void fmpz_poly_mat_pow(fmpz_poly_mat_t B, const fmpz_poly_mat_t A, ulong exp) { long d = fmpz_poly_mat_nrows(A); if (exp == 0 || d == 0) { fmpz_poly_mat_one(B); } else if (exp == 1) { fmpz_poly_mat_set(B, A); } else if (exp == 2) { fmpz_poly_mat_sqr(B, A); } else if (d == 1) { fmpz_poly_pow(fmpz_poly_mat_entry(B, 0, 0), fmpz_poly_mat_entry(A, 0, 0), exp); } else { fmpz_poly_mat_t T, U; long i; fmpz_poly_mat_init_set(T, A); fmpz_poly_mat_init(U, d, d); for (i = ((long) FLINT_BIT_COUNT(exp)) - 2; i >= 0; i--) { fmpz_poly_mat_sqr(U, T); if (exp & (1L << i)) fmpz_poly_mat_mul(T, U, A); else fmpz_poly_mat_swap(T, U); } fmpz_poly_mat_swap(B, T); fmpz_poly_mat_clear(T); fmpz_poly_mat_clear(U); } }
void fmpz_poly_q_mul(fmpz_poly_q_t rop, const fmpz_poly_q_t op1, const fmpz_poly_q_t op2) { if (fmpz_poly_q_is_zero(op1) || fmpz_poly_q_is_zero(op2)) { fmpz_poly_q_zero(rop); return; } if (op1 == op2) { fmpz_poly_pow(rop->num, op1->num, 2); fmpz_poly_pow(rop->den, op1->den, 2); return; } if (rop == op1 || rop == op2) { fmpz_poly_q_t t; fmpz_poly_q_init(t); fmpz_poly_q_mul(t, op1, op2); fmpz_poly_q_swap(rop, t); fmpz_poly_q_clear(t); return; } /* From here on, we may assume that rop, op1 and op2 refer to distinct objects in memory, and that op1 and op2 are non-zero */ /* Polynomials? */ if (fmpz_poly_length(op1->den) == 1 && fmpz_poly_length(op2->den) == 1) { const slong len1 = fmpz_poly_length(op1->num); const slong len2 = fmpz_poly_length(op2->num); fmpz_poly_fit_length(rop->num, len1 + len2 - 1); if (len1 >= len2) { _fmpq_poly_mul(rop->num->coeffs, rop->den->coeffs, op1->num->coeffs, op1->den->coeffs, len1, op2->num->coeffs, op2->den->coeffs, len2); } else { _fmpq_poly_mul(rop->num->coeffs, rop->den->coeffs, op2->num->coeffs, op2->den->coeffs, len2, op1->num->coeffs, op1->den->coeffs, len1); } _fmpz_poly_set_length(rop->num, len1 + len2 - 1); _fmpz_poly_set_length(rop->den, 1); return; } fmpz_poly_gcd(rop->num, op1->num, op2->den); if (fmpz_poly_is_one(rop->num)) { fmpz_poly_gcd(rop->den, op2->num, op1->den); if (fmpz_poly_is_one(rop->den)) { fmpz_poly_mul(rop->num, op1->num, op2->num); fmpz_poly_mul(rop->den, op1->den, op2->den); } else { fmpz_poly_div(rop->num, op2->num, rop->den); fmpz_poly_mul(rop->num, op1->num, rop->num); fmpz_poly_div(rop->den, op1->den, rop->den); fmpz_poly_mul(rop->den, rop->den, op2->den); } } else { fmpz_poly_gcd(rop->den, op2->num, op1->den); if (fmpz_poly_is_one(rop->den)) { fmpz_poly_div(rop->den, op2->den, rop->num); fmpz_poly_mul(rop->den, op1->den, rop->den); fmpz_poly_div(rop->num, op1->num, rop->num); fmpz_poly_mul(rop->num, rop->num, op2->num); } else { fmpz_poly_t t, u; fmpz_poly_init(t); fmpz_poly_init(u); fmpz_poly_div(t, op1->num, rop->num); fmpz_poly_div(u, op2->den, rop->num); fmpz_poly_div(rop->num, op2->num, rop->den); fmpz_poly_mul(rop->num, t, rop->num); fmpz_poly_div(rop->den, op1->den, rop->den); fmpz_poly_mul(rop->den, rop->den, u); fmpz_poly_clear(t); fmpz_poly_clear(u); } } }
int main() { slong iter; flint_rand_t state; flint_printf("pow_ui...."); fflush(stdout); flint_randinit(state); /* compare with fmpz_poly */ for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++) { slong zbits1, rbits1, rbits2; ulong e; fmpz_poly_t A, B; arb_poly_t a, b; zbits1 = 2 + n_randint(state, 100); rbits1 = 2 + n_randint(state, 200); rbits2 = 2 + n_randint(state, 200); e = n_randint(state, 30); fmpz_poly_init(A); fmpz_poly_init(B); arb_poly_init(a); arb_poly_init(b); fmpz_poly_randtest(A, state, 1 + n_randint(state, 8), zbits1); fmpz_poly_pow(B, A, e); arb_poly_set_fmpz_poly(a, A, rbits1); arb_poly_pow_ui(b, a, e, rbits2); if (!arb_poly_contains_fmpz_poly(b, B)) { flint_printf("FAIL\n\n"); flint_printf("bits2 = %wd\n", rbits2); flint_printf("e = %wu\n", e); flint_printf("A = "); fmpz_poly_print(A); flint_printf("\n\n"); flint_printf("B = "); fmpz_poly_print(B); flint_printf("\n\n"); flint_printf("a = "); arb_poly_printd(a, 15); flint_printf("\n\n"); flint_printf("b = "); arb_poly_printd(b, 15); flint_printf("\n\n"); abort(); } arb_poly_pow_ui(a, a, e, rbits2); if (!arb_poly_equal(a, b)) { flint_printf("FAIL (aliasing)\n\n"); abort(); } fmpz_poly_clear(A); fmpz_poly_clear(B); arb_poly_clear(a); arb_poly_clear(b); } flint_randclear(state); flint_cleanup(); flint_printf("PASS\n"); return EXIT_SUCCESS; }
int main(int argc, char *argv[]) { fmpz_poly_t f, g; fmpz_poly_factor_t fac; fmpz_t t; slong compd, printd, i, j; if (argc < 2) { flint_printf("poly_roots [-refine d] [-print d] <poly>\n\n"); flint_printf("Isolates all the complex roots of a polynomial with integer coefficients.\n\n"); flint_printf("If -refine d is passed, the roots are refined to an absolute tolerance\n"); flint_printf("better than 10^(-d). By default, the roots are only computed to sufficient\n"); flint_printf("accuracy to isolate them. The refinement is not currently done efficiently.\n\n"); flint_printf("If -print d is passed, the computed roots are printed to d decimals.\n"); flint_printf("By default, the roots are not printed.\n\n"); flint_printf("The polynomial can be specified by passing the following as <poly>:\n\n"); flint_printf("a <n> Easy polynomial 1 + 2x + ... + (n+1)x^n\n"); flint_printf("t <n> Chebyshev polynomial T_n\n"); flint_printf("u <n> Chebyshev polynomial U_n\n"); flint_printf("p <n> Legendre polynomial P_n\n"); flint_printf("c <n> Cyclotomic polynomial Phi_n\n"); flint_printf("s <n> Swinnerton-Dyer polynomial S_n\n"); flint_printf("b <n> Bernoulli polynomial B_n\n"); flint_printf("w <n> Wilkinson polynomial W_n\n"); flint_printf("e <n> Taylor series of exp(x) truncated to degree n\n"); flint_printf("m <n> <m> The Mignotte-like polynomial x^n + (100x+1)^m, n > m\n"); flint_printf("coeffs <c0 c1 ... cn> c0 + c1 x + ... + cn x^n\n\n"); flint_printf("Concatenate to multiply polynomials, e.g.: p 5 t 6 coeffs 1 2 3\n"); flint_printf("for P_5(x)*T_6(x)*(1+2x+3x^2)\n\n"); return 1; } compd = 0; printd = 0; fmpz_poly_init(f); fmpz_poly_init(g); fmpz_init(t); fmpz_poly_one(f); for (i = 1; i < argc; i++) { if (!strcmp(argv[i], "-refine")) { compd = atol(argv[i+1]); i++; } else if (!strcmp(argv[i], "-print")) { printd = atol(argv[i+1]); i++; } else if (!strcmp(argv[i], "a")) { slong n = atol(argv[i+1]); fmpz_poly_zero(g); for (j = 0; j <= n; j++) fmpz_poly_set_coeff_ui(g, j, j+1); fmpz_poly_mul(f, f, g); i++; } else if (!strcmp(argv[i], "t")) { arith_chebyshev_t_polynomial(g, atol(argv[i+1])); fmpz_poly_mul(f, f, g); i++; } else if (!strcmp(argv[i], "u")) { arith_chebyshev_u_polynomial(g, atol(argv[i+1])); fmpz_poly_mul(f, f, g); i++; } else if (!strcmp(argv[i], "p")) { fmpq_poly_t h; fmpq_poly_init(h); arith_legendre_polynomial(h, atol(argv[i+1])); fmpq_poly_get_numerator(g, h); fmpz_poly_mul(f, f, g); fmpq_poly_clear(h); i++; } else if (!strcmp(argv[i], "c")) { arith_cyclotomic_polynomial(g, atol(argv[i+1])); fmpz_poly_mul(f, f, g); i++; } else if (!strcmp(argv[i], "s")) { arith_swinnerton_dyer_polynomial(g, atol(argv[i+1])); fmpz_poly_mul(f, f, g); i++; } else if (!strcmp(argv[i], "b")) { fmpq_poly_t h; fmpq_poly_init(h); arith_bernoulli_polynomial(h, atol(argv[i+1])); fmpq_poly_get_numerator(g, h); fmpz_poly_mul(f, f, g); fmpq_poly_clear(h); i++; } else if (!strcmp(argv[i], "w")) { slong n = atol(argv[i+1]); fmpz_poly_zero(g); fmpz_poly_fit_length(g, n+2); arith_stirling_number_1_vec(g->coeffs, n+1, n+2); _fmpz_poly_set_length(g, n+2); fmpz_poly_shift_right(g, g, 1); fmpz_poly_mul(f, f, g); i++; } else if (!strcmp(argv[i], "e")) { fmpq_poly_t h; fmpq_poly_init(h); fmpq_poly_set_coeff_si(h, 0, 0); fmpq_poly_set_coeff_si(h, 1, 1); fmpq_poly_exp_series(h, h, atol(argv[i+1]) + 1); fmpq_poly_get_numerator(g, h); fmpz_poly_mul(f, f, g); fmpq_poly_clear(h); i++; } else if (!strcmp(argv[i], "m")) { fmpz_poly_zero(g); fmpz_poly_set_coeff_ui(g, 0, 1); fmpz_poly_set_coeff_ui(g, 1, 100); fmpz_poly_pow(g, g, atol(argv[i+2])); fmpz_poly_set_coeff_ui(g, atol(argv[i+1]), 1); fmpz_poly_mul(f, f, g); i += 2; } else if (!strcmp(argv[i], "coeffs")) { fmpz_poly_zero(g); i++; j = 0; while (i < argc) { if (fmpz_set_str(t, argv[i], 10) != 0) { i--; break; } fmpz_poly_set_coeff_fmpz(g, j, t); i++; j++; } fmpz_poly_mul(f, f, g); } } fmpz_poly_factor_init(fac); flint_printf("computing squarefree factorization...\n"); TIMEIT_ONCE_START fmpz_poly_factor_squarefree(fac, f); TIMEIT_ONCE_STOP TIMEIT_ONCE_START for (i = 0; i < fac->num; i++) { flint_printf("roots with multiplicity %wd\n", fac->exp[i]); fmpz_poly_complex_roots_squarefree(fac->p + i, 32, compd * 3.32193 + 2, printd); } TIMEIT_ONCE_STOP fmpz_poly_factor_clear(fac); fmpz_poly_clear(f); fmpz_poly_clear(g); fmpz_clear(t); flint_cleanup(); return EXIT_SUCCESS; }
int main(void) { int i, result; flint_rand_t state; printf("pow_multinomial...."); fflush(stdout); flint_randinit(state); /* Check aliasing of a and b */ for (i = 0; i < 2000; i++) { fmpz_poly_t a, b; ulong exp; fmpz_poly_init(a); fmpz_poly_init(b); fmpz_poly_randtest(b, state, n_randint(state, 10), 100); exp = n_randtest(state) % 20UL; fmpz_poly_pow_multinomial(a, b, exp); fmpz_poly_pow_multinomial(b, b, exp); result = (fmpz_poly_equal(a, b)); if (!result) { printf("FAIL(1):\n"); printf("exp = %lu\n", exp); printf("a = "), fmpz_poly_print(a), printf("\n\n"); printf("b = "), fmpz_poly_print(b), printf("\n\n"); abort(); } fmpz_poly_clear(a); fmpz_poly_clear(b); } /* Compare with fmpz_poly_pow */ for (i = 0; i < 2000; i++) { fmpz_poly_t a, b; ulong exp; fmpz_poly_init(a); fmpz_poly_init(b); fmpz_poly_randtest(b, state, n_randint(state, 10), 100); exp = n_randtest(state) % 20UL; fmpz_poly_pow_multinomial(a, b, exp); fmpz_poly_pow(b, b, exp); result = (fmpz_poly_equal(a, b)); if (!result) { printf("FAIL(2):\n"); printf("exp = %lu\n", exp); printf("a = "), fmpz_poly_print(a), printf("\n\n"); printf("b = "), fmpz_poly_print(b), printf("\n\n"); abort(); } fmpz_poly_clear(a); fmpz_poly_clear(b); } flint_randclear(state); _fmpz_cleanup(); printf("PASS\n"); return 0; }
var PowerPolyZ(const Tuple& x,uint e){ fmpz_poly_t res; fmpz_poly_init(res); fmpz_poly_pow(res,to_fmpz_poly(x),e); return from_fmpz_poly(res); }