int _fmpq_poly_is_squarefree(const fmpz * poly, const fmpz_t den, long len) { if (len < 3) return 1; else if (len == 3) { int ans; fmpz_t lhs, rhs; fmpz_init(lhs); fmpz_init(rhs); fmpz_mul(lhs, poly + 1, poly + 1); fmpz_mul(rhs, poly, poly + 2); fmpz_mul_ui(rhs, rhs, 4); ans = !fmpz_equal(lhs, rhs); fmpz_clear(lhs); fmpz_clear(rhs); return ans; } else { long gdeg; fmpz * w = _fmpz_vec_init(2 * len); _fmpz_poly_derivative(w, poly, len); _fmpz_poly_gcd(w + len, poly, len, w, len - 1L); for (gdeg = len - 2L; w[gdeg] == 0L; gdeg--) ; _fmpz_vec_clear(w, 2 * len); return (gdeg == 0); } }
void _fmpq_poly_derivative(fmpz * rpoly, fmpz_t rden, const fmpz * poly, const fmpz_t den, slong len) { _fmpz_poly_derivative(rpoly, poly, len); fmpz_set(rden, den); _fmpq_poly_canonicalise(rpoly, rden, len - 1); }
void _fmpz_poly_revert_series_newton(fmpz * Qinv, const fmpz * Q, long n) { if (n <= 2) { _fmpz_vec_set(Qinv, Q, n); return; } else { long *a, i, k; fmpz *T, *U, *V; T = _fmpz_vec_init(n); U = _fmpz_vec_init(n); V = _fmpz_vec_init(n); k = n; for (i = 1; (1L << i) < k; i++); a = (long *) flint_malloc(i * sizeof(long)); a[i = 0] = k; while (k >= FLINT_REVERSE_NEWTON_CUTOFF) a[++i] = (k = (k + 1) / 2); _fmpz_poly_revert_series_lagrange(Qinv, Q, k); _fmpz_vec_zero(Qinv + k, n - k); for (i--; i >= 0; i--) { k = a[i]; _fmpz_poly_compose_series(T, Q, k, Qinv, k, k); _fmpz_poly_derivative(U, T, k); fmpz_zero(U + k - 1); fmpz_zero(T + 1); _fmpz_poly_div_series(V, T, U, k); _fmpz_poly_derivative(T, Qinv, k); _fmpz_poly_mullow(U, V, k, T, k, k); _fmpz_vec_sub(Qinv, Qinv, U, k); } flint_free(a); _fmpz_vec_clear(T, n); _fmpz_vec_clear(U, n); _fmpz_vec_clear(V, n); } }
void arb_rising2_ui_rs(arb_t u, arb_t v, const arb_t x, ulong n, ulong m, slong prec) { if (n == 0) { arb_zero(v); arb_one(u); } else if (n == 1) { arb_set(u, x); arb_one(v); } else { slong wp; ulong i, j, a, b; arb_ptr xs; arb_t S, T, U, V; fmpz *A, *B; wp = ARF_PREC_ADD(prec, FLINT_BIT_COUNT(n)); if (m == 0) { ulong m1, m2; m1 = 0.6 * pow(wp, 0.4); m2 = n_sqrt(n); m = FLINT_MIN(m1, m2); } m = FLINT_MAX(m, 1); xs = _arb_vec_init(m + 1); A = _fmpz_vec_init(2 * m + 1); B = A + (m + 1); arb_init(S); arb_init(T); arb_init(U); arb_init(V); _arb_vec_set_powers(xs, x, m + 1, wp); for (i = 0; i < n; i += m) { a = i; b = FLINT_MIN(n, a + m); if (a == 0 || b != a + m) { _gamma_rf_bsplit(A, a, b); } else { fmpz tt = m; _fmpz_poly_taylor_shift(A, &tt, m + 1); } _fmpz_poly_derivative(B, A, b - a + 1); arb_set_fmpz(S, A); for (j = 1; j <= b - a; j++) arb_addmul_fmpz(S, xs + j, A + j, wp); arb_set_fmpz(T, B); for (j = 1; j < b - a; j++) arb_addmul_fmpz(T, xs + j, B + j, wp); if (i == 0) { arb_set(U, S); arb_set(V, T); } else { arb_mul(V, V, S, wp); arb_addmul(V, U, T, wp); arb_mul(U, U, S, wp); } } arb_set(u, U); arb_set(v, V); _arb_vec_clear(xs, m + 1); _fmpz_vec_clear(A, 2 * m + 1); arb_clear(S); arb_clear(T); arb_clear(U); arb_clear(V); } }
void _fmpz_poly_signature(long * r1, long * r2, fmpz * poly, long len) { fmpz *A, *B, *f, *g, *h, *w; long lenA, lenB; int s, t; if (len <= 2) { *r1 = (len == 2); *r2 = 0; return; } w = _fmpz_vec_init(2 * len + 2); A = w; B = w + len; lenA = len; lenB = lenA - 1; f = w + 2 * len - 1; g = w + 2 * len; h = w + 2 * len + 1; _fmpz_poly_primitive_part(A, poly, lenA); _fmpz_poly_derivative(B, A, lenA); _fmpz_poly_primitive_part(B, B, lenB); fmpz_one(g); fmpz_one(h); s = 1; t = (lenA & 1L) ? -s : s; *r1 = 1; while (1) { long delta = lenA - lenB; int sgnA; _fmpz_poly_pseudo_rem_cohen(A, A, lenA, B, lenB); lenA = lenB; FMPZ_VEC_NORM(A, lenA); if (lenA == 0) { printf("Exception: non-squarefree polynomial detected in fmpz_poly_signature\n"); _fmpz_vec_clear(w, 2 * len + 2); abort(); } if ((fmpz_sgn(B + (lenB - 1)) > 0) || (delta & 1L)) _fmpz_vec_neg(A, A, lenA); sgnA = fmpz_sgn(A + (lenA - 1)); if (sgnA != s) { s = -s; (*r1)--; } if (sgnA != ((lenA & 1L) ? t : -t)) { t = -t; (*r1)++; } if (lenA == 1) { *r2 = ((len - 1) - *r1) / 2; _fmpz_vec_clear(w, 2 * len + 2); return; } else { { fmpz * temp = A; A = B; B = temp; } { long temp = lenA; lenA = lenB; lenB = temp; } if (delta == 1) { fmpz_mul(f, g, h); _fmpz_vec_scalar_divexact_fmpz(B, B, lenB, f); fmpz_set(g, A + (lenA - 1)); fmpz_set(h, g); } else { fmpz_pow_ui(f, h, delta); fmpz_mul(f, f, g); _fmpz_vec_scalar_divexact_fmpz(B, B, lenB, f); fmpz_pow_ui(f, h, delta - 1); fmpz_pow_ui(g, A + (lenA - 1), delta); fmpz_divexact(h, g, f); fmpz_set(g, A + (lenA - 1)); } } } }
int fmpq_poly_check_unique_real_root(const fmpq_poly_t pol, const arb_t a, slong prec) { if (pol->length < 2) return 0; else if (pol->length == 2) { /* linear polynomial */ fmpq_t root; int ans; fmpq_init(root); fmpq_set_fmpz_frac(root, fmpq_poly_numref(pol), fmpq_poly_numref(pol) + 1); fmpq_neg(root, root); ans = arb_contains_fmpq(a, root); fmpq_clear(root); return ans; } else { arb_t b, c; arf_t l, r; fmpz * der; int lsign, rsign; fmpz_poly_t pol2; slong n; /* 1 - cheap test: */ /* - sign(left) * sign(right) = -1 */ /* - no zero of the derivative */ arb_init(b); arb_init(c); arf_init(l); arf_init(r); arb_get_interval_arf(l, r, a, prec); arb_set_arf(b, l); _fmpz_poly_evaluate_arb(c, pol->coeffs, pol->length, b, 2*prec); lsign = arb_sgn2(c); arb_set_arf(b, r); _fmpz_poly_evaluate_arb(c, pol->coeffs, pol->length, b, 2*prec); rsign = arb_sgn2(c); arb_clear(c); if (lsign * rsign == -1) { der = _fmpz_vec_init(pol->length - 1); _fmpz_poly_derivative(der, pol->coeffs, pol->length); _fmpz_poly_evaluate_arb(b, der, pol->length - 1, a, prec); _fmpz_vec_clear(der, pol->length - 1); if (!arb_contains_zero(b)) { arf_clear(l); arf_clear(r); arb_clear(b); return 1; } } else return 0; arb_clear(b); /* 2 - expensive testing */ fmpq_t ql, qr; fmpq_init(ql); fmpq_init(qr); arf_get_fmpq(ql, l); arf_get_fmpq(qr, r); fmpz_poly_init(pol2); fmpz_poly_fit_length(pol2, pol->length); _fmpz_vec_set(pol2->coeffs, pol->coeffs, pol->length); pol2->length = pol->length; _fmpz_poly_scale_0_1_fmpq(pol2->coeffs, pol2->length, ql, qr); n = fmpz_poly_num_real_roots_0_1(pol2); fmpz_poly_clear(pol2); fmpq_clear(ql); fmpq_clear(qr); return (n == 1); } }