nx_root.nx

nx_root.nx -- Newton-Raphson nth-root for i64 + Q14 fixed point. Bits-up math primitive sibling of nx_isqrt.nx. Computes floor of the integer nth-root and the Q14-scaled fractional nth-root via the same iteration shape. Unblocks two queued upgrades: * nx_sampler.nx Karras schedule (Karras et al. 2022 EDM): needs (sigma_min / sigma_max)^(1/rho) with rho = 7. * nx_ms_ssim.nx Wang weighted geometric mean (Wang et al. 2003): needs fractional alpha^a * c^b * s^g powers per scale. ===== Math ======================================================= Newton-Raphson iteration for x^n = T, x > 0: x_{k+1} = ((n-1) * x_k + T / x_k^(n-1)) / n Starting from x_0 = max(T, 1) (an upper bound), each iteration strictly DECREASES x until convergence at floor(T^(1/n)). Quadratic convergence near the root: O(log log n) iterations after the initial linear phase, O(log n) total. Integer convergence test: terminate when x_{k+1} >= x_k. ===== Q14 wrapper =============================================== For fractional nth-root with target represented in Q14: nth_root_q14(target_q14, n) ≈ target^(1/n) in Q14 We iterate directly in Q14 using a Q14-aware power helper `nx_pow_q14` that does (x_q14 * x_q14 / Q14) chain. This keeps intermediate values bounded near the target magnitude rather than overflowing via a target * Q14^(n-1) scale-up. genealogy_id: heron_alexandria_iterative_root + newton_raphson_1671 + wikipedia_nth_root_algorithm lineage_id: substrate_root_v1