/* linalg/sytd.c

 * 

 * Copyright (C) 2001, 2007 Brian Gough

 * 

 * This program 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 3 of the License, or (at

 * your option) any later version.

 * 

 * This program is distributed in the hope that it will be useful, but

 * WITHOUT ANY WARRANTY; without even the implied warranty of

 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU

 * General Public License for more details.

 * 

 * You should have received a copy of the GNU General Public License

 * along with this program; if not, write to the Free Software

 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.

 */

/* Factorise a symmetric matrix A into

 *

 * A = Q T Q'

 *

 * where Q is orthogonal and T is symmetric tridiagonal.  Only the

 * diagonal and lower triangular part of A is referenced and modified.

 *

 * On exit, T is stored in the diagonal and first subdiagonal of

 * A. Since T is symmetric the upper diagonal is not stored.

 *

 * Q is stored as a packed set of Householder transformations in the

 * lower triangular part of the input matrix below the first subdiagonal.

 *

 * The full matrix for Q can be obtained as the product

 *

 *       Q = Q_1 Q_2 ... Q_(N-2)

 *

 * where 

 *

 *       Q_i = (I - tau_i * v_i * v_i')

 *

 * and where v_i is a Householder vector

 *

 *       v_i = [0, ... , 0, 1, A(i+1,i), A(i+2,i), ... , A(N,i)]

 *

 * This storage scheme is the same as in LAPACK.  See LAPACK's

 * ssytd2.f for details.

 *

 * See Golub & Van Loan, "Matrix Computations" (3rd ed), Section 8.3 

 *

 * Note: this description uses 1-based indices. The code below uses

 * 0-based indices 

 */

#include <config.h>

#include <stdlib.h>

#include <gsl/gsl_math.h>

#include <gsl/gsl_vector.h>

#include <gsl/gsl_matrix.h>

#include <gsl/gsl_blas.h>

#include <gsl/gsl_linalg.h>

int 

gsl_linalg_symmtd_decomp (gsl_matrix * A, gsl_vector * tau)  

{

  if (A->size1 != A->size2)

    {

      GSL_ERROR ("symmetric tridiagonal decomposition requires square matrix",

                 GSL_ENOTSQR);

    }

  else if (tau->size + 1 != A->size1)

    {

      GSL_ERROR ("size of tau must be (matrix size - 1)", GSL_EBADLEN);

    }

  else

    {

      const size_t N = A->size1;

      size_t i;

      for (i = 0 ; i < N - 2; i++)

        {

          gsl_vector_view c = gsl_matrix_column (A, i);

          gsl_vector_view v = gsl_vector_subvector (&c.vector, i + 1, N - (i + 1));

          double tau_i = gsl_linalg_householder_transform (&v.vector);

          /* Apply the transformation H^T A H to the remaining columns */

          if (tau_i != 0.0) 

            {

              gsl_matrix_view m = gsl_matrix_submatrix (A, i + 1, i + 1, 

                                                        N - (i+1), N - (i+1));

              double ei = gsl_vector_get(&v.vector, 0);

              gsl_vector_view x = gsl_vector_subvector (tau, i, N-(i+1));

              gsl_vector_set (&v.vector, 0, 1.0);

              /* x = tau * A * v */

              gsl_blas_dsymv (CblasLower, tau_i, &m.matrix, &v.vector, 0.0, &x.vector);

              /* w = x - (1/2) tau * (x' * v) * v  */

              {

                double xv, alpha;

                gsl_blas_ddot(&x.vector, &v.vector, &xv);

                alpha = - (tau_i / 2.0) * xv;

                gsl_blas_daxpy(alpha, &v.vector, &x.vector);

              }

              /* apply the transformation A = A - v w' - w v' */

              gsl_blas_dsyr2(CblasLower, -1.0, &v.vector, &x.vector, &m.matrix);

              gsl_vector_set (&v.vector, 0, ei);

            }

          gsl_vector_set (tau, i, tau_i);

        }

      return GSL_SUCCESS;

    }

}  

/*  Form the orthogonal matrix Q from the packed QR matrix */

int

gsl_linalg_symmtd_unpack (const gsl_matrix * A, 

                          const gsl_vector * tau,

                          gsl_matrix * Q, 

                          gsl_vector * diag, 

                          gsl_vector * sdiag)

{

  if (A->size1 !=  A->size2)

    {

      GSL_ERROR ("matrix A must be square", GSL_ENOTSQR);

    }

  else if (tau->size + 1 != A->size1)

    {

      GSL_ERROR ("size of tau must be (matrix size - 1)", GSL_EBADLEN);

    }

  else if (Q->size1 != A->size1 || Q->size2 != A->size1)

    {

      GSL_ERROR ("size of Q must match size of A", GSL_EBADLEN);

    }

  else if (diag->size != A->size1)

    {

      GSL_ERROR ("size of diagonal must match size of A", GSL_EBADLEN);

    }

  else if (sdiag->size + 1 != A->size1)

    {

      GSL_ERROR ("size of subdiagonal must be (matrix size - 1)", GSL_EBADLEN);

    }

  else

    {

      const size_t N = A->size1;

      size_t i;

      /* Initialize Q to the identity */

      gsl_matrix_set_identity (Q);

      for (i = N - 2; i-- > 0;)

        {

          gsl_vector_const_view c = gsl_matrix_const_column (A, i);

          gsl_vector_const_view h = gsl_vector_const_subvector (&c.vector, i + 1, N - (i+1));

          double ti = gsl_vector_get (tau, i);

          gsl_matrix_view m = gsl_matrix_submatrix (Q, i + 1, i + 1, N-(i+1), N-(i+1));

          gsl_linalg_householder_hm (ti, &h.vector, &m.matrix);

        }

      /* Copy diagonal into diag */

      for (i = 0; i < N; i++)

        {

          double Aii = gsl_matrix_get (A, i, i);

          gsl_vector_set (diag, i, Aii);

        }

      /* Copy subdiagonal into sd */

      for (i = 0; i < N - 1; i++)

        {

          double Aji = gsl_matrix_get (A, i+1, i);

          gsl_vector_set (sdiag, i, Aji);

        }

      return GSL_SUCCESS;

    }

}

int

gsl_linalg_symmtd_unpack_T (const gsl_matrix * A, 

                            gsl_vector * diag, 

                            gsl_vector * sdiag)

{

  if (A->size1 !=  A->size2)

    {

      GSL_ERROR ("matrix A must be square", GSL_ENOTSQR);

    }

  else if (diag->size != A->size1)

    {

      GSL_ERROR ("size of diagonal must match size of A", GSL_EBADLEN);

    }

  else if (sdiag->size + 1 != A->size1)

    {

      GSL_ERROR ("size of subdiagonal must be (matrix size - 1)", GSL_EBADLEN);

    }

  else

    {

      const size_t N = A->size1;

      size_t i;

      /* Copy diagonal into diag */

      for (i = 0; i < N; i++)

        {

          double Aii = gsl_matrix_get (A, i, i);

          gsl_vector_set (diag, i, Aii);

        }

      /* Copy subdiagonal into sdiag */

      for (i = 0; i < N - 1; i++)

        {

          double Aij = gsl_matrix_get (A, i+1, i);

          gsl_vector_set (sdiag, i, Aij);

        }

      return GSL_SUCCESS;

    }

}