FLINT3.7.0-dev
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arb_calc_newton_step

Synopsis#include <flint/arb_calc.h>
int arb_calc_newton_step(arb_t xnew, arb_calc_func_t func, void * param, const arb_t x, const arb_t conv_region, const arf_t conv_factor, slong prec)source

Performs one step of interval Newton iteration for a ball x=[m±r]x = [m \pm r] that contains exactly one root of ff, given a ball I⊇xI \supseteq x (conv_region) and the bound CC for 12∣f′′∣/∣f′∣\tfrac12 |f''| / |f'| on II (conv_factor). Computes m′=m−f(m)/f′(m)m' = m - f(m)/f'(m) in ball arithmetic at prec bits 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 ARB_CALC_SUCCESS; otherwise sets xnew to xx and returns ARB_CALC_NO_CONVERGENCE.

For the root ξ∈x\xi \in x, 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\eta between mm and ξ\xi, so ∣m′−ξ∣≤Cr2|m' - \xi| \le C r^2, and the rounding errors in m′m' are included in the ball arithmetic. This is only valid if CC really bounds 12∣f′′∣/∣f′∣\tfrac12 |f''|/|f'| on II.

Cite this page

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