Solution of Real Cubic Equations without Cardano's Formula
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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 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 ρ_q is an approximation to √(q) to within a relative error of five percent, we prove a seed x_0 in {ρ_q, ± .95 ρ_q, -1/3, 1 } can be selected such that in t Newton iterations |x_t - θ_q| ≤√(q)· 2^-2^t for some real root θ_q. While computing a good seed, even for approximation of √(q), is considered to be “somewhat of black art” (see Wikipedia), as we justify, ρ_q is readily computable from mantissa and 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.
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