/* F = 1 + U + U^2 + ... = 1/(1-U) assuming that U[0] is positive; indeterminate if not convergent */ static void arb_poly_geometric_sum(arb_poly_t F, const arb_poly_t U, long len, long prec) { if (U->length == 0) { arb_poly_one(F); return; } arb_poly_add_si(F, U, -1, prec); arb_poly_neg(F, F); if (F->length > 0 && arb_is_positive(F->coeffs)) { arb_poly_inv_series(F, F, len, prec); } else { arb_poly_fit_length(F, len); _arb_vec_indeterminate(F->coeffs, len); _arb_poly_set_length(F, len); } }
void arb_poly_tan_series(arb_poly_t g, const arb_poly_t h, slong n, slong prec) { if (h->length == 0 || n == 0) { arb_poly_zero(g); return; } if (g == h) { arb_poly_t t; arb_poly_init(t); arb_poly_tan_series(t, h, n, prec); arb_poly_swap(g, t); arb_poly_clear(t); return; } arb_poly_fit_length(g, n); _arb_poly_tan_series(g->coeffs, h->coeffs, h->length, n, prec); _arb_poly_set_length(g, n); _arb_poly_normalise(g); }
void arb_poly_mullow_classical(arb_poly_t res, const arb_poly_t poly1, const arb_poly_t poly2, long n, long prec) { long len_out; if (poly1->length == 0 || poly2->length == 0 || n == 0) { arb_poly_zero(res); return; } len_out = poly1->length + poly2->length - 1; if (n > len_out) n = len_out; if (res == poly1 || res == poly2) { arb_poly_t t; arb_poly_init2(t, n); _arb_poly_mullow_classical(t->coeffs, poly1->coeffs, poly1->length, poly2->coeffs, poly2->length, n, prec); arb_poly_swap(res, t); arb_poly_clear(t); } else { arb_poly_fit_length(res, n); _arb_poly_mullow_classical(res->coeffs, poly1->coeffs, poly1->length, poly2->coeffs, poly2->length, n, prec); } _arb_poly_set_length(res, n); _arb_poly_normalise(res); }
/* F = 1 + U + U^2 + U^3 + ... = 1/(1-U) U = product of (1 + |A-B|/(|B[0] - |B[1:]|) product of (1 / (|B[0] - |B[1:]|)) * |Z| */ void acb_hypgeom_pfq_series_bound_factor(arb_poly_t F, const acb_poly_struct * a, long p, const acb_poly_struct * b, long q, const acb_poly_t z, long n, long len, long prec) { long i; arb_poly_t T, U, V; acb_poly_t BN, AB; /* not convergent */ if (p > q) { arb_poly_fit_length(F, len); _arb_vec_indeterminate(F->coeffs, len); _arb_poly_set_length(F, len); return; } arb_poly_init(T); arb_poly_init(U); arb_poly_init(V); acb_poly_init(BN); acb_poly_init(AB); acb_poly_majorant(U, z, prec); for (i = 0; i < q; i++) { acb_poly_add_si(BN, b + i, n, prec); if (acb_poly_length(BN) != 0 && arb_is_positive(acb_realref(BN->coeffs))) { if (i < p) { /* 1 + |a-b|/reciprocal_majorant(b + n) */ acb_poly_sub(AB, a + i, b + i, prec); acb_poly_majorant(T, AB, prec); acb_poly_reciprocal_majorant(V, BN, prec); arb_poly_div_series(T, T, V, len, prec); arb_poly_add_si(T, T, 1, prec); arb_poly_mullow(U, U, T, len, prec); } else { acb_poly_reciprocal_majorant(T, BN, prec); arb_poly_div_series(U, U, T, len, prec); } } else { arb_poly_fit_length(U, len); _arb_vec_indeterminate(U->coeffs, len); _arb_poly_set_length(U, len); break; } } /* F = 1/(1-U) */ arb_poly_geometric_sum(F, U, len, prec); arb_poly_clear(T); arb_poly_clear(U); arb_poly_clear(V); acb_poly_clear(BN); acb_poly_clear(AB); }
int main() { slong iter; flint_rand_t state; flint_printf("rising_ui_series...."); fflush(stdout); flint_randinit(state); /* check rf(f, a) * rf(f + a, b) = rf(f, a + b) */ for (iter = 0; iter < 1000 * arb_test_multiplier(); iter++) { slong bits, trunc; ulong a, b; arb_poly_t f, g, h1, h2, h1h2, h3; bits = 2 + n_randint(state, 200); trunc = 1 + n_randint(state, 20); a = n_randint(state, 10); b = n_randint(state, 10); arb_poly_init(f); arb_poly_init(g); arb_poly_init(h1); arb_poly_init(h2); arb_poly_init(h1h2); arb_poly_init(h3); arb_poly_randtest(f, state, 1 + n_randint(state, 20), bits, 4); arb_poly_set(g, f); /* g = f + 1 */ if (g->length == 0) { arb_poly_fit_length(g, 1); arb_set_ui(g->coeffs, a); _arb_poly_set_length(g, 1); _arb_poly_normalise(g); } else { arb_add_ui(g->coeffs, g->coeffs, a, bits); _arb_poly_normalise(g); } arb_poly_rising_ui_series(h1, f, a, trunc, bits); arb_poly_rising_ui_series(h2, g, b, trunc, bits); arb_poly_rising_ui_series(h3, f, a + b, trunc, bits); arb_poly_mullow(h1h2, h1, h2, trunc, bits); if (!arb_poly_overlaps(h1h2, h3)) { flint_printf("FAIL\n\n"); flint_printf("bits = %wd\n", bits); flint_printf("trunc = %wd\n", trunc); flint_printf("a = %wu\n", a); flint_printf("b = %wu\n", a); flint_printf("f = "); arb_poly_printd(f, 15); flint_printf("\n\n"); flint_printf("g = "); arb_poly_printd(g, 15); flint_printf("\n\n"); flint_printf("h1 = "); arb_poly_printd(h1, 15); flint_printf("\n\n"); flint_printf("h2 = "); arb_poly_printd(h2, 15); flint_printf("\n\n"); flint_printf("h1h2 = "); arb_poly_printd(h1h2, 15); flint_printf("\n\n"); flint_printf("h3 = "); arb_poly_printd(h3, 15); flint_printf("\n\n"); abort(); } arb_poly_rising_ui_series(f, f, a, trunc, bits); if (!arb_poly_equal(f, h1)) { flint_printf("FAIL (aliasing)\n\n"); flint_printf("bits = %wd\n", bits); flint_printf("trunc = %wd\n", trunc); flint_printf("a = %wu\n", a); flint_printf("b = %wu\n", a); flint_printf("f = "); arb_poly_printd(f, 15); flint_printf("\n\n"); flint_printf("h1 = "); arb_poly_printd(h1, 15); flint_printf("\n\n"); abort(); } arb_poly_clear(f); arb_poly_clear(g); arb_poly_clear(h1); arb_poly_clear(h2); arb_poly_clear(h1h2); arb_poly_clear(h3); } flint_randclear(state); flint_cleanup(); flint_printf("PASS\n"); return EXIT_SUCCESS; }
void acb_hypgeom_pfq_series_direct(acb_poly_t res, const acb_poly_struct * a, long p, const acb_poly_struct * b, long q, const acb_poly_t z, int regularized, long n, long len, long prec) { acb_poly_t s, t, err; arb_poly_t C, T; long i; int is_real; int terminating; /* default algorithm to choose number of terms */ if (n < 0) { n = acb_hypgeom_pfq_series_choose_n(a, p, b, q, z, len, prec); } terminating = 0; /* check if it terminates due to a root of the numerator */ for (i = 0; i < p; i++) { if (acb_poly_length(a + i) == 0 && n > 0) { terminating = 1; } else if (acb_poly_length(a + i) == 1) { acb_srcptr c = acb_poly_get_coeff_ptr(a + i, 0); if (acb_is_int(c) && arb_is_negative(acb_realref(c)) && arf_cmpabs_ui(arb_midref(acb_realref(c)), n) < 0) { terminating = 1; } } } /* check if it terminates (to order n) due to z */ /* the following tests could be made stronger... */ if (z->length == 0 && n >= 1) { terminating = 1; } else if (!terminating && z->length > 0 && acb_is_zero(z->coeffs) && n >= len) { if (regularized) { terminating = 1; } else { terminating = 1; for (i = 0; i < q; i++) { acb_srcptr c = acb_poly_get_coeff_ptr(b + i, 0); if (!arb_is_positive(acb_realref(c)) && acb_contains_int(c)) terminating = 0; } } } acb_poly_init(s); acb_poly_init(t); acb_poly_init(err); arb_poly_init(C); arb_poly_init(T); acb_hypgeom_pfq_series_sum_forward(s, t, a, p, b, q, z, regularized, n, len, prec); if (!terminating) { is_real = acb_poly_is_real(z); for (i = 0; i < p; i++) is_real = is_real && acb_poly_is_real(a + i); for (i = 0; i < q; i++) is_real = is_real && acb_poly_is_real(b + i); acb_poly_majorant(T, t, MAG_BITS); acb_hypgeom_pfq_series_bound_factor(C, a, p, b, q, z, n, len, MAG_BITS); if (!_arb_vec_is_finite(T->coeffs, T->length) || !_arb_vec_is_finite(C->coeffs, C->length)) { arb_poly_fit_length(T, len); _arb_vec_indeterminate(T->coeffs, len); _arb_poly_set_length(T, len); } else { arb_poly_mullow(T, T, C, len, MAG_BITS); } /* create polynomial of errors */ acb_poly_fit_length(err, len); for (i = 0; i < FLINT_MIN(len, T->length); i++) { arb_add_error(acb_realref(err->coeffs + i), T->coeffs + i); if (!is_real) arb_add_error(acb_imagref(err->coeffs + i), T->coeffs + i); } _acb_poly_set_length(err, len); _acb_poly_normalise(err); acb_poly_add(s, s, err, prec); } acb_poly_set(res, s); acb_poly_clear(s); acb_poly_clear(t); acb_poly_clear(err); arb_poly_clear(C); arb_poly_clear(T); }