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/* eigen/hermv.c
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*
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* Copyright (C) 2001, 2007 Brian Gough
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation; either version 3 of the License, or (at
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* your option) any later version.
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*
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* This program is distributed in the hope that it will be useful, but
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* WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program; if not, write to the Free Software
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* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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*/
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#include <config.h>
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#include <stdlib.h>
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#include <gsl/gsl_math.h>
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#include <gsl/gsl_vector.h>
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#include <gsl/gsl_matrix.h>
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#include <gsl/gsl_complex_math.h>
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#include <gsl/gsl_linalg.h>
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#include <gsl/gsl_eigen.h>
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/* Compute eigenvalues/eigenvectors of complex hermitian matrix using
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reduction to real symmetric tridiagonal form, followed by QR
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iteration with implicit shifts.
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See Golub & Van Loan, "Matrix Computations" (3rd ed), Section 8.3 */
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#include "qrstep.c"
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gsl_eigen_hermv_workspace *
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gsl_eigen_hermv_alloc (const size_t n)
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{
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gsl_eigen_hermv_workspace * w ;
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if (n == 0)
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{
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GSL_ERROR_NULL ("matrix dimension must be positive integer", GSL_EINVAL);
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}
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w = (gsl_eigen_hermv_workspace *) malloc (sizeof(gsl_eigen_hermv_workspace));
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if (w == 0)
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{
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GSL_ERROR_NULL ("failed to allocate space for workspace", GSL_ENOMEM);
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}
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w->d = (double *) malloc (n * sizeof (double));
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if (w->d == 0)
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{
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free (w);
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GSL_ERROR_NULL ("failed to allocate space for diagonal", GSL_ENOMEM);
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}
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w->sd = (double *) malloc (n * sizeof (double));
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if (w->sd == 0)
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{
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free (w->d);
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free (w);
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GSL_ERROR_NULL ("failed to allocate space for subdiagonal", GSL_ENOMEM);
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}
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w->tau = (double *) malloc (2 * n * sizeof (double));
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if (w->tau == 0)
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{
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free (w->sd);
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free (w->d);
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free (w);
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GSL_ERROR_NULL ("failed to allocate space for tau", GSL_ENOMEM);
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}
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w->gc = (double *) malloc (n * sizeof (double));
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if (w->gc == 0)
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{
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free (w->tau);
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free (w->sd);
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free (w->d);
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free (w);
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GSL_ERROR_NULL ("failed to allocate space for cosines", GSL_ENOMEM);
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}
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w->gs = (double *) malloc (n * sizeof (double));
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if (w->gs == 0)
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{
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free (w->gc);
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free (w->tau);
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free (w->sd);
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free (w->d);
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free (w);
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GSL_ERROR_NULL ("failed to allocate space for sines", GSL_ENOMEM);
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}
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w->size = n;
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return w;
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}
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void
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gsl_eigen_hermv_free (gsl_eigen_hermv_workspace * w)
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{
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RETURN_IF_NULL (w);
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free (w->gs);
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free (w->gc);
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free (w->tau);
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free (w->sd);
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free (w->d);
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free (w);
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}
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int
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gsl_eigen_hermv (gsl_matrix_complex * A, gsl_vector * eval,
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gsl_matrix_complex * evec,
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gsl_eigen_hermv_workspace * w)
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{
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if (A->size1 != A->size2)
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{
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GSL_ERROR ("matrix must be square to compute eigenvalues", GSL_ENOTSQR);
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}
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else if (eval->size != A->size1)
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{
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GSL_ERROR ("eigenvalue vector must match matrix size", GSL_EBADLEN);
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}
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else if (evec->size1 != A->size1 || evec->size2 != A->size1)
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{
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GSL_ERROR ("eigenvector matrix must match matrix size", GSL_EBADLEN);
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}
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else
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{
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const size_t N = A->size1;
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double *const d = w->d;
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double *const sd = w->sd;
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size_t a, b;
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/* handle special case */
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if (N == 1)
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{
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gsl_complex A00 = gsl_matrix_complex_get (A, 0, 0);
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gsl_vector_set (eval, 0, GSL_REAL(A00));
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gsl_matrix_complex_set (evec, 0, 0, GSL_COMPLEX_ONE);
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return GSL_SUCCESS;
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}
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/* Transform the matrix into a symmetric tridiagonal form */
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{
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gsl_vector_view d_vec = gsl_vector_view_array (d, N);
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gsl_vector_view sd_vec = gsl_vector_view_array (sd, N - 1);
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gsl_vector_complex_view tau_vec = gsl_vector_complex_view_array (w->tau, N-1);
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gsl_linalg_hermtd_decomp (A, &tau_vec.vector);
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gsl_linalg_hermtd_unpack (A, &tau_vec.vector, evec, &d_vec.vector, &sd_vec.vector);
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}
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/* Make an initial pass through the tridiagonal decomposition
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to remove off-diagonal elements which are effectively zero */
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chop_small_elements (N, d, sd);
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/* Progressively reduce the matrix until it is diagonal */
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b = N - 1;
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while (b > 0)
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{
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if (sd[b - 1] == 0.0 || isnan(sd[b - 1]))
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{
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b--;
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continue;
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}
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/* Find the largest unreduced block (a,b) starting from b
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and working backwards */
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a = b - 1;
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while (a > 0)
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{
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if (sd[a - 1] == 0.0)
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{
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break;
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}
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a--;
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}
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{
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size_t i;
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const size_t n_block = b - a + 1;
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double *d_block = d + a;
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double *sd_block = sd + a;
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double * const gc = w->gc;
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double * const gs = w->gs;
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/* apply QR reduction with implicit deflation to the
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unreduced block */
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qrstep (n_block, d_block, sd_block, gc, gs);
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/* Apply Givens rotation Gij(c,s) to matrix Q, Q <- Q G */
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for (i = 0; i < n_block - 1; i++)
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{
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const double c = gc[i], s = gs[i];
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size_t k;
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for (k = 0; k < N; k++)
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{
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gsl_complex qki = gsl_matrix_complex_get (evec, k, a + i);
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gsl_complex qkj = gsl_matrix_complex_get (evec, k, a + i + 1);
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/* qki <= qki * c - qkj * s */
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/* qkj <= qki * s + qkj * c */
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gsl_complex x1 = gsl_complex_mul_real(qki, c);
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gsl_complex y1 = gsl_complex_mul_real(qkj, -s);
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gsl_complex x2 = gsl_complex_mul_real(qki, s);
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gsl_complex y2 = gsl_complex_mul_real(qkj, c);
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gsl_complex qqki = gsl_complex_add(x1, y1);
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gsl_complex qqkj = gsl_complex_add(x2, y2);
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gsl_matrix_complex_set (evec, k, a + i, qqki);
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gsl_matrix_complex_set (evec, k, a + i + 1, qqkj);
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}
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}
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/* remove any small off-diagonal elements */
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chop_small_elements (n_block, d_block, sd_block);
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}
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}
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{
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gsl_vector_view d_vec = gsl_vector_view_array (d, N);
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gsl_vector_memcpy (eval, &d_vec.vector);
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}
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return GSL_SUCCESS;
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}
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}
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