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/* linalg/householdercomplex.c
*
* Copyright (C) 2001 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 2 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.
*/
/* Computes a householder transformation matrix H such that
*
* H' v = -/+ |v| e_1
*
* where e_1 is the first unit vector. On exit the matrix H can be
* computed from the return values (tau, v)
*
* H = I - tau * w * w'
*
* where w = (1, v(2), ..., v(N)). The nonzero element of the result
* vector -/+|v| e_1 is stored in v(1).
*
* Note that the matrix H' in the householder transformation is the
* hermitian conjugate of H. To compute H'v, pass the conjugate of
* tau as the first argument to gsl_linalg_householder_hm() rather
* than tau itself. See the LAPACK function CLARFG for details of this
* convention. */
#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_complex_math.h>
#include <gsl/gsl_linalg.h>
gsl_complex
gsl_linalg_complex_householder_transform (gsl_vector_complex * 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)
{
gsl_complex alpha = gsl_vector_complex_get (v, 0) ;
double absa = gsl_complex_abs (alpha);
double beta_r = - (GSL_REAL(alpha) >= 0 ? +1 : -1) * absa ;
gsl_complex tau;
if (beta_r == 0.0)
{
GSL_REAL(tau) = 0.0;
GSL_IMAG(tau) = 0.0;
}
else
{
GSL_REAL(tau) = (beta_r - GSL_REAL(alpha)) / beta_r ;
GSL_IMAG(tau) = - GSL_IMAG(alpha) / beta_r ;
{
gsl_complex beta = gsl_complex_rect (beta_r, 0.0);
gsl_vector_complex_set (v, 0, beta) ;
}
}
return tau;
}
else
{
gsl_complex tau ;
double beta_r;
gsl_vector_complex_view x = gsl_vector_complex_subvector (v, 1, n - 1) ;
gsl_complex alpha = gsl_vector_complex_get (v, 0) ;
double absa = gsl_complex_abs (alpha);
double xnorm = gsl_blas_dznrm2 (&x.vector);
if (xnorm == 0 && GSL_IMAG(alpha) == 0)
{
gsl_complex zero = gsl_complex_rect(0.0, 0.0);
return zero; /* tau = 0 */
}
beta_r = - (GSL_REAL(alpha) >= 0 ? +1 : -1) * hypot(absa, xnorm) ;
GSL_REAL(tau) = (beta_r - GSL_REAL(alpha)) / beta_r ;
GSL_IMAG(tau) = - GSL_IMAG(alpha) / beta_r ;
{
gsl_complex amb = gsl_complex_sub_real(alpha, beta_r);
gsl_complex s = gsl_complex_inverse(amb);
gsl_blas_zscal (s, &x.vector);
}
{
gsl_complex beta = gsl_complex_rect (beta_r, 0.0);
gsl_vector_complex_set (v, 0, beta) ;
}
return tau;
}
}
int
gsl_linalg_complex_householder_hm (gsl_complex tau, const gsl_vector_complex * v, gsl_matrix_complex * A)
{
/* applies a householder transformation v,tau to matrix m */
size_t i, j;
if (GSL_REAL(tau) == 0.0 && GSL_IMAG(tau) == 0.0)
{
return GSL_SUCCESS;
}
/* w = (v' A)^T */
for (j = 0; j < A->size2; j++)
{
gsl_complex tauwj;
gsl_complex wj = gsl_matrix_complex_get(A,0,j);
for (i = 1; i < A->size1; i++) /* note, computed for v(0) = 1 above */
{
gsl_complex Aij = gsl_matrix_complex_get(A,i,j);
gsl_complex vi = gsl_vector_complex_get(v,i);
gsl_complex Av = gsl_complex_mul (Aij, gsl_complex_conjugate(vi));
wj = gsl_complex_add (wj, Av);
}
tauwj = gsl_complex_mul (tau, wj);
/* A = A - v w^T */
{
gsl_complex A0j = gsl_matrix_complex_get (A, 0, j);
gsl_complex Atw = gsl_complex_sub (A0j, tauwj);
/* store A0j - tau * wj */
gsl_matrix_complex_set (A, 0, j, Atw);
}
for (i = 1; i < A->size1; i++)
{
gsl_complex vi = gsl_vector_complex_get (v, i);
gsl_complex tauvw = gsl_complex_mul(vi, tauwj);
gsl_complex Aij = gsl_matrix_complex_get (A, i, j);
gsl_complex Atwv = gsl_complex_sub (Aij, tauvw);
/* store Aij - tau * vi * wj */
gsl_matrix_complex_set (A, i, j, Atwv);
}
}
return GSL_SUCCESS;
}
int
gsl_linalg_complex_householder_hv (gsl_complex tau, const gsl_vector_complex * v, gsl_vector_complex * w)
{
const size_t N = v->size;
if (GSL_REAL(tau) == 0.0 && GSL_IMAG(tau) == 0.0)
return GSL_SUCCESS;
{
/* compute z = v'w */
gsl_complex z0 = gsl_vector_complex_get(w,0);
gsl_complex z1, z;
gsl_complex tz, ntz;
gsl_vector_complex_const_view v1 = gsl_vector_complex_const_subvector(v, 1, N-1);
gsl_vector_complex_view w1 = gsl_vector_complex_subvector(w, 1, N-1);
gsl_blas_zdotc(&v1.vector, &w1.vector, &z1);
z = gsl_complex_add (z0, z1);
tz = gsl_complex_mul(tau, z);
ntz = gsl_complex_negative (tz);
/* compute w = w - tau * (v'w) * v */
{
gsl_complex w0 = gsl_vector_complex_get(w, 0);
gsl_complex w0ntz = gsl_complex_add (w0, ntz);
gsl_vector_complex_set (w, 0, w0ntz);
}
gsl_blas_zaxpy(ntz, &v1.vector, &w1.vector);
}
return GSL_SUCCESS;
}
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