/* linalg/householder.c

 * 

 * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2004, 2007, 2010 Gerard Jungman, 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.

 */

#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>

/*

gsl_linalg_householder_transform()

  Compute a householder transformation (tau,v) of a vector

x so that P x = [ I - tau*v*v' ] x annihilates x(1:n-1)

Inputs: v - on input, x vector

            on output, householder vector v

Notes:

1) on output, v is normalized so that v[0] = 1. The 1 is

not actually stored; instead v[0] = -sign(x[0])*||x|| so

that:

P x = v[0] * e_1

Therefore external routines should take care when applying

the projection matrix P to vectors, taking into account

that v[0] should be 1 when doing so.

*/

double

gsl_linalg_householder_transform (gsl_vector * v)

{

  /* replace v[0:n-1] with a householder vector (v[0:n-1]) and

     coefficient tau that annihilate v[1:n-1] */

  const size_t n = v->size ;

  if (n == 1)

    {

      return 0.0; /* tau = 0 */

    }

  else

    { 

      double alpha, beta, tau ;

      gsl_vector_view x = gsl_vector_subvector (v, 1, n - 1) ; 

      double xnorm = gsl_blas_dnrm2 (&x.vector);

      if (xnorm == 0) 

        {

          return 0.0; /* tau = 0 */

        }

      alpha = gsl_vector_get (v, 0) ;

      beta = - (alpha >= 0.0 ? +1.0 : -1.0) * hypot(alpha, xnorm) ;

      tau = (beta - alpha) / beta ;

      {

        double s = (alpha - beta);

        if (fabs(s) > GSL_DBL_MIN) 

          {

            gsl_blas_dscal (1.0 / s, &x.vector);

            gsl_vector_set (v, 0, beta) ;

          }

        else

          {

            gsl_blas_dscal (GSL_DBL_EPSILON / s, &x.vector);

            gsl_blas_dscal (1.0 / GSL_DBL_EPSILON, &x.vector);

            gsl_vector_set (v, 0, beta) ;

          }

      }

      return tau;

    }

}

int

gsl_linalg_householder_hm (double tau, const gsl_vector * v, gsl_matrix * A)

{

  /* applies a householder transformation v,tau to matrix m */

  if (tau == 0.0)

    {

      return GSL_SUCCESS;

    }

#ifdef USE_BLAS

  {

    gsl_vector_const_view v1 = gsl_vector_const_subvector (v, 1, v->size - 1);

    gsl_matrix_view A1 = gsl_matrix_submatrix (A, 1, 0, A->size1 - 1, A->size2);

    size_t j;

    for (j = 0; j < A->size2; j++)

      {

        double wj = 0.0;

        gsl_vector_view A1j = gsl_matrix_column(&A1.matrix, j);

        gsl_blas_ddot (&A1j.vector, &v1.vector, &wj;;

        wj += gsl_matrix_get(A,0,j);

        {

          double A0j = gsl_matrix_get (A, 0, j);

          gsl_matrix_set (A, 0, j, A0j - tau *  wj);

        }

        gsl_blas_daxpy (-tau * wj, &v1.vector, &A1j.vector);

      }

  }

#else

  {

    size_t i, j;

    for (j = 0; j < A->size2; j++)

      {

        /* Compute wj = Akj vk */

        double wj = gsl_matrix_get(A,0,j);  

        for (i = 1; i < A->size1; i++)  /* note, computed for v(0) = 1 above */

          {

            wj += gsl_matrix_get(A,i,j) * gsl_vector_get(v,i);

          }

        /* Aij = Aij - tau vi wj */

        /* i = 0 */

        {

          double A0j = gsl_matrix_get (A, 0, j);

          gsl_matrix_set (A, 0, j, A0j - tau *  wj);

        }

        /* i = 1 .. M-1 */

        for (i = 1; i < A->size1; i++)

          {

            double Aij = gsl_matrix_get (A, i, j);

            double vi = gsl_vector_get (v, i);

            gsl_matrix_set (A, i, j, Aij - tau * vi * wj);

          }

      }

  }

#endif

  return GSL_SUCCESS;

}

int

gsl_linalg_householder_mh (double tau, const gsl_vector * v, gsl_matrix * A)

{

  /* applies a householder transformation v,tau to matrix m from the

     right hand side in order to zero out rows */

  if (tau == 0)

    return GSL_SUCCESS;

  /* A = A - tau w v' */

#ifdef USE_BLAS

  {

    gsl_vector_const_view v1 = gsl_vector_const_subvector (v, 1, v->size - 1);

    gsl_matrix_view A1 = gsl_matrix_submatrix (A, 0, 1, A->size1, A->size2-1);

    size_t i;

    for (i = 0; i < A->size1; i++)

      {

        double wi = 0.0;

        gsl_vector_view A1i = gsl_matrix_row(&A1.matrix, i);

        gsl_blas_ddot (&A1i.vector, &v1.vector, &wi;;

        wi += gsl_matrix_get(A,i,0);  

        {

          double Ai0 = gsl_matrix_get (A, i, 0);

          gsl_matrix_set (A, i, 0, Ai0 - tau *  wi);

        }

        gsl_blas_daxpy(-tau * wi, &v1.vector, &A1i.vector);

      }

  }

#else

  {

    size_t i, j;

    for (i = 0; i < A->size1; i++)

      {

        double wi = gsl_matrix_get(A,i,0);  

        for (j = 1; j < A->size2; j++)  /* note, computed for v(0) = 1 above */

          {

            wi += gsl_matrix_get(A,i,j) * gsl_vector_get(v,j);

          }

        /* j = 0 */

        {

          double Ai0 = gsl_matrix_get (A, i, 0);

          gsl_matrix_set (A, i, 0, Ai0 - tau *  wi);

        }

        /* j = 1 .. N-1 */

        for (j = 1; j < A->size2; j++) 

          {

            double vj = gsl_vector_get (v, j);

            double Aij = gsl_matrix_get (A, i, j);

            gsl_matrix_set (A, i, j, Aij - tau * wi * vj);

          }

      }

  }

#endif

  return GSL_SUCCESS;

}

int

gsl_linalg_householder_hv (double tau, const gsl_vector * v, gsl_vector * w)

{

  /* applies a householder transformation v to vector w */

  const size_t N = v->size;

  if (tau == 0)

    return GSL_SUCCESS ;

  {

    /* compute d = v'w */

    double w0 = gsl_vector_get(w,0);

    double d1, d;

    gsl_vector_const_view v1 = gsl_vector_const_subvector(v, 1, N-1);

    gsl_vector_view w1 = gsl_vector_subvector(w, 1, N-1);

    /* compute d1 = v(2:n)'w(2:n) */

    gsl_blas_ddot (&v1.vector, &w1.vector, &d1;;

    /* compute d = v'w = w(1) + d1 since v(1) = 1 */

    d = w0 + d1;

    /* compute w = w - tau (v) (v'w) */

    gsl_vector_set (w, 0, w0 - tau * d);

    gsl_blas_daxpy (-tau * d, &v1.vector, &w1.vector);

  }

  return GSL_SUCCESS;

}

int

gsl_linalg_householder_hm1 (double tau, gsl_matrix * A)

{

  /* applies a householder transformation v,tau to a matrix being

     build up from the identity matrix, using the first column of A as

     a householder vector */

  if (tau == 0)

    {

      size_t i,j;

      gsl_matrix_set (A, 0, 0, 1.0);

      for (j = 1; j < A->size2; j++)

        {

          gsl_matrix_set (A, 0, j, 0.0);

        }

      for (i = 1; i < A->size1; i++)

        {

          gsl_matrix_set (A, i, 0, 0.0);

        }

      return GSL_SUCCESS;

    }

  /* w = A' v */

#ifdef USE_BLAS

  {

    gsl_matrix_view A1 = gsl_matrix_submatrix (A, 1, 0, A->size1 - 1, A->size2);

    gsl_vector_view v1 = gsl_matrix_column (&A1.matrix, 0);

    size_t j;

    for (j = 1; j < A->size2; j++)

      {

        double wj = 0.0;   /* A0j * v0 */

        gsl_vector_view A1j = gsl_matrix_column(&A1.matrix, j);

        gsl_blas_ddot (&A1j.vector, &v1.vector, &wj;;

        /* A = A - tau v w' */

        gsl_matrix_set (A, 0, j, - tau *  wj);

        gsl_blas_daxpy(-tau*wj, &v1.vector, &A1j.vector);

      }

    gsl_blas_dscal(-tau, &v1.vector);

    gsl_matrix_set (A, 0, 0, 1.0 - tau);

  }

#else

  {

    size_t i, j;

    for (j = 1; j < A->size2; j++)

      {

        double wj = 0.0;   /* A0j * v0 */

        for (i = 1; i < A->size1; i++)

          {

            double vi = gsl_matrix_get(A, i, 0);

            wj += gsl_matrix_get(A,i,j) * vi;

          }

        /* A = A - tau v w' */

        gsl_matrix_set (A, 0, j, - tau *  wj);

        for (i = 1; i < A->size1; i++)

          {

            double vi = gsl_matrix_get (A, i, 0);

            double Aij = gsl_matrix_get (A, i, j);

            gsl_matrix_set (A, i, j, Aij - tau * vi * wj);

          }

      }

    for (i = 1; i < A->size1; i++)

      {

        double vi = gsl_matrix_get(A, i, 0);

        gsl_matrix_set(A, i, 0, -tau * vi);

      }

    gsl_matrix_set (A, 0, 0, 1.0 - tau);

  }

#endif

  return GSL_SUCCESS;

}