void arb_mat_mul_classical(arb_mat_t C, const arb_mat_t A, const arb_mat_t B, long prec) { long ar, ac, br, bc, i, j, k; ar = arb_mat_nrows(A); ac = arb_mat_ncols(A); br = arb_mat_nrows(B); bc = arb_mat_ncols(B); if (ac != br || ar != arb_mat_nrows(C) || bc != arb_mat_ncols(C)) { printf("arb_mat_mul: incompatible dimensions\n"); abort(); } if (br == 0) { arb_mat_zero(C); return; } if (A == C || B == C) { arb_mat_t T; arb_mat_init(T, ar, bc); arb_mat_mul(T, A, B, prec); arb_mat_swap(T, C); arb_mat_clear(T); return; } for (i = 0; i < ar; i++) { for (j = 0; j < bc; j++) { arb_mul(arb_mat_entry(C, i, j), arb_mat_entry(A, i, 0), arb_mat_entry(B, 0, j), prec); for (k = 1; k < br; k++) { arb_addmul(arb_mat_entry(C, i, j), arb_mat_entry(A, i, k), arb_mat_entry(B, k, j), prec); } } } }
void arb_mat_pow_ui(arb_mat_t B, const arb_mat_t A, ulong exp, slong prec) { slong d = arb_mat_nrows(A); if (exp <= 2 || d <= 1) { if (exp == 0 || d == 0) { arb_mat_one(B); } else if (d == 1) { arb_pow_ui(arb_mat_entry(B, 0, 0), arb_mat_entry(A, 0, 0), exp, prec); } else if (exp == 1) { arb_mat_set(B, A); } else if (exp == 2) { arb_mat_sqr(B, A, prec); } } else { arb_mat_t T, U; slong i; arb_mat_init(T, d, d); arb_mat_set(T, A); arb_mat_init(U, d, d); for (i = ((slong) FLINT_BIT_COUNT(exp)) - 2; i >= 0; i--) { arb_mat_sqr(U, T, prec); if (exp & (WORD(1) << i)) arb_mat_mul(T, U, A, prec); else arb_mat_swap(T, U); } arb_mat_swap(B, T); arb_mat_clear(T); arb_mat_clear(U); } }
int arb_mat_generalized_eigenproblem_symmetric_positive_definite(arb_mat_t D, arb_mat_t P, const arb_mat_t A, const arb_mat_t B, slong prec) { // solve the generalized eigenvalue problem Ax = lambda * Bx, where B // is symmetric positive definite and A is symmetric. // // Returns 0 on success. // // If there is a problem inverting the Cholesky factorization of B, returns -1. // If the eigenvalues are not distinct (or cannot be certified as distinct), // returns 1. In this case the returned eigenvalues will be correct, but // some eigenvectors will be nonsense, or will be accurate eigenvectors but with // infinite radius, or something like that. int dim = arb_mat_nrows(B); arb_mat_t L; arb_mat_t X; arb_mat_init(L, dim, dim); arb_mat_init(X, dim, dim); arb_mat_cholesky(L, B, prec); int result = arb_mat_inv(L, L, prec); if(!result) { arb_mat_clear(L); arb_mat_clear(X); return -1; } arb_mat_mul(X, L, A, prec); arb_mat_transpose(L, L); arb_mat_mul(X, X, L, prec); result = arb_mat_jacobi(D, P, X, prec); arb_mat_mul(P, L, P, prec); //arb_mat_transpose(P, P); arb_mat_clear(L); arb_mat_clear(X); return result; }
void evaluate_site_frechet( arb_struct *lhood_scaled_edge_expectations, const arb_mat_struct *lhood_node_vectors, const arb_mat_struct *forward_edge_vectors, const arb_mat_struct *frechet_matrices, const csr_graph_t g, int *preorder, int node_count, int state_count, slong prec) { slong u, idx, state; arb_mat_t fvec; arb_mat_init(fvec, state_count, 1); for (u = 0; u < node_count; u++) { slong a = preorder[u]; slong start = g->indptr[a]; slong stop = g->indptr[a+1]; for (idx = start; idx < stop; idx++) { slong b = g->indices[idx]; const arb_mat_struct *lvec = lhood_node_vectors + b; const arb_mat_struct *evec = forward_edge_vectors + idx; arb_zero(lhood_scaled_edge_expectations + idx); arb_mat_mul(fvec, frechet_matrices + idx, lvec, prec); for (state = 0; state < state_count; state++) { arb_addmul(lhood_scaled_edge_expectations + idx, arb_mat_entry(fvec, state, 0), arb_mat_entry(evec, state, 0), prec); } } } arb_mat_clear(fvec); }
int main() { slong iter; flint_rand_t state; flint_printf("ldl...."); fflush(stdout); flint_randinit(state); /* check special matrices */ { slong n; for (n = 1; n < 10; n++) { slong lprec; arb_mat_t L, A; arb_mat_init(L, n, n); arb_mat_init(A, n, n); for (lprec = 2; lprec < 10; lprec++) { int result; slong prec; prec = 1 << lprec; /* zero */ arb_mat_zero(A); result = arb_mat_ldl(L, A, prec); if (result) { flint_printf("FAIL (zero):\n"); flint_printf("n = %wd, prec = %wd\n", n, prec); flint_printf("L = \n"); arb_mat_printd(L, 15); flint_printf("\n\n"); } /* negative identity */ arb_mat_one(A); arb_mat_neg(A, A); result = arb_mat_ldl(L, A, prec); if (result) { flint_printf("FAIL (negative identity):\n"); flint_printf("n = %wd, prec = %wd\n", n, prec); flint_printf("L = \n"); arb_mat_printd(L, 15); flint_printf("\n\n"); } /* identity */ arb_mat_one(A); result = arb_mat_ldl(L, A, prec); if (!result || !arb_mat_equal(L, A)) { flint_printf("FAIL (identity):\n"); flint_printf("n = %wd, prec = %wd\n", n, prec); flint_printf("L = \n"); arb_mat_printd(L, 15); flint_printf("\n\n"); } } arb_mat_clear(L); arb_mat_clear(A); } } for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++) { fmpq_mat_t Q; arb_mat_t A, L, D, U, T; slong n, qbits, prec; int q_invertible, r_invertible; n = n_randint(state, 8); qbits = 1 + n_randint(state, 100); prec = 2 + n_randint(state, 202); fmpq_mat_init(Q, n, n); arb_mat_init(A, n, n); arb_mat_init(L, n, n); arb_mat_init(D, n, n); arb_mat_init(U, n, n); arb_mat_init(T, n, n); _fmpq_mat_randtest_positive_semidefinite(Q, state, qbits); q_invertible = fmpq_mat_is_invertible(Q); if (!q_invertible) { arb_mat_set_fmpq_mat(A, Q, prec); r_invertible = arb_mat_ldl(L, A, prec); if (r_invertible) { flint_printf("FAIL: matrix is singular over Q but not over R\n"); flint_printf("n = %wd, prec = %wd\n", n, prec); flint_printf("\n"); flint_printf("Q = \n"); fmpq_mat_print(Q); flint_printf("\n\n"); flint_printf("A = \n"); arb_mat_printd(A, 15); flint_printf("\n\n"); flint_printf("L = \n"); arb_mat_printd(L, 15); flint_printf("\n\n"); } } else { /* now this must converge */ while (1) { arb_mat_set_fmpq_mat(A, Q, prec); r_invertible = arb_mat_ldl(L, A, prec); if (r_invertible) { break; } else { if (prec > 10000) { flint_printf("FAIL: failed to converge at 10000 bits\n"); flint_printf("n = %wd, prec = %wd\n", n, prec); flint_printf("Q = \n"); fmpq_mat_print(Q); flint_printf("\n\n"); flint_printf("A = \n"); arb_mat_printd(A, 15); flint_printf("\n\n"); abort(); } prec *= 2; } } /* multiply out the decomposition */ { slong i; arb_mat_zero(D); arb_mat_transpose(U, L); for (i = 0; i < n; i++) { arb_set(arb_mat_entry(D, i, i), arb_mat_entry(L, i, i)); arb_one(arb_mat_entry(L, i, i)); arb_one(arb_mat_entry(U, i, i)); } arb_mat_mul(T, L, D, prec); arb_mat_mul(T, T, U, prec); } if (!arb_mat_contains_fmpq_mat(T, Q)) { flint_printf("FAIL (containment, iter = %wd)\n", iter); flint_printf("n = %wd, prec = %wd\n", n, prec); flint_printf("\n"); flint_printf("Q = \n"); fmpq_mat_print(Q); flint_printf("\n\n"); flint_printf("A = \n"); arb_mat_printd(A, 15); flint_printf("\n\n"); flint_printf("L = \n"); arb_mat_printd(L, 15); flint_printf("\n\n"); flint_printf("U = \n"); arb_mat_printd(U, 15); flint_printf("\n\n"); flint_printf("L*U = \n"); arb_mat_printd(T, 15); flint_printf("\n\n"); abort(); } } fmpq_mat_clear(Q); arb_mat_clear(A); arb_mat_clear(L); arb_mat_clear(D); arb_mat_clear(U); arb_mat_clear(T); } flint_randclear(state); flint_cleanup(); flint_printf("PASS\n"); return EXIT_SUCCESS; }
int arb_mat_jacobi(arb_mat_t D, arb_mat_t P, const arb_mat_t A, slong prec) { // // Given a d x d real symmetric matrix A, compute an orthogonal matrix // P and a diagonal D such that A = P D P^t = P D P^(-1). // // D should have already been initialized as a d x 1 matrix, and Pp // should have already been initialized as a d x d matrix. // // If the eigenvalues can be certified as unique, then a nonzero int is // returned, and the eigenvectors should have reasonable error bounds. If // the eigenvalues cannot be certified as unique, then some of the // eigenvectors will have infinite error radius. #define B(i,j) arb_mat_entry(B, i, j) #define D(i) arb_mat_entry(D, i, 0) #define P(i,j) arb_mat_entry(P, i, j) int dim = arb_mat_nrows(A); if(dim == 1) { arb_mat_set(D, A); arb_mat_one(P); return 0; } arb_mat_t B; arb_mat_init(B, dim, dim); arf_t * B1 = (arf_t*)malloc(dim * sizeof(arf_t)); arf_t * B2 = (arf_t*)malloc(dim * sizeof(arf_t)); arf_t * row_max = (arf_t*)malloc((dim - 1) * sizeof(arf_t)); int * row_max_indices = (int*)malloc((dim - 1) * sizeof(int)); for(int k = 0; k < dim; k++) { arf_init(B1[k]); arf_init(B2[k]); } for(int k = 0; k < dim - 1; k++) { arf_init(row_max[k]); } arf_t x1, x2; arf_init(x1); arf_init(x2); arf_t Gii, Gij, Gji, Gjj; arf_init(Gii); arf_init(Gij); arf_init(Gji); arf_init(Gjj); arb_mat_set(B, A); arb_mat_one(P); for(int i = 0; i < dim - 1; i++) { for(int j = i + 1; j < dim; j++) { arf_abs(x1, arb_midref(B(i,j))); if(arf_cmp(row_max[i], x1) < 0) { arf_set(row_max[i], x1); row_max_indices[i] = j; } } } int finished = 0; while(!finished) { arf_zero(x1); int i = 0; int j = 0; for(int k = 0; k < dim - 1; k++) { if(arf_cmp(x1, row_max[k]) < 0) { arf_set(x1, row_max[k]); i = k; } } j = row_max_indices[i]; slong bound = arf_abs_bound_lt_2exp_si(x1); if(bound < -prec * .9) { finished = 1; break; } else { //printf("%ld\n", arf_abs_bound_lt_2exp_si(x1)); //arb_mat_printd(B, 10); //printf("\n"); } arf_twobytwo_diag(Gii, Gij, arb_midref(B(i,i)), arb_midref(B(i,j)), arb_midref(B(j,j)), 2*prec); arf_neg(Gji, Gij); arf_set(Gjj, Gii); //printf("%d %d\n", i, j); //arf_printd(Gii, 100); //printf(" "); //arf_printd(Gij, 100); //printf("\n"); if(arf_is_zero(Gij)) { // If this happens, we're finished = 1; // not going to do any better break; // without increasing the precision. } for(int k = 0; k < dim; k++) { arf_mul(B1[k], Gii, arb_midref(B(i,k)), prec, ARF_RND_NEAR); arf_addmul(B1[k], Gji, arb_midref(B(j,k)), prec, ARF_RND_NEAR); arf_mul(B2[k], Gij, arb_midref(B(i,k)), prec, ARF_RND_NEAR); arf_addmul(B2[k], Gjj, arb_midref(B(j,k)), prec, ARF_RND_NEAR); } for(int k = 0; k < dim; k++) { arf_set(arb_midref(B(i,k)), B1[k]); arf_set(arb_midref(B(j,k)), B2[k]); } for(int k = 0; k < dim; k++) { arf_mul(B1[k], Gii, arb_midref(B(k,i)), prec, ARF_RND_NEAR); arf_addmul(B1[k], Gji, arb_midref(B(k,j)), prec, ARF_RND_NEAR); arf_mul(B2[k], Gij, arb_midref(B(k,i)), prec, ARF_RND_NEAR); arf_addmul(B2[k], Gjj, arb_midref(B(k,j)), prec, ARF_RND_NEAR); } for(int k = 0; k < dim; k++) { arf_set(arb_midref(B(k,i)), B1[k]); arf_set(arb_midref(B(k,j)), B2[k]); } for(int k = 0; k < dim; k++) { arf_mul(B1[k], Gii, arb_midref(P(k,i)), prec, ARF_RND_NEAR); arf_addmul(B1[k], Gji, arb_midref(P(k,j)), prec, ARF_RND_NEAR); arf_mul(B2[k], Gij, arb_midref(P(k,i)), prec, ARF_RND_NEAR); arf_addmul(B2[k], Gjj, arb_midref(P(k,j)), prec, ARF_RND_NEAR); } for(int k = 0; k < dim; k++) { arf_set(arb_midref(P(k,i)), B1[k]); arf_set(arb_midref(P(k,j)), B2[k]); } if(i < dim - 1) arf_set_ui(row_max[i], 0); if(j < dim - 1) arf_set_ui(row_max[j], 0); // Update the max in any row where the maximum // was in a column that changed. for(int k = 0; k < dim - 1; k++) { if(row_max_indices[k] == j || row_max_indices[k] == i) { arf_abs(row_max[k], arb_midref(B(k,k+1))); row_max_indices[k] = k+1; for(int l = k+2; l < dim; l++) { arf_abs(x1, arb_midref(B(k,l))); if(arf_cmp(row_max[k], x1) < 0) { arf_set(row_max[k], x1); row_max_indices[k] = l; } } } } // Update the max in the ith row. for(int k = i + 1; k < dim; k++) { arf_abs(x1, arb_midref(B(i, k))); if(arf_cmp(row_max[i], x1) < 0) { arf_set(row_max[i], x1); row_max_indices[i] = k; } } // Update the max in the jth row. for(int k = j + 1; k < dim; k++) { arf_abs(x1, arb_midref(B(j, k))); if(arf_cmp(row_max[j], x1) < 0) { arf_set(row_max[j], x1); row_max_indices[j] = k; } } // Go through column i to see if any of // the new entries are larger than the // max of their row. for(int k = 0; k < i; k++) { if(k == dim) continue; arf_abs(x1, arb_midref(B(k, i))); if(arf_cmp(row_max[k], x1) < 0) { arf_set(row_max[k], x1); row_max_indices[k] = i; } } // And then column j. for(int k = 0; k < j; k++) { if(k == dim) continue; arf_abs(x1, arb_midref(B(k, j))); if(arf_cmp(row_max[k], x1) < 0) { arf_set(row_max[k], x1); row_max_indices[k] = j; } } } for(int k = 0; k < dim; k++) { arb_set(D(k), B(k,k)); arb_set_exact(D(k)); } // At this point we've done that diagonalization and all that remains is // to certify the correctness and compute error bounds. arb_mat_t e; arb_t error_norms[dim]; for(int k = 0; k < dim; k++) arb_init(error_norms[k]); arb_mat_init(e, dim, 1); arb_t z1, z2; arb_init(z1); arb_init(z2); for(int j = 0; j < dim; j++) { arb_mat_set(B, A); for(int k = 0; k < dim; k++) { arb_sub(B(k, k), B(k, k), D(j), prec); } for(int k = 0; k < dim; k++) { arb_set(arb_mat_entry(e, k, 0), P(k, j)); } arb_mat_L2norm(z2, e, prec); arb_mat_mul(e, B, e, prec); arb_mat_L2norm(error_norms[j], e, prec); arb_div(z2, error_norms[j], z2, prec); // and now z1 is an upper bound for the // error in the eigenvalue arb_add_error(D(j), z2); } int unique_eigenvalues = 1; for(int j = 0; j < dim; j++) { if(j == 0) { arb_sub(z1, D(j), D(1), prec); } else { arb_sub(z1, D(j), D(0), prec); } arb_get_abs_lbound_arf(x1, z1, prec); for(int k = 1; k < dim; k++) { if(k == j) continue; arb_sub(z1, D(j), D(k), prec); arb_get_abs_lbound_arf(x2, z1, prec); if(arf_cmp(x2, x1) < 0) { arf_set(x1, x2); } } if(arf_is_zero(x1)) { unique_eigenvalues = 0; } arb_div_arf(z1, error_norms[j], x1, prec); for(int k = 0; k < dim; k++) { arb_add_error(P(k, j), z1); } } arb_mat_clear(e); arb_clear(z1); arb_clear(z2); for(int k = 0; k < dim; k++) arb_clear(error_norms[k]); arf_clear(x1); arf_clear(x2); arb_mat_clear(B); for(int k = 0; k < dim; k++) { arf_clear(B1[k]); arf_clear(B2[k]); } for(int k = 0; k < dim - 1; k++) { arf_clear(row_max[k]); } arf_clear(Gii); arf_clear(Gij); arf_clear(Gji); arf_clear(Gjj); free(B1); free(B2); free(row_max); free(row_max_indices); if(unique_eigenvalues) return 0; else return 1; #undef B #undef D #undef P }
int main() { slong iter; flint_rand_t state; flint_printf("solve_tril...."); fflush(stdout); flint_randinit(state); for (iter = 0; iter < 2000 * arb_test_multiplier(); iter++) { arb_mat_t A, X, B, Y; slong rows, cols, prec, i, j; int unit; prec = 2 + n_randint(state, 200); if (n_randint(state, 10) == 0) { rows = n_randint(state, 60); cols = n_randint(state, 60); } else { rows = n_randint(state, 10); cols = n_randint(state, 10); } unit = n_randint(state, 2); arb_mat_init(A, rows, rows); arb_mat_init(B, rows, cols); arb_mat_init(X, rows, cols); arb_mat_init(Y, rows, cols); arb_mat_randtest(A, state, prec, 10); arb_mat_randtest(X, state, prec, 10); arb_mat_randtest(Y, state, prec, 10); for (i = 0; i < rows; i++) { if (unit) arb_one(arb_mat_entry(A, i, i)); else arb_set_ui(arb_mat_entry(A, i, i), 1 + n_randint(state, 100)); for (j = i + 1; j < rows; j++) arb_zero(arb_mat_entry(A, i, j)); } arb_mat_mul(B, A, X, prec); if (unit) /* check that diagonal entries are ignored */ { for (i = 0; i < rows; i++) arb_set_ui(arb_mat_entry(A, i, i), 1 + n_randint(state, 100)); } /* Check Y = A^(-1) * (A * X) = X */ arb_mat_solve_tril(Y, A, B, unit, prec); if (!arb_mat_overlaps(Y, X)) { flint_printf("FAIL\n"); flint_printf("A = \n"); arb_mat_printd(A, 10); flint_printf("\n\n"); flint_printf("B = \n"); arb_mat_printd(B, 10); flint_printf("\n\n"); flint_printf("X = \n"); arb_mat_printd(X, 10); flint_printf("\n\n"); flint_printf("Y = \n"); arb_mat_printd(Y, 10); flint_printf("\n\n"); flint_abort(); } /* Check aliasing */ arb_mat_solve_tril(B, A, B, unit, prec); if (!arb_mat_equal(B, Y)) { flint_printf("FAIL (aliasing)\n"); flint_printf("A = \n"); arb_mat_printd(A, 10); flint_printf("\n\n"); flint_printf("B = \n"); arb_mat_printd(B, 10); flint_printf("\n\n"); flint_printf("X = \n"); arb_mat_printd(X, 10); flint_printf("\n\n"); flint_printf("Y = \n"); arb_mat_printd(Y, 10); flint_printf("\n\n"); flint_abort(); } arb_mat_clear(A); arb_mat_clear(B); arb_mat_clear(X); arb_mat_clear(Y); } flint_randclear(state); flint_cleanup(); flint_printf("PASS\n"); return EXIT_SUCCESS; }
/* evaluates the truncated Taylor series (assumes no aliasing) */ void _arb_mat_exp_taylor(arb_mat_t S, const arb_mat_t A, slong N, slong prec) { if (N == 1) { arb_mat_one(S); } else if (N == 2) { arb_mat_one(S); arb_mat_add(S, S, A, prec); } else if (N == 3) { arb_mat_t T; arb_mat_init(T, arb_mat_nrows(A), arb_mat_nrows(A)); arb_mat_mul(T, A, A, prec); arb_mat_scalar_mul_2exp_si(T, T, -1); arb_mat_add(S, A, T, prec); arb_mat_one(T); arb_mat_add(S, S, T, prec); arb_mat_clear(T); } else { slong i, lo, hi, m, w, dim; arb_mat_struct * pows; arb_mat_t T, U; fmpz_t c, f; dim = arb_mat_nrows(A); m = n_sqrt(N); w = (N + m - 1) / m; fmpz_init(c); fmpz_init(f); pows = flint_malloc(sizeof(arb_mat_t) * (m + 1)); arb_mat_init(T, dim, dim); arb_mat_init(U, dim, dim); for (i = 0; i <= m; i++) { arb_mat_init(pows + i, dim, dim); if (i == 0) arb_mat_one(pows + i); else if (i == 1) arb_mat_set(pows + i, A); else arb_mat_mul(pows + i, pows + i - 1, A, prec); } arb_mat_zero(S); fmpz_one(f); for (i = w - 1; i >= 0; i--) { lo = i * m; hi = FLINT_MIN(N - 1, lo + m - 1); arb_mat_zero(T); fmpz_one(c); while (hi >= lo) { arb_mat_scalar_addmul_fmpz(T, pows + hi - lo, c, prec); if (hi != 0) fmpz_mul_ui(c, c, hi); hi--; } arb_mat_mul(U, pows + m, S, prec); arb_mat_scalar_mul_fmpz(S, T, f, prec); arb_mat_add(S, S, U, prec); fmpz_mul(f, f, c); } arb_mat_scalar_div_fmpz(S, S, f, prec); fmpz_clear(c); fmpz_clear(f); for (i = 0; i <= m; i++) arb_mat_clear(pows + i); flint_free(pows); arb_mat_clear(T); arb_mat_clear(U); } }
void arb_mat_exp(arb_mat_t B, const arb_mat_t A, slong prec) { slong i, j, dim, wp, N, q, r; mag_t norm, err; arb_mat_t T; dim = arb_mat_nrows(A); if (dim != arb_mat_ncols(A)) { flint_printf("arb_mat_exp: a square matrix is required!\n"); abort(); } if (dim == 0) { return; } else if (dim == 1) { arb_exp(arb_mat_entry(B, 0, 0), arb_mat_entry(A, 0, 0), prec); return; } wp = prec + 3 * FLINT_BIT_COUNT(prec); mag_init(norm); mag_init(err); arb_mat_init(T, dim, dim); arb_mat_bound_inf_norm(norm, A); if (mag_is_zero(norm)) { arb_mat_one(B); } else { q = pow(wp, 0.25); /* wanted magnitude */ if (mag_cmp_2exp_si(norm, 2 * wp) > 0) /* too big */ r = 2 * wp; else if (mag_cmp_2exp_si(norm, -q) < 0) /* tiny, no need to reduce */ r = 0; else r = FLINT_MAX(0, q + MAG_EXP(norm)); /* reduce to magnitude 2^(-r) */ arb_mat_scalar_mul_2exp_si(T, A, -r); mag_mul_2exp_si(norm, norm, -r); N = _arb_mat_exp_choose_N(norm, wp); mag_exp_tail(err, norm, N); _arb_mat_exp_taylor(B, T, N, wp); for (i = 0; i < dim; i++) for (j = 0; j < dim; j++) arb_add_error_mag(arb_mat_entry(B, i, j), err); for (i = 0; i < r; i++) { arb_mat_mul(T, B, B, wp); arb_mat_swap(T, B); } for (i = 0; i < dim; i++) for (j = 0; j < dim; j++) arb_set_round(arb_mat_entry(B, i, j), arb_mat_entry(B, i, j), prec); } mag_clear(norm); mag_clear(err); arb_mat_clear(T); }
int main() { slong iter; flint_rand_t state; flint_printf("frobenius_norm...."); fflush(stdout); flint_randinit(state); /* compare to the exact rational norm */ for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++) { fmpq_mat_t Q; fmpq_t q; arb_mat_t A; slong n, qbits, prec; n = n_randint(state, 8); qbits = 1 + n_randint(state, 100); prec = 2 + n_randint(state, 200); fmpq_mat_init(Q, n, n); fmpq_init(q); arb_mat_init(A, n, n); fmpq_mat_randtest(Q, state, qbits); _fmpq_mat_sum_of_squares(q, Q); arb_mat_set_fmpq_mat(A, Q, prec); /* check that the arb interval contains the exact value */ { arb_t a; arb_init(a); arb_mat_frobenius_norm(a, A, prec); arb_mul(a, a, a, prec); if (!arb_contains_fmpq(a, q)) { flint_printf("FAIL (containment, iter = %wd)\n", iter); flint_printf("n = %wd, prec = %wd\n", n, prec); flint_printf("\n"); flint_printf("Q = \n"); fmpq_mat_print(Q); flint_printf("\n\n"); flint_printf("frobenius_norm(Q)^2 = \n"); fmpq_print(q); flint_printf("\n\n"); flint_printf("A = \n"); arb_mat_printd(A, 15); flint_printf("\n\n"); flint_printf("frobenius_norm(A)^2 = \n"); arb_printd(a, 15); flint_printf("\n\n"); flint_printf("frobenius_norm(A)^2 = \n"); arb_print(a); flint_printf("\n\n"); abort(); } arb_clear(a); } /* check that the upper bound is not less than the exact value */ { mag_t b; fmpq_t y; mag_init(b); fmpq_init(y); arb_mat_bound_frobenius_norm(b, A); mag_mul(b, b, b); mag_get_fmpq(y, b); if (fmpq_cmp(q, y) > 0) { flint_printf("FAIL (bound, iter = %wd)\n", iter); flint_printf("n = %wd, prec = %wd\n", n, prec); flint_printf("\n"); flint_printf("Q = \n"); fmpq_mat_print(Q); flint_printf("\n\n"); flint_printf("frobenius_norm(Q)^2 = \n"); fmpq_print(q); flint_printf("\n\n"); flint_printf("A = \n"); arb_mat_printd(A, 15); flint_printf("\n\n"); flint_printf("bound_frobenius_norm(A)^2 = \n"); mag_printd(b, 15); flint_printf("\n\n"); flint_printf("bound_frobenius_norm(A)^2 = \n"); mag_print(b); flint_printf("\n\n"); abort(); } mag_clear(b); fmpq_clear(y); } fmpq_mat_clear(Q); fmpq_clear(q); arb_mat_clear(A); } /* check trace(A^T A) = frobenius_norm(A)^2 */ for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++) { slong m, n, prec; arb_mat_t A, AT, ATA; arb_t t; prec = 2 + n_randint(state, 200); m = n_randint(state, 10); n = n_randint(state, 10); arb_mat_init(A, m, n); arb_mat_init(AT, n, m); arb_mat_init(ATA, n, n); arb_init(t); arb_mat_randtest(A, state, 2 + n_randint(state, 100), 10); arb_mat_transpose(AT, A); arb_mat_mul(ATA, AT, A, prec); arb_mat_trace(t, ATA, prec); arb_sqrt(t, t, prec); /* check the norm bound */ { mag_t low, frobenius; mag_init(low); arb_get_mag_lower(low, t); mag_init(frobenius); arb_mat_bound_frobenius_norm(frobenius, A); if (mag_cmp(low, frobenius) > 0) { flint_printf("FAIL (bound)\n", iter); flint_printf("m = %wd, n = %wd, prec = %wd\n", m, n, prec); flint_printf("\n"); flint_printf("A = \n"); arb_mat_printd(A, 15); flint_printf("\n\n"); flint_printf("lower(sqrt(trace(A^T A))) = \n"); mag_printd(low, 15); flint_printf("\n\n"); flint_printf("bound_frobenius_norm(A) = \n"); mag_printd(frobenius, 15); flint_printf("\n\n"); abort(); } mag_clear(low); mag_clear(frobenius); } /* check the norm interval */ { arb_t frobenius; arb_init(frobenius); arb_mat_frobenius_norm(frobenius, A, prec); if (!arb_overlaps(t, frobenius)) { flint_printf("FAIL (overlap)\n", iter); flint_printf("m = %wd, n = %wd, prec = %wd\n", m, n, prec); flint_printf("\n"); flint_printf("A = \n"); arb_mat_printd(A, 15); flint_printf("\n\n"); flint_printf("sqrt(trace(A^T A)) = \n"); arb_printd(t, 15); flint_printf("\n\n"); flint_printf("frobenius_norm(A) = \n"); arb_printd(frobenius, 15); flint_printf("\n\n"); abort(); } arb_clear(frobenius); } arb_mat_clear(A); arb_mat_clear(AT); arb_mat_clear(ATA); arb_clear(t); } flint_randclear(state); flint_cleanup(); flint_printf("PASS\n"); return EXIT_SUCCESS; }
int main() { long iter; flint_rand_t state; printf("exp...."); fflush(stdout); flint_randinit(state); /* check exp(A)*exp(c*A) = exp((1+c)*A) */ for (iter = 0; iter < 1000; iter++) { arb_mat_t A, E, F, EF, G; fmpq_mat_t Q; arb_t c, d; long n, qbits, prec; n = n_randint(state, 5); qbits = 2 + n_randint(state, 300); prec = 2 + n_randint(state, 300); arb_init(c); arb_init(d); fmpq_mat_init(Q, n, n); arb_mat_init(A, n, n); arb_mat_init(E, n, n); arb_mat_init(F, n, n); arb_mat_init(EF, n, n); arb_mat_init(G, n, n); fmpq_mat_randtest(Q, state, qbits); arb_mat_set_fmpq_mat(A, Q, prec); arb_mat_exp(E, A, prec); arb_randtest(c, state, prec, 10); arb_mat_scalar_mul_arb(F, A, c, prec); arb_mat_exp(F, F, prec); arb_add_ui(d, c, 1, prec); arb_mat_scalar_mul_arb(G, A, d, prec); arb_mat_exp(G, G, prec); arb_mat_mul(EF, E, F, prec); if (!arb_mat_overlaps(EF, G)) { printf("FAIL\n\n"); printf("n = %ld, prec = %ld\n", n, prec); printf("c = \n"); arb_printd(c, 15); printf("\n\n"); printf("A = \n"); arb_mat_printd(A, 15); printf("\n\n"); printf("E = \n"); arb_mat_printd(E, 15); printf("\n\n"); printf("F = \n"); arb_mat_printd(F, 15); printf("\n\n"); printf("E*F = \n"); arb_mat_printd(EF, 15); printf("\n\n"); printf("G = \n"); arb_mat_printd(G, 15); printf("\n\n"); abort(); } arb_clear(c); arb_clear(d); fmpq_mat_clear(Q); arb_mat_clear(A); arb_mat_clear(E); arb_mat_clear(F); arb_mat_clear(EF); arb_mat_clear(G); } flint_randclear(state); flint_cleanup(); printf("PASS\n"); return EXIT_SUCCESS; }
void Lib_Arb_Mat_Mul(ArbMatPtr X, ArbMatPtr Y, ArbMatPtr Z, int32_t digits) { arb_mat_mul( (arb_mat_struct*) X, (arb_mat_struct*) Y, (arb_mat_struct*) Z, digits); }
int main() { slong iter; flint_rand_t state; flint_printf("lu...."); fflush(stdout); flint_randinit(state); for (iter = 0; iter < 100000; iter++) { fmpq_mat_t Q; arb_mat_t A, LU, P, L, U, T; slong i, j, n, qbits, prec, *perm; int q_invertible, r_invertible; n = n_randint(state, 8); qbits = 1 + n_randint(state, 100); prec = 2 + n_randint(state, 202); fmpq_mat_init(Q, n, n); arb_mat_init(A, n, n); arb_mat_init(LU, n, n); arb_mat_init(P, n, n); arb_mat_init(L, n, n); arb_mat_init(U, n, n); arb_mat_init(T, n, n); perm = _perm_init(n); fmpq_mat_randtest(Q, state, qbits); q_invertible = fmpq_mat_is_invertible(Q); if (!q_invertible) { arb_mat_set_fmpq_mat(A, Q, prec); r_invertible = arb_mat_lu(perm, LU, A, prec); if (r_invertible) { flint_printf("FAIL: matrix is singular over Q but not over R\n"); flint_printf("n = %wd, prec = %wd\n", n, prec); flint_printf("\n"); flint_printf("Q = \n"); fmpq_mat_print(Q); flint_printf("\n\n"); flint_printf("A = \n"); arb_mat_printd(A, 15); flint_printf("\n\n"); flint_printf("LU = \n"); arb_mat_printd(LU, 15); flint_printf("\n\n"); } } else { /* now this must converge */ while (1) { arb_mat_set_fmpq_mat(A, Q, prec); r_invertible = arb_mat_lu(perm, LU, A, prec); if (r_invertible) { break; } else { if (prec > 10000) { flint_printf("FAIL: failed to converge at 10000 bits\n"); abort(); } prec *= 2; } } arb_mat_one(L); for (i = 0; i < n; i++) for (j = 0; j < i; j++) arb_set(arb_mat_entry(L, i, j), arb_mat_entry(LU, i, j)); for (i = 0; i < n; i++) for (j = i; j < n; j++) arb_set(arb_mat_entry(U, i, j), arb_mat_entry(LU, i, j)); for (i = 0; i < n; i++) arb_one(arb_mat_entry(P, perm[i], i)); arb_mat_mul(T, P, L, prec); arb_mat_mul(T, T, U, prec); if (!arb_mat_contains_fmpq_mat(T, Q)) { flint_printf("FAIL (containment, iter = %wd)\n", iter); flint_printf("n = %wd, prec = %wd\n", n, prec); flint_printf("\n"); flint_printf("Q = \n"); fmpq_mat_print(Q); flint_printf("\n\n"); flint_printf("A = \n"); arb_mat_printd(A, 15); flint_printf("\n\n"); flint_printf("LU = \n"); arb_mat_printd(LU, 15); flint_printf("\n\n"); flint_printf("L = \n"); arb_mat_printd(L, 15); flint_printf("\n\n"); flint_printf("U = \n"); arb_mat_printd(U, 15); flint_printf("\n\n"); flint_printf("P*L*U = \n"); arb_mat_printd(T, 15); flint_printf("\n\n"); abort(); } } fmpq_mat_clear(Q); arb_mat_clear(A); arb_mat_clear(LU); arb_mat_clear(P); arb_mat_clear(L); arb_mat_clear(U); arb_mat_clear(T); _perm_clear(perm); } flint_randclear(state); flint_cleanup(); flint_printf("PASS\n"); return EXIT_SUCCESS; }