FLINT3.7.0-dev
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_mp_real_inv_newton_basecase, _mp_real_inv_newton

Synopsis#include <flint/mp_real.h>
void _mp_real_inv_newton_basecase(nn_ptr q, nn_srcptr a, slong an, slong n)source
void _mp_real_inv_newton(nn_ptr q, nn_srcptr a, slong an, slong n)source

For a∈[1/B,1)a \in [1/B, 1) with aan−1≠0a_{a_n - 1} \ne 0, set (q,n+2)(q, n + 2) to an approximation of 1/a∈(1,B]1/a \in (1, B] with absolute error at most 4B−n/a4 B^{-n}/a.

The basecase takes the truncated reciprocal of the top min⁡(an,n+1)\min(a_n, n + 1) limbs of aa by _flint_mpn_inv_basecase. The main function runs a Newton iteration on middle products, each step of second order, T′=T+T(1−aT)T' = T + T(1 - aT), or third order, T′=T(1−e+e2)T' = T(1 - e + e^2) with e=aT−1e = aT - 1; the third-order step is used for 24≤n≤8024 \le n \le 80 and n≥1000n \ge 1000. See Integer division and Newton iteration.

Cite this page

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