Flint integral polynomials utilities#

Real root isolation

void _fmpz_poly_scale_0_1_fmpq(fmpz *pol, slong len, fmpq_t a, fmpq_t b)#: Precompose the polynomial pol by an affine transformation so that the interval [a, b] becomes the interval [0,1]

int _fmpz_poly_has_real_root(fmpz *pol, slong len)#: Return 1 if the polynomial pol has a real root and 0 otherwise.

slong fmpz_poly_positive_root_upper_bound_2exp(const fmpz_poly_t pol)#: Return an upper bound on the bitsize of largest real root of pol.

slong _fmpz_poly_positive_root_upper_bound_2exp(const fmpz *pol, slong len)#: Return an upper bound on the bitsize of largest real root of (pol, len).

slong _fmpz_poly_positive_root_upper_bound_2exp_local_max(const fmpz *pol, slong len)#

slong fmpz_poly_num_real_roots_upper_bound(fmpz_poly_t pol)#: Return an upper bound on the number of real roots of the polynomial pol (currently using Descartes’ rule of sign).

slong _fmpz_poly_descartes_bound_0_1(fmpz *p, slong len, slong bound)#: Return an upper bound on the number of real roots between 0 and 1 of the polynomial (p, len) using Descartes’ rule of sign. If the result is larger than bound then WORD_MAX is returned.

void _fmpz_poly_isolate_real_roots_0_1_vca(fmpq *exact_roots, slong *n_exact_roots, fmpz *c_array, slong *k_array, slong *n_intervals, fmpz *pol, slong len)#: Isolate the real roots of (pol, len) contained in the interval \((0, 1)\). The array exact_roots will be set by the exact dyadic roots found by the algorithm and n_exact_roots updated accordingly. The arrays c_array and k_array are set to be interval data that enclose the remaining roots and n_interval is updated accordingly. A data c = c_array + i and k = k_array[i] represents the open interval \((c 2^k, (c + 1) 2^k)\).

void fmpz_poly_isolate_real_roots(fmpq *exact_roots, slong *n_exact, fmpz *c_array, slong *k_array, slong *n_interval, fmpz_poly_t pol)#: Isolate the real roots of pol. The array exact_roots will be set by the exact dyadic roots found by the algorithm and n_exact_roots updated accordingly. The arrays c_array and k_array are set to be interval data that enclose the remaining roots and n_interval is updated accordingly. A data c = c_array + i and k = k_array[i] represents the open interval \((c 2^k, (c + 1) 2^k)\).

int _fmpz_poly_newton_step_arb(arb_t res, const fmpz *pol, const fmpz *der, slong len, arb_t a, slong prec)#

int _fmpz_poly_bisection_step_arb(arb_t res, fmpz *pol, slong len, arb_t a, slong prec)#

int fmpz_poly_newton_step_arb(arb_t res, const fmpz_poly_t pol, const fmpz_poly_t der, arb_t a, slong prec)#

void _fmpz_poly_bisection_step_arf(arf_t l, arf_t r, const fmpz *pol, slong len, int sl, int sr, slong prec)#

int fmpz_poly_bisection_step_arb(arb_t res, const fmpz_poly_t pol, arb_t a, slong prec)#

Miscellaneous

FLINT, Arb extra

void fmpz_poly_abs(fmpz_poly_t res, fmpz_poly_t p)#: Set res to be the polynomial whose coefficients are the absolute values of the ones in p

void _fmpz_poly_evaluate_arb(arb_t res, const fmpz *pol, slong len, const arb_t a, slong prec)#: Evaluate the polynomial pol at the ball a and set result in res

void fmpz_poly_evaluate_arb(arb_t b, const fmpz_poly_t pol, const arb_t a, slong prec)#

void _fmpz_poly_evaluate_arf(arf_t res, const fmpz *pol, slong len, const arf_t a, slong prec)#

void fmpz_poly_evaluate_arf(arf_t res, const fmpz_poly_t pol, const arf_t a, slong prec)#

int _fmpz_poly_relative_condition_number_2exp(slong *cond, fmpz *p, slong len, arb_t x, slong prec)#

int fmpz_poly_relative_condition_number_2exp(slong *cond, fmpz_poly_t p, arb_t x, slong prec)#

Functions

void fmpz_poly_randtest_irreducible(fmpz_poly_t p, flint_rand_t state, slong len, mp_bitcnt_t bits)#: Set p to be a random irreducible polynomial.

int fmpz_poly_set_str_pretty(fmpz_poly_t p, const char *s, const char *var)#: Set the polynomial p from the string s using var as variable name.