_arb_fmpz_poly_evaluate_arb_horner res poly len x prec

arb_fmpz_poly_evaluate_arb_horner res poly x prec

_arb_fmpz_poly_evaluate_arb_rectangular res poly len x prec

arb_fmpz_poly_evaluate_arb_rectangular res poly x prec

_arb_fmpz_poly_evaluate_arb res poly len x prec

arb_fmpz_poly_evaluate_arb res poly x prec

_arb_fmpz_poly_evaluate_acb_horner res poly len x prec

arb_fmpz_poly_evaluate_acb_horner res poly x prec

_arb_fmpz_poly_evaluate_acb_rectangular res poly len x prec

arb_fmpz_poly_evaluate_acb_rectangular res poly x prec

_arb_fmpz_poly_evaluate_acb res poly len x prec

arb_fmpz_poly_evaluate_acb res poly x prec

Evaluates poly (given by a polynomial object or an array with len coefficients) at the given real or complex number, respectively using Horner's rule, rectangular splitting, or a default algorithm choice.

arb_fmpz_poly_deflation poly

Finds the maximal exponent by which poly can be deflated.

arb_fmpz_poly_deflate res poly deflation

Sets res to a copy of poly deflated by the exponent deflation.

arb_fmpz_poly_complex_roots roots poly flags prec

Writes to roots all the real and complex roots of the polynomial poly, computed to at least prec accurate bits. The root enclosures are guaranteed to be disjoint, so that all roots are isolated.

The real roots are written first in ascending order (with the imaginary parts set exactly to zero). The following nonreal roots are written in arbitrary order, but with conjugate pairs grouped together (the root in the upper plane leading the root in the lower plane).

The input polynomial must be squarefree. For a general polynomial, compute the squarefree part \(f / \gcd(f,f')\) or do a full squarefree factorization to obtain the multiplicities of the roots:

fmpz_poly_factor_t fac;
fmpz_poly_factor_init(fac);
fmpz_poly_factor_squarefree(fac, poly);

for (i = 0; i < fac->num; i++)
{
    deg = fmpz_poly_degree(fac->p + i);
    flint_printf("%wd roots of multiplicity %wd\n", deg, fac->exp[i]);
    roots = _acb_vec_init(deg);
    arb_fmpz_poly_complex_roots(roots, fac->p + i, 0, prec);
    _acb_vec_clear(roots, deg);
}

fmpz_poly_factor_clear(fac);

All roots are refined to a relative accuracy of at least prec bits. The output values will generally have higher actual precision, depending on the precision needed for isolation and the precision used internally by the algorithm.

This implementation should be adequate for general use, but it is not currently competitive with state-of-the-art isolation methods for finding real roots alone.

The following flags are supported:

• arb_fmpz_poly_roots_verbose

arb_fmpz_poly_cos_minpoly res n

Sets res to the monic minimal polynomial of \(2 \cos(2 \pi / n)\). This is a wrapper of FLINT's fmpz_poly_cos_minpoly, provided here for backward compatibility.

arb_fmpz_poly_gauss_period_minpoly res q n

Sets res to the minimal polynomial of the Gaussian periods \(\sum_{a \in H} \zeta^a\) where \(\zeta = \exp(2 \pi i / q)\) and H are the cosets of the subgroups of order \(d = (q - 1) / n\) of \((\mathbb{Z}/q\mathbb{Z})^{\times}\). The resulting polynomial has degree n. When \(d = 1\), the result is the cyclotomic polynomial \(\Phi_q\).

The implementation assumes that q is prime, and that n is a divisor of \(q - 1\) such that n is coprime with d. If any condition is not met, res is set to the zero polynomial.

This method provides a fast (in practice) way to construct finite field extensions of prescribed degree. If q satisfies the conditions stated above and \((q-1)/f\) additionally is coprime with n, where f is the multiplicative order of p mod q, then the Gaussian period minimal polynomial is irreducible over \(\operatorname{GF}(p)\) [CP2005].