Solution of Real Cubic Equations without Cardano's Formula

Abstract

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 $0$, $\pm 1$ coefficients, except for the constant term as $\pm q$, $q \geq 0$. Next, given any form, if $\rho_q$ is an approximation to $\sqrt[3]{q}$ to within a relative error of five percent, we prove a {\it seed} $x_0$ in $\{ \rho_q, \pm .95 \rho_q, -\frac{1}{3}, 1 \}$ can be selected such that in $t$ Newton iterations $|x_t - \theta_q| \leq \sqrt[3]{q}\cdot 2^{-2^{t}}$ for some real root $\theta_q$. While computing a good seed, even for approximation of $\sqrt[3]{q}$, is considered to be ``somewhat of black art'' (see Wikipedia), as we justify, $\rho_q$ 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.