2023/03/31 by Kalantari, Bahman
#26C10 #65H04 #FOS: Mathematics #G.1.5 #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2303.17747
Building on a classification of zeros of cubic equations due to the 12-th century Persian mathematician Sharaf al-Din Tusi, together with Smale's theory of \it point estimation, we derive an efficient recipe for computing high-precision approximation to a real root of an arbitrary real cubic equation. First, via reversible transformations we reduce any real cubic equation into one of four canonical forms with 0, ± 1 coefficients, except for the constant term as ± q, q ≥ 0. Next, given any form, if ρq is an approximation to √[3]q to within a relative error of five percent, we prove a \it seed x0 in \ ρq, ± .95 ρq, -(1)/(3), 1 \ can be selected such that in t Newton iterations |xt - θq| ≤ √[3]q⋅ 2^-2t for some real root θq. While computing a good seed, even for approximation of √[3]q, is considered to be ``somewhat of black art'' (see Wikipedia), as we justify, ρq is readily computable from \it mantissa and \it exponent of q. It follows that the above approach gives a simple recipe for numerical approximation of solutions of real cubic equations independent of Cardano's formula.