FLINT3.7.0-dev
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mp_real_root_ui, mp_real_rroot_ui, _mp_real_root_ui_order

Synopsis#include <flint/mp_real.h>
void mp_real_root_ui(mp_real_t res, const mp_real_t x, ulong k, slong n)source
void mp_real_rroot_ui(mp_real_t res, const mp_real_t x, ulong k, slong n)source
void _mp_real_root_ui_order(mp_real_t res, const mp_real_t x, ulong k, slong n, int r, int recip)source

Set res to x1/kx^{1/k} resp. x−1/kx^{-1/k} for x>0x > 0 and 1≤k<1 \le k < MP_REAL_ROOT_K_MAX. They throw for k=0k = 0, for xx not positive (the third function for k<2k < 2 as well). The third function takes the order rr of the iteration (2≤r≤152 \le r \le 15, or 0 for the default) and recip = 1 for the reciprocal root.

For k=1k = 1 the result is xx resp. 1/x1/x, and for k=2k = 2 these call mp_real_sqrt and mp_real_rsqrt. Otherwise one step of order rr of a binomial-series iteration is taken from an approximation zz of x−1/kx^{-1/k} accurate to about 1/r1/r of the precision, obtained recursively (the recursion starts from a double):

x−1/k=z(1−u)−1/k,x1/k=xzk−1(1−u)−(k−1)/k,u=1−xzk,x^{-1/k} = z (1 - u)^{-1/k}, \qquad x^{1/k} = x z^{k-1} (1 - u)^{-(k-1)/k}, \qquad u = 1 - x z^k,

with the series (1−u)−b=∑jcjuj(1 - u)^{-b} = \sum_j c_j u^j truncated after ur−1u^{r-1} and its tail, below 2cr∣u∣r2 c_r |u|^r for ∣u∣≤1/2|u| \le 1/2, added to the radius. The series is summed over a common denominator of the cjc_j, so that it costs one division by an integer; the coefficient tables are cached per thread. The default order is 3 for k<8k < 8 and 4 otherwise. Only the size of the final uu is computed rigorously, so the radius of the recursive approximation is discarded.

Cite this page

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