/* 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 #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include /****************************************** * 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() */ /* test if A is orthogonal */ int test_eigen_orthog(gsl_matrix *A, size_t count, const char *desc, const char *desc2) { size_t N = A->size1; gsl_matrix *M = gsl_matrix_alloc(A->size1, A->size2); size_t i, j; gsl_blas_dgemm(CblasNoTrans, CblasTrans, 1.0, A, A, 0.0, M); for (i = 0; i < A->size1; ++i) { for (j = 0; j < A->size2; ++j) { double val; double mij = gsl_matrix_get(M, i, j); if (i == j) val = 1.0; else val = 0.0; gsl_test_abs(mij, val, 1.0e8 * GSL_DBL_EPSILON, "%s(N=%u,cnt=%u), %s, orthog(%d,%d)", desc, N, count, desc2, i, j); } } gsl_matrix_free(M); return 1; } /* test_eigen_orthog() */ 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, 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, 0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 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 }; gsl_matrix_view m; m = gsl_matrix_view_array (dat, 27, 27); test_eigen_symm_matrix(&m.matrix, 0, "symm(27)"); }; } /* test_eigen_symm() */ /****************************************** * herm test code * ******************************************/ void test_eigen_herm_results (const gsl_matrix_complex * A, const gsl_vector * eval, const gsl_matrix_complex * evec, size_t count, const char * desc, const char * desc2) { const size_t N = A->size1; size_t i, j; gsl_vector_complex * x = gsl_vector_complex_alloc(N); gsl_vector_complex * y = gsl_vector_complex_alloc(N); /* check eigenvalues */ for (i = 0; i < N; i++) { double ei = gsl_vector_get (eval, i); gsl_vector_complex_const_view vi = gsl_matrix_complex_const_column(evec, i); gsl_vector_complex_memcpy(x, &vi.vector); /* compute y = m x (should = lambda v) */ gsl_blas_zgemv (CblasNoTrans, GSL_COMPLEX_ONE, A, x, GSL_COMPLEX_ZERO, y); for (j = 0; j < N; j++) { gsl_complex xj = gsl_vector_complex_get (x, j); gsl_complex yj = gsl_vector_complex_get (y, j); gsl_test_rel(GSL_REAL(yj), ei * GSL_REAL(xj), 1e8*GSL_DBL_EPSILON, "%s, eigenvalue(%d,%d), real, %s", desc, i, j, desc2); gsl_test_rel(GSL_IMAG(yj), ei * GSL_IMAG(xj), 1e8*GSL_DBL_EPSILON, "%s, eigenvalue(%d,%d), imag, %s", desc, i, j, desc2); } } /* check eigenvectors are orthonormal */ for (i = 0; i < N; i++) { gsl_vector_complex_const_view vi = gsl_matrix_complex_const_column(evec, i); double nrm_v = gsl_blas_dznrm2(&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_complex_const_view vi = gsl_matrix_complex_const_column(evec, i); for (j = i + 1; j < N; j++) { gsl_vector_complex_const_view vj = gsl_matrix_complex_const_column(evec, j); gsl_complex vivj; gsl_blas_zdotc (&vi.vector, &vj.vector, &vivj); gsl_test_abs (gsl_complex_abs(vivj), 0.0, 10.0 * N * GSL_DBL_EPSILON, "%s, orthogonal(%d,%d), %s", desc, i, j, desc2); } } gsl_vector_complex_free(x); gsl_vector_complex_free(y); } /* test_eigen_herm_results() */ void test_eigen_herm_matrix(const gsl_matrix_complex * m, size_t count, const char * desc) { const size_t N = m->size1; gsl_matrix_complex * A = gsl_matrix_complex_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_complex * evec = gsl_matrix_complex_alloc(N, N); gsl_eigen_herm_workspace * w = gsl_eigen_herm_alloc(N); gsl_eigen_hermv_workspace * wv = gsl_eigen_hermv_alloc(N); gsl_matrix_complex_memcpy(A, m); gsl_eigen_hermv(A, evalv, evec, wv); test_eigen_herm_results(m, evalv, evec, count, desc, "unsorted"); gsl_matrix_complex_memcpy(A, m); gsl_eigen_herm(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_hermv_sort(evalv, evec, GSL_EIGEN_SORT_VAL_ASC); test_eigen_herm_results(m, evalv, evec, count, desc, "val/asc"); gsl_eigen_hermv_sort(evalv, evec, GSL_EIGEN_SORT_VAL_DESC); test_eigen_herm_results(m, evalv, evec, count, desc, "val/desc"); gsl_eigen_hermv_sort(evalv, evec, GSL_EIGEN_SORT_ABS_ASC); test_eigen_herm_results(m, evalv, evec, count, desc, "abs/asc"); gsl_eigen_hermv_sort(evalv, evec, GSL_EIGEN_SORT_ABS_DESC); test_eigen_herm_results(m, evalv, evec, count, desc, "abs/desc"); gsl_matrix_complex_free(A); gsl_vector_free(eval); gsl_vector_free(evalv); gsl_vector_free(x); gsl_vector_free(y); gsl_matrix_complex_free(evec); gsl_eigen_herm_free(w); gsl_eigen_hermv_free(wv); } /* test_eigen_herm_matrix() */ void test_eigen_herm(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_complex * A = gsl_matrix_complex_alloc(n, n); for (i = 0; i < 5; ++i) { create_random_herm_matrix(A, r, -10, 10); test_eigen_herm_matrix(A, i, "herm random"); } gsl_matrix_complex_free(A); } gsl_rng_free(r); { double dat1[] = { 0,0, 0,0, -1,0, 0,0, 0,0, 1,0, 0,0, 1,0, -1,0, 0,0, 0,0, 0,0, 0,0, 1,0, 0,0, 0,0 }; double dat2[] = { 1,0, 0,0, 0,0, 0,0, 0,0, 2,0, 0,0, 0,0, 0,0, 0,0, 3,0, 0,0, 0,0, 0,0, 0,0, 4,0 }; gsl_matrix_complex_view m; m = gsl_matrix_complex_view_array (dat1, 4, 4); test_eigen_herm_matrix(&m.matrix, 0, "herm(4)"); m = gsl_matrix_complex_view_array (dat2, 4, 4); test_eigen_herm_matrix(&m.matrix, 1, "herm(4) diag"); } } /* test_eigen_herm() */ /****************************************** * nonsymm test code * ******************************************/ void test_eigen_nonsymm_results (const gsl_matrix * m, const gsl_vector_complex * eval, const gsl_matrix_complex * evec, size_t count, const char * desc, const char * desc2) { size_t i,j; size_t N = m->size1; gsl_vector_complex * x = gsl_vector_complex_alloc(N); gsl_vector_complex * y = gsl_vector_complex_alloc(N); gsl_matrix_complex * A = gsl_matrix_complex_alloc(N, N); /* we need a complex matrix for the blas routines, so copy m into A */ for (i = 0; i < N; ++i) { for (j = 0; j < N; ++j) { gsl_complex z; GSL_SET_COMPLEX(&z, gsl_matrix_get(m, i, j), 0.0); gsl_matrix_complex_set(A, i, j, z); } } for (i = 0; i < N; i++) { gsl_complex ei = gsl_vector_complex_get (eval, i); gsl_vector_complex_const_view vi = gsl_matrix_complex_const_column(evec, i); double norm = gsl_blas_dznrm2(&vi.vector); /* check that eigenvector is normalized */ gsl_test_rel(norm, 1.0, N * GSL_DBL_EPSILON, "nonsymm(N=%u,cnt=%u), %s, normalized(%d), %s", N, count, desc, i, desc2); gsl_vector_complex_memcpy(x, &vi.vector); /* compute y = m x (should = lambda v) */ gsl_blas_zgemv (CblasNoTrans, GSL_COMPLEX_ONE, A, x, GSL_COMPLEX_ZERO, y); /* compute x = lambda v */ gsl_blas_zscal(ei, x); /* now test if y = x */ for (j = 0; j < N; j++) { gsl_complex xj = gsl_vector_complex_get (x, j); gsl_complex yj = gsl_vector_complex_get (y, j); /* use abs here in case the values are close to 0 */ gsl_test_abs(GSL_REAL(yj), GSL_REAL(xj), 1e8*GSL_DBL_EPSILON, "nonsymm(N=%u,cnt=%u), %s, eigenvalue(%d,%d), real, %s", N, count, desc, i, j, desc2); gsl_test_abs(GSL_IMAG(yj), GSL_IMAG(xj), 1e8*GSL_DBL_EPSILON, "nonsymm(N=%u,cnt=%u), %s, eigenvalue(%d,%d), imag, %s", N, count, desc, i, j, desc2); } } gsl_matrix_complex_free(A); gsl_vector_complex_free(x); gsl_vector_complex_free(y); } void test_eigen_nonsymm_matrix(const gsl_matrix * m, size_t count, const char * desc, gsl_eigen_nonsymmv_workspace *w) { const size_t N = m->size1; gsl_matrix * A = gsl_matrix_alloc(N, N); gsl_matrix * Z = gsl_matrix_alloc(N, N); gsl_matrix_complex * evec = gsl_matrix_complex_alloc(N, N); gsl_vector_complex * eval = gsl_vector_complex_alloc(N); /* * calculate eigenvalues and eigenvectors - it is sufficient to * test gsl_eigen_nonsymmv() since that function calls * gsl_eigen_nonsymm() for the eigenvalues */ gsl_matrix_memcpy(A, m); gsl_eigen_nonsymmv(A, eval, evec, w); test_eigen_nonsymm_results (m, eval, evec, count, desc, "unsorted"); /* test sort routines */ gsl_eigen_nonsymmv_sort (eval, evec, GSL_EIGEN_SORT_ABS_ASC); test_eigen_nonsymm_results (m, eval, evec, count, desc, "abs/asc"); gsl_eigen_nonsymmv_sort (eval, evec, GSL_EIGEN_SORT_ABS_DESC); test_eigen_nonsymm_results (m, eval, evec, count, desc, "abs/desc"); /* test Schur vectors */ gsl_matrix_memcpy(A, m); gsl_eigen_nonsymmv_Z(A, eval, evec, Z, w); gsl_linalg_hessenberg_set_zero(A); test_eigen_schur(m, A, Z, Z, count, "nonsymm", desc); /* test if Z is an orthogonal matrix */ if (w->nonsymm_workspace_p->do_balance == 0) test_eigen_orthog(Z, count, "nonsymm", desc); gsl_matrix_free(A); gsl_matrix_free(Z); gsl_matrix_complex_free(evec); gsl_vector_complex_free(eval); } void test_eigen_nonsymm(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 * m = gsl_matrix_alloc(n, n); gsl_eigen_nonsymmv_workspace * w = gsl_eigen_nonsymmv_alloc(n); for (i = 0; i < 5; ++i) { create_random_nonsymm_matrix(m, r, -10, 10); gsl_eigen_nonsymmv_params(0, w); test_eigen_nonsymm_matrix(m, i, "random, unbalanced", w); gsl_eigen_nonsymmv_params(1, w); test_eigen_nonsymm_matrix(m, i, "random, balanced", w); } gsl_matrix_free(m); gsl_eigen_nonsymmv_free(w); } gsl_rng_free(r); { double dat1[] = { 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 }; double dat2[] = { 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0 }; gsl_matrix_view v; gsl_eigen_nonsymmv_workspace * w = gsl_eigen_nonsymmv_alloc(4); v = gsl_matrix_view_array (dat1, 4, 4); test_eigen_nonsymm_matrix(&v.matrix, 0, "integer", w); v = gsl_matrix_view_array (dat2, 4, 4); test_eigen_nonsymm_matrix(&v.matrix, 1, "integer", w); gsl_eigen_nonsymmv_free(w); } } /* test_eigen_nonsymm() */ /****************************************** * gensymm test code * ******************************************/ void test_eigen_gensymm_results (const gsl_matrix * A, const gsl_matrix * B, 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; gsl_vector * x = gsl_vector_alloc(N); gsl_vector * y = gsl_vector_alloc(N); gsl_vector * z = gsl_vector_alloc(N); /* check A v = lambda B v */ for (i = 0; i < N; i++) { double ei = gsl_vector_get (eval, i); gsl_vector_const_view vi = gsl_matrix_const_column(evec, i); double norm = gsl_blas_dnrm2(&vi.vector); /* check that eigenvector is normalized */ gsl_test_rel(norm, 1.0, N * GSL_DBL_EPSILON, "gensymm(N=%u,cnt=%u), %s, normalized(%d), %s", N, count, desc, i, desc2); gsl_vector_memcpy(z, &vi.vector); /* compute y = A z */ gsl_blas_dgemv (CblasNoTrans, 1.0, A, z, 0.0, y); /* compute x = B z */ gsl_blas_dgemv (CblasNoTrans, 1.0, B, z, 0.0, x); /* compute x = lambda B z */ gsl_blas_dscal(ei, x); /* now test if y = x */ for (j = 0; j < N; j++) { double xj = gsl_vector_get (x, j); double yj = gsl_vector_get (y, j); gsl_test_rel(yj, xj, 1e9 * GSL_DBL_EPSILON, "gensymm(N=%u,cnt=%u), %s, eigenvalue(%d,%d), real, %s", N, count, desc, i, j, desc2); } } gsl_vector_free(x); gsl_vector_free(y); gsl_vector_free(z); } void test_eigen_gensymm(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); gsl_matrix * B = gsl_matrix_alloc(n, n); gsl_matrix * ma = gsl_matrix_alloc(n, n); gsl_matrix * mb = 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_gensymm_workspace * w = gsl_eigen_gensymm_alloc(n); gsl_eigen_gensymmv_workspace * wv = gsl_eigen_gensymmv_alloc(n); for (i = 0; i < 5; ++i) { create_random_symm_matrix(A, r, -10, 10); create_random_posdef_matrix(B, r); gsl_matrix_memcpy(ma, A); gsl_matrix_memcpy(mb, B); gsl_eigen_gensymmv(ma, mb, evalv, evec, wv); test_eigen_gensymm_results(A, B, evalv, evec, i, "random", "unsorted"); gsl_matrix_memcpy(ma, A); gsl_matrix_memcpy(mb, B); gsl_eigen_gensymm(ma, mb, eval, w); /* eval and evalv have to be sorted? not sure why */ gsl_vector_memcpy(x, eval); gsl_vector_memcpy(y, evalv); gsl_sort_vector(x); gsl_sort_vector(y); test_eigenvalues_real(y, x, "gensymm, random", "unsorted"); gsl_eigen_gensymmv_sort(evalv, evec, GSL_EIGEN_SORT_VAL_ASC); test_eigen_gensymm_results(A, B, evalv, evec, i, "random", "val/asc"); gsl_eigen_gensymmv_sort(evalv, evec, GSL_EIGEN_SORT_VAL_DESC); test_eigen_gensymm_results(A, B, evalv, evec, i, "random", "val/desc"); gsl_eigen_gensymmv_sort(evalv, evec, GSL_EIGEN_SORT_ABS_ASC); test_eigen_gensymm_results(A, B, evalv, evec, i, "random", "abs/asc"); gsl_eigen_gensymmv_sort(evalv, evec, GSL_EIGEN_SORT_ABS_DESC); test_eigen_gensymm_results(A, B, evalv, evec, i, "random", "abs/desc"); } gsl_matrix_free(A); gsl_matrix_free(B); gsl_matrix_free(ma); gsl_matrix_free(mb); gsl_vector_free(eval); gsl_vector_free(evalv); gsl_vector_free(x); gsl_vector_free(y); gsl_matrix_free(evec); gsl_eigen_gensymm_free(w); gsl_eigen_gensymmv_free(wv); } gsl_rng_free(r); } /* test_eigen_gensymm() */ /****************************************** * genherm test code * ******************************************/ void test_eigen_genherm_results (const gsl_matrix_complex * A, const gsl_matrix_complex * B, const gsl_vector * eval, const gsl_matrix_complex * evec, size_t count, const char * desc, const char * desc2) { const size_t N = A->size1; size_t i, j; gsl_vector_complex * x = gsl_vector_complex_alloc(N); gsl_vector_complex * y = gsl_vector_complex_alloc(N); /* check A v = lambda B v */ for (i = 0; i < N; i++) { double ei = gsl_vector_get (eval, i); gsl_vector_complex_const_view vi = gsl_matrix_complex_const_column(evec, i); double norm = gsl_blas_dznrm2(&vi.vector); /* check that eigenvector is normalized */ gsl_test_rel(norm, 1.0, N * GSL_DBL_EPSILON, "genherm(N=%u,cnt=%u), %s, normalized(%d), %s", N, count, desc, i, desc2); /* compute y = A z */ gsl_blas_zgemv (CblasNoTrans, GSL_COMPLEX_ONE, A, &vi.vector, GSL_COMPLEX_ZERO, y); /* compute x = B z */ gsl_blas_zgemv (CblasNoTrans, GSL_COMPLEX_ONE, B, &vi.vector, GSL_COMPLEX_ZERO, x); /* compute x = lambda B z */ gsl_blas_zdscal(ei, x); /* now test if y = x */ for (j = 0; j < N; j++) { gsl_complex xj = gsl_vector_complex_get (x, j); gsl_complex yj = gsl_vector_complex_get (y, j); gsl_test_rel(GSL_REAL(yj), GSL_REAL(xj), 1e9 * GSL_DBL_EPSILON, "genherm(N=%u,cnt=%u), %s, eigenvalue(%d,%d), real, %s", N, count, desc, i, j, desc2); gsl_test_abs(GSL_IMAG(yj), GSL_IMAG(xj), 1e9 * GSL_DBL_EPSILON, "genherm(N=%u,cnt=%u), %s, eigenvalue(%d,%d), imag, %s", N, count, desc, i, j, desc2); } } gsl_vector_complex_free(x); gsl_vector_complex_free(y); } void test_eigen_genherm(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_complex * A = gsl_matrix_complex_alloc(n, n); gsl_matrix_complex * B = gsl_matrix_complex_alloc(n, n); gsl_matrix_complex * ma = gsl_matrix_complex_alloc(n, n); gsl_matrix_complex * mb = gsl_matrix_complex_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_vector_complex * work = gsl_vector_complex_alloc(n); gsl_matrix_complex * evec = gsl_matrix_complex_alloc(n, n); gsl_eigen_genherm_workspace * w = gsl_eigen_genherm_alloc(n); gsl_eigen_genhermv_workspace * wv = gsl_eigen_genhermv_alloc(n); for (i = 0; i < 5; ++i) { create_random_herm_matrix(A, r, -10, 10); create_random_complex_posdef_matrix(B, r, work); gsl_matrix_complex_memcpy(ma, A); gsl_matrix_complex_memcpy(mb, B); gsl_eigen_genhermv(ma, mb, evalv, evec, wv); test_eigen_genherm_results(A, B, evalv, evec, i, "random", "unsorted"); gsl_matrix_complex_memcpy(ma, A); gsl_matrix_complex_memcpy(mb, B); gsl_eigen_genherm(ma, mb, eval, w); /* eval and evalv have to be sorted? not sure why */ gsl_vector_memcpy(x, eval); gsl_vector_memcpy(y, evalv); gsl_sort_vector(x); gsl_sort_vector(y); test_eigenvalues_real(y, x, "genherm, random", "unsorted"); gsl_eigen_genhermv_sort(evalv, evec, GSL_EIGEN_SORT_VAL_ASC); test_eigen_genherm_results(A, B, evalv, evec, i, "random", "val/asc"); gsl_eigen_genhermv_sort(evalv, evec, GSL_EIGEN_SORT_VAL_DESC); test_eigen_genherm_results(A, B, evalv, evec, i, "random", "val/desc"); gsl_eigen_genhermv_sort(evalv, evec, GSL_EIGEN_SORT_ABS_ASC); test_eigen_genherm_results(A, B, evalv, evec, i, "random", "abs/asc"); gsl_eigen_genhermv_sort(evalv, evec, GSL_EIGEN_SORT_ABS_DESC); test_eigen_genherm_results(A, B, evalv, evec, i, "random", "abs/desc"); } gsl_matrix_complex_free(A); gsl_matrix_complex_free(B); gsl_matrix_complex_free(ma); gsl_matrix_complex_free(mb); gsl_vector_free(eval); gsl_vector_free(evalv); gsl_vector_free(x); gsl_vector_free(y); gsl_vector_complex_free(work); gsl_matrix_complex_free(evec); gsl_eigen_genherm_free(w); gsl_eigen_genhermv_free(wv); } gsl_rng_free(r); } /* test_eigen_genherm() */ /****************************************** * gen test code * ******************************************/ typedef struct { gsl_matrix *A; gsl_matrix *B; gsl_vector_complex *alpha; gsl_vector *beta; gsl_vector_complex *alphav; gsl_vector *betav; gsl_vector_complex *eval; gsl_vector_complex *evalv; gsl_vector *x; gsl_vector *y; gsl_matrix *Q; gsl_matrix *Z; gsl_matrix_complex *evec; gsl_eigen_gen_workspace *gen_p; gsl_eigen_genv_workspace *genv_p; } test_eigen_gen_workspace; test_eigen_gen_workspace * test_eigen_gen_alloc(const size_t n) { test_eigen_gen_workspace *w; w = (test_eigen_gen_workspace *) calloc(1, sizeof(test_eigen_gen_workspace)); w->A = gsl_matrix_alloc(n, n); w->B = gsl_matrix_alloc(n, n); w->alpha = gsl_vector_complex_alloc(n); w->beta = gsl_vector_alloc(n); w->alphav = gsl_vector_complex_alloc(n); w->betav = gsl_vector_alloc(n); w->eval = gsl_vector_complex_alloc(n); w->evalv = gsl_vector_complex_alloc(n); w->x = gsl_vector_alloc(n); w->y = gsl_vector_alloc(n); w->Q = gsl_matrix_alloc(n, n); w->Z = gsl_matrix_alloc(n, n); w->evec = gsl_matrix_complex_alloc(n, n); w->gen_p = gsl_eigen_gen_alloc(n); w->genv_p = gsl_eigen_genv_alloc(n); return (w); } /* test_eigen_gen_alloc() */ void test_eigen_gen_free(test_eigen_gen_workspace *w) { gsl_matrix_free(w->A); gsl_matrix_free(w->B); gsl_vector_complex_free(w->alpha); gsl_vector_free(w->beta); gsl_vector_complex_free(w->alphav); gsl_vector_free(w->betav); gsl_vector_complex_free(w->eval); gsl_vector_complex_free(w->evalv); gsl_vector_free(w->x); gsl_vector_free(w->y); gsl_matrix_free(w->Q); gsl_matrix_free(w->Z); gsl_matrix_complex_free(w->evec); gsl_eigen_gen_free(w->gen_p); gsl_eigen_genv_free(w->genv_p); free(w); } /* test_eigen_gen_free() */ void test_eigen_gen_results (const gsl_matrix * A, const gsl_matrix * B, const gsl_vector_complex * alpha, const gsl_vector * beta, const gsl_matrix_complex * evec, size_t count, const char * desc, const char * desc2) { const size_t N = A->size1; size_t i, j; gsl_matrix_complex *ma, *mb; gsl_vector_complex *x, *y; gsl_complex z_one, z_zero; ma = gsl_matrix_complex_alloc(N, N); mb = gsl_matrix_complex_alloc(N, N); y = gsl_vector_complex_alloc(N); x = gsl_vector_complex_alloc(N); /* ma <- A, mb <- B */ for (i = 0; i < N; ++i) { for (j = 0; j < N; ++j) { gsl_complex z; GSL_SET_COMPLEX(&z, gsl_matrix_get(A, i, j), 0.0); gsl_matrix_complex_set(ma, i, j, z); GSL_SET_COMPLEX(&z, gsl_matrix_get(B, i, j), 0.0); gsl_matrix_complex_set(mb, i, j, z); } } GSL_SET_COMPLEX(&z_one, 1.0, 0.0); GSL_SET_COMPLEX(&z_zero, 0.0, 0.0); /* check eigenvalues */ for (i = 0; i < N; ++i) { gsl_vector_complex_const_view vi = gsl_matrix_complex_const_column(evec, i); gsl_complex ai = gsl_vector_complex_get(alpha, i); double bi = gsl_vector_get(beta, i); /* compute x = alpha * B * v */ gsl_blas_zgemv(CblasNoTrans, z_one, mb, &vi.vector, z_zero, x); gsl_blas_zscal(ai, x); /* compute y = beta * A v */ gsl_blas_zgemv(CblasNoTrans, z_one, ma, &vi.vector, z_zero, y); gsl_blas_zdscal(bi, y); /* now test if y = x */ for (j = 0; j < N; ++j) { gsl_complex xj = gsl_vector_complex_get(x, j); gsl_complex yj = gsl_vector_complex_get(y, j); gsl_test_abs(GSL_REAL(yj), GSL_REAL(xj), 1e8*GSL_DBL_EPSILON, "gen(N=%u,cnt=%u), %s, eigenvalue(%d,%d), real, %s", N, count, desc, i, j, desc2); gsl_test_abs(GSL_IMAG(yj), GSL_IMAG(xj), 1e8*GSL_DBL_EPSILON, "gen(N=%u,cnt=%u), %s, eigenvalue(%d,%d), real, %s", N, count, desc, i, j, desc2); } } gsl_matrix_complex_free(ma); gsl_matrix_complex_free(mb); gsl_vector_complex_free(y); gsl_vector_complex_free(x); } /* test_eigen_gen_results() */ void test_eigen_gen_pencil(const gsl_matrix * A, const gsl_matrix * B, size_t count, const char * desc, int test_schur, test_eigen_gen_workspace *w) { const size_t N = A->size1; size_t i; gsl_matrix_memcpy(w->A, A); gsl_matrix_memcpy(w->B, B); if (test_schur) { gsl_eigen_genv_QZ(w->A, w->B, w->alphav, w->betav, w->evec, w->Q, w->Z, w->genv_p); test_eigen_schur(A, w->A, w->Q, w->Z, count, "genv/A", desc); test_eigen_schur(B, w->B, w->Q, w->Z, count, "genv/B", desc); } else gsl_eigen_genv(w->A, w->B, w->alphav, w->betav, w->evec, w->genv_p); test_eigen_gen_results(A, B, w->alphav, w->betav, w->evec, count, desc, "unsorted"); gsl_matrix_memcpy(w->A, A); gsl_matrix_memcpy(w->B, B); if (test_schur) { gsl_eigen_gen_params(1, 1, 0, w->gen_p); gsl_eigen_gen_QZ(w->A, w->B, w->alpha, w->beta, w->Q, w->Z, w->gen_p); test_eigen_schur(A, w->A, w->Q, w->Z, count, "gen/A", desc); test_eigen_schur(B, w->B, w->Q, w->Z, count, "gen/B", desc); } else { gsl_eigen_gen_params(0, 0, 0, w->gen_p); gsl_eigen_gen(w->A, w->B, w->alpha, w->beta, w->gen_p); } /* compute eval = alpha / beta values */ for (i = 0; i < N; ++i) { gsl_complex z, ai; double bi; ai = gsl_vector_complex_get(w->alpha, i); bi = gsl_vector_get(w->beta, i); GSL_SET_COMPLEX(&z, GSL_REAL(ai) / bi, GSL_IMAG(ai) / bi); gsl_vector_complex_set(w->eval, i, z); ai = gsl_vector_complex_get(w->alphav, i); bi = gsl_vector_get(w->betav, i); GSL_SET_COMPLEX(&z, GSL_REAL(ai) / bi, GSL_IMAG(ai) / bi); gsl_vector_complex_set(w->evalv, i, z); } /* sort eval and evalv and test them */ gsl_eigen_nonsymmv_sort(w->eval, NULL, GSL_EIGEN_SORT_VAL_ASC); gsl_eigen_nonsymmv_sort(w->evalv, NULL, GSL_EIGEN_SORT_VAL_ASC); test_eigenvalues_complex(w->evalv, w->eval, "gen", desc); gsl_eigen_genv_sort(w->alphav, w->betav, w->evec, GSL_EIGEN_SORT_ABS_ASC); test_eigen_gen_results(A, B, w->alphav, w->betav, w->evec, count, desc, "abs/asc"); gsl_eigen_genv_sort(w->alphav, w->betav, w->evec, GSL_EIGEN_SORT_ABS_DESC); test_eigen_gen_results(A, B, w->alphav, w->betav, w->evec, count, desc, "abs/desc"); } /* test_eigen_gen_pencil() */ void test_eigen_gen(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); gsl_matrix * B = gsl_matrix_alloc(n, n); test_eigen_gen_workspace * w = test_eigen_gen_alloc(n); for (i = 0; i < 5; ++i) { create_random_nonsymm_matrix(A, r, -10, 10); create_random_nonsymm_matrix(B, r, -10, 10); test_eigen_gen_pencil(A, B, i, "random", 0, w); test_eigen_gen_pencil(A, B, i, "random", 1, w); } gsl_matrix_free(A); gsl_matrix_free(B); test_eigen_gen_free(w); } gsl_rng_free(r); /* this system will test the exceptional shift code */ { double datA[] = { 1, 1, 0, 0, 0, -1, 1, 0, 0 }; double datB[] = { -1, 0, -1, 0, -1, 0, 0, 0, -1 }; gsl_matrix_view va = gsl_matrix_view_array (datA, 3, 3); gsl_matrix_view vb = gsl_matrix_view_array (datB, 3, 3); test_eigen_gen_workspace * w = test_eigen_gen_alloc(3); test_eigen_gen_pencil(&va.matrix, &vb.matrix, 0, "integer", 0, w); test_eigen_gen_pencil(&va.matrix, &vb.matrix, 0, "integer", 1, w); test_eigen_gen_free(w); } } /* test_eigen_gen() */ int main() { gsl_ieee_env_setup (); gsl_rng_env_setup (); test_eigen_symm(); test_eigen_herm(); test_eigen_nonsymm(); test_eigen_gensymm(); test_eigen_genherm(); test_eigen_gen(); exit (gsl_test_summary()); }