francis.c

math library from gnu

   */

  dat[0] = (h_tmp1*h_tmp2 - h_cross) / gsl_matrix_get(H, 1, 0) +
           gsl_matrix_get(H, 0, 1);
  dat[1] = gsl_matrix_get(H, 1, 1) - gsl_matrix_get(H, 0, 0) - h_tmp1 -
           h_tmp2;
  dat[2] = gsl_matrix_get(H, 2, 1);

  scale = fabs(dat[0]) + fabs(dat[1]) + fabs(dat[2]);
  if (scale != 0.0)
    {
      /* scale to prevent overflow or underflow */
      dat[0] /= scale;
      dat[1] /= scale;
      dat[2] /= scale;
    }

  if (w->Z || w->compute_t)
    {
      /*
       * get absolute indices of this (sub)matrix relative to the
       * original Hessenberg matrix
       */
      top = francis_get_submatrix(w->H, H);
    }

  for (i = 0; i < N - 2; ++i)
    {
      tau_i = gsl_linalg_householder_transform(&v3.vector);

      if (tau_i != 0.0)
        {
          /* q = max(1, i - 1) */
          q = (1 > ((int)i - 1)) ? 0 : (i - 1);

          /* r = min(i + 3, N - 1) */
          r = ((i + 3) < (N - 1)) ? (i + 3) : (N - 1);

          if (w->compute_t)
            {
              /*
               * We are computing the Schur form T, so we
               * need to transform the whole matrix H
               *
               * H -> P_k^t H P_k
               *
               * where P_k is the current Householder matrix
               */

              /* apply left householder matrix (I - tau_i v v') to H */
              m = gsl_matrix_submatrix(w->H,
                                       top + i,
                                       top + q,
                                       3,
                                       w->size - top - q);
              gsl_linalg_householder_hm(tau_i, &v3.vector, &m.matrix);

              /* apply right householder matrix (I - tau_i v v') to H */
              m = gsl_matrix_submatrix(w->H,
                                       0,
                                       top + i,
                                       top + r + 1,
                                       3);
              gsl_linalg_householder_mh(tau_i, &v3.vector, &m.matrix);
            }
          else
            {
              /*
               * We are not computing the Schur form T, so we
               * only need to transform the active block
               */

              /* apply left householder matrix (I - tau_i v v') to H */
              m = gsl_matrix_submatrix(H, i, q, 3, N - q);
              gsl_linalg_householder_hm(tau_i, &v3.vector, &m.matrix);

              /* apply right householder matrix (I - tau_i v v') to H */
              m = gsl_matrix_submatrix(H, 0, i, r + 1, 3);
              gsl_linalg_householder_mh(tau_i, &v3.vector, &m.matrix);
            }

          if (w->Z)
            {
              /* accumulate the similarity transformation into Z */
              m = gsl_matrix_submatrix(w->Z, 0, top + i, w->size, 3);
              gsl_linalg_householder_mh(tau_i, &v3.vector, &m.matrix);
            }
        } /* if (tau_i != 0.0) */

      dat[0] = gsl_matrix_get(H, i + 1, i);
      dat[1] = gsl_matrix_get(H, i + 2, i);
      if (i < (N - 3))
        {
          dat[2] = gsl_matrix_get(H, i + 3, i);
        }

      scale = fabs(dat[0]) + fabs(dat[1]) + fabs(dat[2]);
      if (scale != 0.0)
        {
          /* scale to prevent overflow or underflow */
          dat[0] /= scale;
          dat[1] /= scale;
          dat[2] /= scale;
        }
    } /* for (i = 0; i < N - 2; ++i) */

  scale = fabs(dat[0]) + fabs(dat[1]);
  if (scale != 0.0)
    {
      /* scale to prevent overflow or underflow */
      dat[0] /= scale;
      dat[1] /= scale;
    }

  tau_i = gsl_linalg_householder_transform(&v2.vector);

  if (w->compute_t)
    {
      m = gsl_matrix_submatrix(w->H,
                               top + N - 2,
                               top + N - 3,
                               2,
                               w->size - top - N + 3);
      gsl_linalg_householder_hm(tau_i, &v2.vector, &m.matrix);

      m = gsl_matrix_submatrix(w->H,
                               0,
                               top + N - 2,
                               top + N,
                               2);
      gsl_linalg_householder_mh(tau_i, &v2.vector, &m.matrix);
    }
  else
    {
      m = gsl_matrix_submatrix(H, N - 2, N - 3, 2, 3);
      gsl_linalg_householder_hm(tau_i, &v2.vector, &m.matrix);

      m = gsl_matrix_submatrix(H, 0, N - 2, N, 2);
      gsl_linalg_householder_mh(tau_i, &v2.vector, &m.matrix);
    }

  if (w->Z)
    {
      /* accumulate transformation into Z */
      m = gsl_matrix_submatrix(w->Z, 0, top + N - 2, w->size, 2);
      gsl_linalg_householder_mh(tau_i, &v2.vector, &m.matrix);
    }

  return GSL_SUCCESS;
} /* francis_qrstep() */

/*
francis_search_subdiag_small_elements()
  Search for a small subdiagonal element starting from the bottom
of a matrix A. A small element is one that satisfies:

|A_{i,i-1}| <= eps * (|A_{i,i}| + |A_{i-1,i-1}|)

Inputs: A - matrix (must be at least 3-by-3)

Return: row index of small subdiagonal element or 0 if not found

Notes: the first small element that is found (starting from bottom)
       is set to zero
*/

static inline size_t
francis_search_subdiag_small_elements(gsl_matrix * A)
{
  const size_t N = A->size1;
  size_t i;
  double dpel = gsl_matrix_get(A, N - 2, N - 2);

  for (i = N - 1; i > 0; --i)
    {
      double sel = gsl_matrix_get(A, i, i - 1);
      double del = gsl_matrix_get(A, i, i);

      if ((sel == 0.0) ||
          (fabs(sel) < GSL_DBL_EPSILON * (fabs(del) + fabs(dpel))))
        {
          gsl_matrix_set(A, i, i - 1, 0.0);
          return (i);
        }

      dpel = del;
    }

  return (0);
} /* francis_search_subdiag_small_elements() */

/*
francis_schur_standardize()
  Convert a 2-by-2 diagonal block in the Schur form to standard form
and update the rest of T and Z matrices if required.

Inputs: A     - 2-by-2 matrix
        eval1 - where to store eigenvalue 1
        eval2 - where to store eigenvalue 2
        w     - francis workspace
*/

static inline void
francis_schur_standardize(gsl_matrix *A, gsl_complex *eval1,
                          gsl_complex *eval2,
                          gsl_eigen_francis_workspace *w)
{
  const size_t N = w->size;
  double cs, sn;
  size_t top;

  /*
   * figure out where the submatrix A resides in the
   * original matrix H
   */
  top = francis_get_submatrix(w->H, A);

  /* convert 2-by-2 block to standard form */
  francis_standard_form(A, &cs, &sn);

  /* set eigenvalues */

  GSL_SET_REAL(eval1, gsl_matrix_get(A, 0, 0));
  GSL_SET_REAL(eval2, gsl_matrix_get(A, 1, 1));
  if (gsl_matrix_get(A, 1, 0) == 0.0)
    {
      GSL_SET_IMAG(eval1, 0.0);
      GSL_SET_IMAG(eval2, 0.0);
    }
  else
    {
      double tmp = sqrt(fabs(gsl_matrix_get(A, 0, 1)) *
                        fabs(gsl_matrix_get(A, 1, 0)));
      GSL_SET_IMAG(eval1, tmp);
      GSL_SET_IMAG(eval2, -tmp);
    }

  if (w->compute_t)
    {
      gsl_vector_view xv, yv;

      /*
       * The above call to francis_standard_form transformed a 2-by-2 block
       * of T into upper triangular form via the transformation
       *
       * U = [ CS -SN ]
       *     [ SN  CS ]
       *
       * The original matrix T was
       *
       * T = [ T_{11} | T_{12} | T_{13} ]
       *     [   0*   |   A    | T_{23} ]
       *     [   0    |   0*   | T_{33} ]
       *
       * where 0* indicates all zeros except for possibly
       * one subdiagonal element next to A.
       *
       * After francis_standard_form, T looks like this:
       *
       * T = [ T_{11} | T_{12}  | T_{13} ]
       *     [   0*   | U^t A U | T_{23} ]
       *     [   0    |    0*   | T_{33} ]
       *
       * since only the 2-by-2 block of A was changed. However,
       * in order to be able to back transform T at the end,
       * we need to apply the U transformation to the rest
       * of the matrix T since there is no way to apply a
       * similarity transformation to T and change only the
       * middle 2-by-2 block. In other words, let
       *
       * M = [ I 0 0 ]
       *     [ 0 U 0 ]
       *     [ 0 0 I ]
       *
       * and compute
       *
       * M^t T M = [ T_{11} | T_{12} U |   T_{13}   ]
       *           [ U^t 0* | U^t A U  | U^t T_{23} ]
       *           [   0    |   0* U   |   T_{33}   ]
       *
       * So basically we need to apply the transformation U
       * to the i x 2 matrix T_{12} and the 2 x (n - i + 2)
       * matrix T_{23}, where i is the index of the top of A
       * in T.
       *
       * The BLAS routine drot() is suited for this.
       */

      if (top < (N - 2))
        {
          /* transform the 2 rows of T_{23} */

          xv = gsl_matrix_subrow(w->H, top, top + 2, N - top - 2);
          yv = gsl_matrix_subrow(w->H, top + 1, top + 2, N - top - 2);
          gsl_blas_drot(&xv.vector, &yv.vector, cs, sn);
        }

      if (top > 0)
        {
          /* transform the 2 columns of T_{12} */

          xv = gsl_matrix_subcolumn(w->H, top, 0, top);
          yv = gsl_matrix_subcolumn(w->H, top + 1, 0, top);
          gsl_blas_drot(&xv.vector, &yv.vector, cs, sn);
        }
    } /* if (w->compute_t) */

  if (w->Z)
    {
      gsl_vector_view xv, yv;

      /*
       * Accumulate the transformation in Z. Here, Z -> Z * M
       *
       * So:
       *
       * Z -> [ Z_{11} | Z_{12} U | Z_{13} ]
       *      [ Z_{21} | Z_{22} U | Z_{23} ]
       *      [ Z_{31} | Z_{32} U | Z_{33} ]
       *
       * So we just need to apply drot() to the 2 columns
       * starting at index 'top'
       */

      xv = gsl_matrix_column(w->Z, top);
      yv = gsl_matrix_column(w->Z, top + 1);

      gsl_blas_drot(&xv.vector, &yv.vector, cs, sn);
    } /* if (w->Z) */
} /* francis_schur_standardize() */

/*
francis_get_submatrix()
  B is a submatrix of A. The goal of this function is to
compute the indices in A of where the matrix B resides
*/

static inline size_t
francis_get_submatrix(gsl_matrix *A, gsl_matrix *B)
{
  size_t diff;
  double ratio;
  size_t top;

  diff = (size_t) (B->data - A->data);

  ratio = (double)diff / ((double) (A->tda + 1));

  top = (size_t) floor(ratio);

  return top;
} /* francis_get_submatrix() */

/*
francis_standard_form()
  Compute the Schur factorization of a real 2-by-2 matrix in
standard form:

[ A B ] = [ CS -SN ] [ T11 T12 ] [ CS SN ]
[ C D ]   [ SN  CS ] [ T21 T22 ] [-SN CS ]

where either:
1) T21 = 0 so that T11 and T22 are real eigenvalues of the matrix, or
2) T11 = T22 and T21*T12 < 0, so that T11 +/- sqrt(|T21*T12|) are
   complex conjugate eigenvalues

Inputs: A  - 2-by-2 matrix
        cs - where to store cosine parameter of rotation matrix
        sn - where to store sine parameter of rotation matrix

Notes: 1) based on LAPACK routine DLANV2
       2) On output, A is modified to contain the matrix in standard form
*/

static void
francis_standard_form(gsl_matrix *A, double *cs, double *sn)
{
  double a, b, c, d; /* input matrix values */
  double tmp;
  double p, z;
  double bcmax, bcmis, scale;
  double tau, sigma;
  double cs1, sn1;
  double aa, bb, cc, dd;
  double sab, sac;

  a = gsl_matrix_get(A, 0, 0);
  b = gsl_matrix_get(A, 0, 1);
  c = gsl_matrix_get(A, 1, 0);
  d = gsl_matrix_get(A, 1, 1);

  if (c == 0.0)
    {
      /*
       * matrix is already upper triangular - set rotation matrix
       * to the identity
       */
      *cs = 1.0;
      *sn = 0.0;
    }
  else if (b == 0.0)
    {
      /* swap rows and columns to make it upper triangular */

      *cs = 0.0;
      *sn = 1.0;

      tmp = d;
      d = a;
      a = tmp;
      b = -c;
      c = 0.0;
    }
  else if (((a - d) == 0.0) && (GSL_SIGN(b) != GSL_SIGN(c)))
    {
      /* the matrix has complex eigenvalues with a == d */
      *cs = 1.0;
      *sn = 0.0;
    }
  else
    {
      tmp = a - d;
      p = 0.5 * tmp;
      bcmax = GSL_MAX(fabs(b), fabs(c));
      bcmis = GSL_MIN(fabs(b), fabs(c)) * GSL_SIGN(b) * GSL_SIGN(c);
      scale = GSL_MAX(fabs(p), bcmax);
      z = (p / scale) * p + (bcmax / scale) * bcmis;

      if (z >= 4.0 * GSL_DBL_EPSILON)
        {
          /* real eigenvalues, compute a and d */

          z = p + GSL_SIGN(p) * fabs(sqrt(scale) * sqrt(z));
          a = d + z;
          d -= (bcmax / z) * bcmis;

          /* compute b and the rotation matrix */

          tau = gsl_hypot(c, z);
          *cs = z / tau;
          *sn = c / tau;
          b -= c;
          c = 0.0;
        }
      else
        {
          /*
           * complex eigenvalues, or real (almost) equal eigenvalues -
           * make diagonal elements equal
           */

          sigma = b + c;
          tau = gsl_hypot(sigma, tmp);
          *cs = sqrt(0.5 * (1.0 + fabs(sigma) / tau));
          *sn = -(p / (tau * (*cs))) * GSL_SIGN(sigma);

          /*
           * Compute [ AA BB ] = [ A B ] [ CS -SN ]
           *         [ CC DD ]   [ C D ] [ SN  CS ]
           */
          aa = a * (*cs) + b * (*sn);
          bb = -a * (*sn) + b * (*cs);
          cc = c * (*cs) + d * (*sn);
          dd = -c * (*sn) + d * (*cs);

          /*
           * Compute [ A B ] = [ CS SN ] [ AA BB ]
           *         [ C D ]   [-SN CS ] [ CC DD ]
           */
          a = aa * (*cs) + cc * (*sn);
          b = bb * (*cs) + dd * (*sn);
          c = -aa * (*sn) + cc * (*cs);
          d = -bb * (*sn) + dd * (*cs);

          tmp = 0.5 * (a + d);
          a = d = tmp;

          if (c != 0.0)
            {
              if (b != 0.0)
                {
                  if (GSL_SIGN(b) == GSL_SIGN(c))
                    {
                      /*
                       * real eigenvalues: reduce to upper triangular
                       * form
                       */
                      sab = sqrt(fabs(b));
                      sac = sqrt(fabs(c));
                      p = GSL_SIGN(c) * fabs(sab * sac);
                      tau = 1.0 / sqrt(fabs(b + c));
                      a = tmp + p;
                      d = tmp - p;
                      b -= c;
                      c = 0.0;

                      cs1 = sab * tau;
                      sn1 = sac * tau;
                      tmp = (*cs) * cs1 - (*sn) * sn1;
                      *sn = (*cs) * sn1 + (*sn) * cs1;
                      *cs = tmp;
                    }
                }
              else
                {
                  b = -c;
                  c = 0.0;
                  tmp = *cs;
                  *cs = -(*sn);
                  *sn = tmp;
                }
            }
        }
    }

  /* set new matrix elements */

  gsl_matrix_set(A, 0, 0, a);
  gsl_matrix_set(A, 0, 1, b);
  gsl_matrix_set(A, 1, 0, c);
  gsl_matrix_set(A, 1, 1, d);
} /* francis_standard_form() */