT &>(eigval_);
    MAT2 &eigvect = const_cast<MAT2 &>(eigvect_);
    typedef typename linalg_traits<MAT1>::value_type T;
    typedef typename number_traits<T>::magnitude_type R;

    size_type n = mat_nrows(A), q = 0, p, ite = 0;
    if (compvect) gmm::copy(identity_matrix(), eigvect);
    if (n == 0) return;
    if (n == 1) { eigval[0]=gmm::real(A(0,0)); return; }
    dense_matrix<T> Tri(n, n);
    gmm::copy(A, Tri);

    Householder_tridiagonalization(Tri, eigvect, compvect);

    std::vector<R> diag(n);
    std::vector<T> sdiag(n);
    for (size_type i = 0; i < n; ++i)
      { diag[i] = gmm::real(Tri(i, i)); if (i+1 < n) sdiag[i] = Tri(i+1, i); }
    
    symmetric_qr_stop_criterion(diag, sdiag, p, q, tol);
    
    while (q < n) {
      sub_interval SUBI(p, n-p-q), SUBJ(0, mat_ncols(eigvect)), SUBK(p, n-p-q);
      if (!compvect) SUBK = sub_interval(0,0);
      
      symmetric_Wilkinson_qr_step(sub_vector(diag, SUBI),
				  sub_vector(sdiag, SUBI),
				  sub_matrix(eigvect, SUBJ, SUBK), compvect);

      symmetric_qr_stop_criterion(diag, sdiag, p, q, tol*R(3));
      ++ite;
      GMM_ASSERT1(ite < n*100, "QR algorithm failed.");
    }
    
    gmm::copy(diag, eigval);
  }


  template <typename MAT1, typename VECT>
    void symmetric_qr_algorithm(const MAT1 &a, VECT &eigval,
				tol_type_for_qr tol = default_tol_for_qr) {
    dense_matrix<typename linalg_traits<MAT1>::value_type> m(0,0);
    symmetric_qr_algorithm(a, eigval, m, tol, false);
  }


}

#endif