/* eigen/genherm.c

 * 

 * Copyright (C) 2007 Patrick Alken

 * 

 * 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.

 */

#include <stdlib.h>

#include <config.h>

#include <gsl/gsl_eigen.h>

#include <gsl/gsl_linalg.h>

#include <gsl/gsl_math.h>

#include <gsl/gsl_blas.h>

#include <gsl/gsl_vector.h>

#include <gsl/gsl_matrix.h>

#include <gsl/gsl_complex.h>

#include <gsl/gsl_complex_math.h>

/*

 * This module computes the eigenvalues of a complex generalized

 * hermitian-definite eigensystem A x = \lambda B x, where A and

 * B are hermitian, and B is positive-definite.

 */

/*

gsl_eigen_genherm_alloc()

Allocate a workspace for solving the generalized hermitian-definite

eigenvalue problem. The size of this workspace is O(3n).

Inputs: n - size of matrices

Return: pointer to workspace

*/

gsl_eigen_genherm_workspace *

gsl_eigen_genherm_alloc(const size_t n)

{

  gsl_eigen_genherm_workspace *w;

  if (n == 0)

    {

      GSL_ERROR_NULL ("matrix dimension must be positive integer",

                      GSL_EINVAL);

    }

  w = (gsl_eigen_genherm_workspace *) calloc (1, sizeof (gsl_eigen_genherm_workspace));

  if (w == 0)

    {

      GSL_ERROR_NULL ("failed to allocate space for workspace", GSL_ENOMEM);

    }

  w->size = n;

  w->herm_workspace_p = gsl_eigen_herm_alloc(n);

  if (!w->herm_workspace_p)

    {

      gsl_eigen_genherm_free(w);

      GSL_ERROR_NULL("failed to allocate space for herm workspace", GSL_ENOMEM);

    }

  return (w);

} /* gsl_eigen_genherm_alloc() */

/*

gsl_eigen_genherm_free()

  Free workspace w

*/

void

gsl_eigen_genherm_free (gsl_eigen_genherm_workspace * w)

{

  RETURN_IF_NULL (w);

  if (w->herm_workspace_p)

    gsl_eigen_herm_free(w->herm_workspace_p);

  free(w);

} /* gsl_eigen_genherm_free() */

/*

gsl_eigen_genherm()

Solve the generalized hermitian-definite eigenvalue problem

A x = \lambda B x

for the eigenvalues \lambda.

Inputs: A    - complex hermitian matrix

        B    - complex hermitian and positive definite matrix

        eval - where to store eigenvalues

        w    - workspace

Return: success or error

*/

int

gsl_eigen_genherm (gsl_matrix_complex * A, gsl_matrix_complex * B,

                   gsl_vector * eval, gsl_eigen_genherm_workspace * w)

{

  const size_t N = A->size1;

  /* check matrix and vector sizes */

  if (N != A->size2)

    {

      GSL_ERROR ("matrix must be square to compute eigenvalues", GSL_ENOTSQR);

    }

  else if ((N != B->size1) || (N != B->size2))

    {

      GSL_ERROR ("B matrix dimensions must match A", GSL_EBADLEN);

    }

  else if (eval->size != N)

    {

      GSL_ERROR ("eigenvalue vector must match matrix size", GSL_EBADLEN);

    }

  else if (w->size != N)

    {

      GSL_ERROR ("matrix size does not match workspace", GSL_EBADLEN);

    }

  else

    {

      int s;

      /* compute Cholesky factorization of B */

      s = gsl_linalg_complex_cholesky_decomp(B);

      if (s != GSL_SUCCESS)

        return s; /* B is not positive definite */

      /* transform to standard hermitian eigenvalue problem */

      gsl_eigen_genherm_standardize(A, B);

      s = gsl_eigen_herm(A, eval, w->herm_workspace_p);

      return s;

    }

} /* gsl_eigen_genherm() */

/*

gsl_eigen_genherm_standardize()

  Reduce the generalized hermitian-definite eigenproblem to

the standard hermitian eigenproblem by computing

C = L^{-1} A L^{-H}

where L L^H is the Cholesky decomposition of B

Inputs: A - (input/output) complex hermitian matrix

        B - complex hermitian, positive definite matrix in Cholesky form

Return: success

Notes: A is overwritten by L^{-1} A L^{-H}

*/

int

gsl_eigen_genherm_standardize(gsl_matrix_complex *A,

                              const gsl_matrix_complex *B)

{

  const size_t N = A->size1;

  size_t i;

  double a, b;

  gsl_complex y, z;

  GSL_SET_IMAG(&z, 0.0);

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

    {

      /* update lower triangle of A(i:n, i:n) */

      y = gsl_matrix_complex_get(A, i, i);

      a = GSL_REAL(y);

      y = gsl_matrix_complex_get(B, i, i);

      b = GSL_REAL(y);

      a /= b * b;

      GSL_SET_REAL(&z, a);

      gsl_matrix_complex_set(A, i, i, z);

      if (i < N - 1)

        {

          gsl_vector_complex_view ai =

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

          gsl_matrix_complex_view ma =

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

          gsl_vector_complex_const_view bi =

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

          gsl_matrix_complex_const_view mb =

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

          gsl_blas_zdscal(1.0 / b, &ai.vector);

          GSL_SET_REAL(&z, -0.5 * a);

          gsl_blas_zaxpy(z, &bi.vector, &ai.vector);

          gsl_blas_zher2(CblasLower,

                         GSL_COMPLEX_NEGONE,

                         &ai.vector,

                         &bi.vector,

                         &ma.matrix);

          gsl_blas_zaxpy(z, &bi.vector, &ai.vector);

          gsl_blas_ztrsv(CblasLower,

                         CblasNoTrans,

                         CblasNonUnit,

                         &mb.matrix,

                         &ai.vector);

        }

    }

  return GSL_SUCCESS;

} /* gsl_eigen_genherm_standardize() */