FLINT3.7.0-dev
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_arb_poly_newton_step

Synopsis#include <flint/arb_poly.h>
int _arb_poly_newton_step(arb_t xnew, arb_srcptr poly, slong len, const arb_t x, const arb_t convergence_interval, const arf_t convergence_factor, slong prec)source

Performs one step of Newton's method for a ball x=[m±r]x = [m \pm r] that contains exactly one root of ff = {poly, len}, given a ball I⊇xI \supseteq x (convergence_interval) and the factor CC computed by _arb_poly_newton_convergence_factor. Computes m′=m−f(m)/f′(m)m' = m - f(m)/f'(m) in ball arithmetic and adds Cr2C r^2 to its radius. If the new ball x′x' lies in II and has a smaller radius than xx, sets xnew to x′x' and returns nonzero; otherwise sets xnew to xx and returns zero.

If ξ∈x\xi \in x is the root, Taylor's theorem gives m′−ξ=f′′(η)2f′(m)(m−ξ)2m' - \xi = \frac{f''(\eta)}{2 f'(m)} (m - \xi)^2 for some η∈I\eta \in I, hence ∣m′−ξ∣≤Cr2|m' - \xi| \le C r^2 (Isolating and refining real roots).

Cite this page

The FLINT team, “_arb_poly_newton_step (arb_poly.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/arb_poly/_arb_poly_newton_step.html (generated 2026-11-02).