test.c

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/* eigen/test.c
 * 
 * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2006, 2007 Gerard Jungman, Patrick Alken, 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.
 */

/* Author:  G. Jungman
 */

#include <config.h>
#include <stdlib.h>
#include <gsl/gsl_test.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_ieee_utils.h>
#include <gsl/gsl_complex_math.h>
#include <gsl/gsl_eigen.h>
#include <gsl/gsl_linalg.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sort.h>
#include <gsl/gsl_sort_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_vector.h>

/******************************************
 * common test code                       *
 ******************************************/

double 
chop_subnormals (double x) 
{
  /* Chop any subnormal values */
  return fabs(x) < GSL_DBL_MIN ? 0 : x;
}

void
create_random_symm_matrix(gsl_matrix *m, gsl_rng *r, int lower, int upper)
{
  size_t i, j;

  for (i = 0; i < m->size1; ++i)
    {
      for (j = i; j < m->size2; ++j)
      {
        double x = gsl_rng_uniform(r) * (upper - lower) + lower;
        gsl_matrix_set(m, i, j, x);
        gsl_matrix_set(m, j, i, x);
      }
    }
} /* create_random_symm_matrix() */

void
create_random_herm_matrix(gsl_matrix_complex *m, gsl_rng *r, int lower,
                          int upper)
{
  size_t i, j;

  for (i = 0; i < m->size1; ++i)
    {
      for (j = i; j < m->size2; ++j)
      {
        gsl_complex z;

        GSL_REAL(z) = gsl_rng_uniform(r) * (upper - lower) + lower;

        if (i == j)
          GSL_IMAG(z) = 0.0;
        else
          GSL_IMAG(z) = gsl_rng_uniform(r) * (upper - lower) + lower;

        gsl_matrix_complex_set(m, i, j, z);
        gsl_matrix_complex_set(m, j, i, gsl_complex_conjugate(z));
      }
    }
} /* create_random_herm_matrix() */

/* with r \in (0,1) if m_{ij} = r^{|i - j|} then m is positive definite */
void
create_random_posdef_matrix(gsl_matrix *m, gsl_rng *r)
{
  size_t i, j;
  double x = gsl_rng_uniform(r);

  for (i = 0; i < m->size1; ++i)
    {
      for (j = i; j < m->size2; ++j)
      {
        double a = pow(x, (double) (j - i));

        gsl_matrix_set(m, i, j, a);
        gsl_matrix_set(m, j, i, a);
      }
    }
} /* create_random_posdef_matrix() */

void
create_random_complex_posdef_matrix(gsl_matrix_complex *m, gsl_rng *r,
                                    gsl_vector_complex *work)
{
  const size_t N = m->size1;
  size_t i, j;
  double x, y;
  gsl_complex z;
  gsl_complex tau;

  GSL_SET_IMAG(&z, 0.0);

  /* make a positive diagonal matrix */
  gsl_matrix_complex_set_zero(m);
  for (i = 0; i < N; ++i)
    {
      x = gsl_rng_uniform(r);
      GSL_SET_REAL(&z, x);
      gsl_matrix_complex_set(m, i, i, z);
    }

  /* now generate random householder reflections and form P D P^H */
  for (i = 0; i < N; ++i)
    {
      /* form complex vector */
      for (j = 0; j < N; ++j)
        {
          x = 2.0 * gsl_rng_uniform(r) - 1.0;
          y = 2.0 * gsl_rng_uniform(r) - 1.0;
          GSL_SET_COMPLEX(&z, x, y);
          gsl_vector_complex_set(work, j, z);
        }

      tau = gsl_linalg_complex_householder_transform(work);
      gsl_linalg_complex_householder_hm(tau, work, m);
      gsl_linalg_complex_householder_mh(gsl_complex_conjugate(tau), work, m);
    }
} /* create_random_complex_posdef_matrix() */

void
create_random_nonsymm_matrix(gsl_matrix *m, gsl_rng *r, int lower,
                             int upper)
{
  size_t i, j;

  for (i = 0; i < m->size1; ++i)
    {
      for (j = 0; j < m->size2; ++j)
      {
        gsl_matrix_set(m,
                       i,
                       j,
                       gsl_rng_uniform(r) * (upper - lower) + lower);
      }
    }
} /* create_random_nonsymm_matrix() */

/* test if A Z = Q S */
void
test_eigen_schur(const gsl_matrix * A, const gsl_matrix * S,
                 const gsl_matrix * Q, const gsl_matrix * Z,
                 size_t count, const char * desc,
                 const char * desc2)
{
  const size_t N = A->size1;
  size_t i, j;

  gsl_matrix * T1 = gsl_matrix_alloc(N, N);
  gsl_matrix * T2 = gsl_matrix_alloc(N, N);

  /* compute T1 = A Z */
  gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, A, Z, 0.0, T1);

  /* compute T2 = Q S */
  gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, Q, S, 0.0, T2);

  for (i = 0; i < N; ++i)
    {
      for (j = 0; j < N; ++j)
        {
          double x = gsl_matrix_get(T1, i, j);
          double y = gsl_matrix_get(T2, i, j);

          gsl_test_abs(x, y, 1.0e8 * GSL_DBL_EPSILON,
                       "%s(N=%u,cnt=%u), %s, schur(%d,%d)", desc, N, count, desc2, i, j);
        }
    }

  gsl_matrix_free (T1);
  gsl_matrix_free (T2);
} /* test_eigen_schur() */

void
test_eigenvalues_real (const gsl_vector *eval, const gsl_vector * eval2, 
                       const char * desc, const char * desc2)
{
  const size_t N = eval->size;
  size_t i;

  double emax = 0;

  /* check eigenvalues */
  for (i = 0; i < N; i++) 
    {
      double ei = gsl_vector_get (eval, i);
      if (fabs(ei) > emax) emax = fabs(ei);
    }

  for (i = 0; i < N; i++)
    {
      double ei = gsl_vector_get (eval, i);
      double e2i = gsl_vector_get (eval2, i);
      e2i = chop_subnormals(e2i);
      gsl_test_abs(ei, e2i, emax * 1e8 * GSL_DBL_EPSILON, 
                   "%s, direct eigenvalue(%d), %s",
                   desc, i, desc2);
    }
}

void
test_eigenvalues_complex (const gsl_vector_complex * eval,
                          const gsl_vector_complex * eval2, 
                          const char * desc, const char * desc2)
{
  const size_t N = eval->size;
  size_t i;

  for (i = 0; i < N; i++)
    {
      gsl_complex ei = gsl_vector_complex_get (eval, i);
      gsl_complex e2i = gsl_vector_complex_get (eval2, i);
      gsl_test_rel(GSL_REAL(ei), GSL_REAL(e2i), 10*N*GSL_DBL_EPSILON, 
                   "%s, direct eigenvalue(%d) real, %s",
                   desc, i, desc2);
      gsl_test_rel(GSL_IMAG(ei), GSL_IMAG(e2i), 10*N*GSL_DBL_EPSILON, 
                   "%s, direct eigenvalue(%d) imag, %s",
                   desc, i, desc2);
    }
}

/******************************************
 * symm test code                         *
 ******************************************/

void
test_eigen_symm_results (const gsl_matrix * A, 
                         const gsl_vector * eval, 
                         const gsl_matrix * evec, 
                         size_t count,
                         const char * desc,
                         const char * desc2)
{
  const size_t N = A->size1;
  size_t i, j;
  double emax = 0;

  gsl_vector * x = gsl_vector_alloc(N);
  gsl_vector * y = gsl_vector_alloc(N);

  /* check eigenvalues */
  for (i = 0; i < N; i++) 
    {
      double ei = gsl_vector_get (eval, i);
      if (fabs(ei) > emax) emax = fabs(ei);
    }

  for (i = 0; i < N; i++)
    {
      double ei = gsl_vector_get (eval, i);
      gsl_vector_const_view vi = gsl_matrix_const_column(evec, i);
      gsl_vector_memcpy(x, &vi.vector);
      /* compute y = A x (should = lambda v) */
      gsl_blas_dgemv (CblasNoTrans, 1.0, A, x, 0.0, y);
      for (j = 0; j < N; j++)
        {
          double xj = gsl_vector_get (x, j);
          double yj = gsl_vector_get (y, j);
	  double eixj = chop_subnormals(ei * xj);
          gsl_test_abs(yj, eixj,  emax * 1e8 * GSL_DBL_EPSILON, 
                       "%s, eigenvalue(%d,%d), %s", desc, i, j, desc2);
        }
    }

  /* check eigenvectors are orthonormal */

  for (i = 0; i < N; i++)
    {
      gsl_vector_const_view vi = gsl_matrix_const_column(evec, i);
      double nrm_v = gsl_blas_dnrm2(&vi.vector);
      gsl_test_rel (nrm_v, 1.0, N * GSL_DBL_EPSILON, "%s, normalized(%d), %s", 
                    desc, i, desc2);
    }

  for (i = 0; i < N; i++)
    {
      gsl_vector_const_view vi = gsl_matrix_const_column(evec, i);
      for (j = i + 1; j < N; j++)
        {
          gsl_vector_const_view vj = gsl_matrix_const_column(evec, j);
          double vivj;
          gsl_blas_ddot (&vi.vector, &vj.vector, &vivj);
          gsl_test_abs (vivj, 0.0, N * GSL_DBL_EPSILON, 
                        "%s, orthogonal(%d,%d), %s", desc, i, j, desc2);
        }
    }

  gsl_vector_free(x);
  gsl_vector_free(y);
}

void
test_eigen_symm_matrix(const gsl_matrix * m, size_t count,
                       const char * desc)
{
  const size_t N = m->size1;
  gsl_matrix * A = gsl_matrix_alloc(N, N);
  gsl_vector * eval = gsl_vector_alloc(N);
  gsl_vector * evalv = gsl_vector_alloc(N);
  gsl_vector * x = gsl_vector_alloc(N);
  gsl_vector * y = gsl_vector_alloc(N);
  gsl_matrix * evec = gsl_matrix_alloc(N, N);
  gsl_eigen_symm_workspace * w = gsl_eigen_symm_alloc(N);
  gsl_eigen_symmv_workspace * wv = gsl_eigen_symmv_alloc(N);

  gsl_matrix_memcpy(A, m);

  gsl_eigen_symmv(A, evalv, evec, wv);
  test_eigen_symm_results(m, evalv, evec, count, desc, "unsorted");

  gsl_matrix_memcpy(A, m);

  gsl_eigen_symm(A, eval, w);

  /* sort eval and evalv */
  gsl_vector_memcpy(x, eval);
  gsl_vector_memcpy(y, evalv);
  gsl_sort_vector(x);
  gsl_sort_vector(y);
  test_eigenvalues_real(y, x, desc, "unsorted");

  gsl_eigen_symmv_sort(evalv, evec, GSL_EIGEN_SORT_VAL_ASC);
  test_eigen_symm_results(m, evalv, evec, count, desc, "val/asc");

  gsl_eigen_symmv_sort(evalv, evec, GSL_EIGEN_SORT_VAL_DESC);
  test_eigen_symm_results(m, evalv, evec, count, desc, "val/desc");

  gsl_eigen_symmv_sort(evalv, evec, GSL_EIGEN_SORT_ABS_ASC);
  test_eigen_symm_results(m, evalv, evec, count, desc, "abs/asc");

  gsl_eigen_symmv_sort(evalv, evec, GSL_EIGEN_SORT_ABS_DESC);
  test_eigen_symm_results(m, evalv, evec, count, desc, "abs/desc");

  gsl_matrix_free(A);
  gsl_vector_free(eval);
  gsl_vector_free(evalv);
  gsl_vector_free(x);
  gsl_vector_free(y);
  gsl_matrix_free(evec);
  gsl_eigen_symm_free(w);
  gsl_eigen_symmv_free(wv);
} /* test_eigen_symm_matrix() */

void
test_eigen_symm(void)
{
  size_t N_max = 20;
  size_t n, i;
  gsl_rng *r = gsl_rng_alloc(gsl_rng_default);

  for (n = 1; n <= N_max; ++n)
    {
      gsl_matrix * A = gsl_matrix_alloc(n, n);

      for (i = 0; i < 5; ++i)
        {
          create_random_symm_matrix(A, r, -10, 10);
          test_eigen_symm_matrix(A, i, "symm random");
        }

      gsl_matrix_free(A);
    }

  gsl_rng_free(r);

  {
    double dat1[] =  { 0,  0, -1,  0, 
                       0,  1,  0,  1,
                       -1,  0,  0,  0,
                       0,  1,  0,  0 };
    double dat2[] =  { 1,  0,  0,  0, 
                       0,  2,  0,  0,
                       0,  0,  3,  0,
                       0,  0,  0,  4 };
    gsl_matrix_view m;

    m = gsl_matrix_view_array (dat1, 4, 4);
    test_eigen_symm_matrix(&m.matrix, 0, "symm(4)");

    m = gsl_matrix_view_array (dat2, 4, 4);
    test_eigen_symm_matrix(&m.matrix, 0, "symm(4) diag");
  }

  { 
    double dat[27*27] = {
	0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,
	0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,
	0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
	0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,