A rapidly convergent method for solving third-order polynomials

Abstract

We present a rapidly convergent method for solving cubic polynomial equations with real coefficients. The method is based on a power series expansion of a simplified form of Cardano’s formula using Newton’s generalized binomial theorem. Unlike Cardano’s formula and semi-analytical iterative root finders, the method is free from round-off error amplification when the polynomial coefficients differ by several orders of magnitude or when they do not differ much from each other, but are all large or small by many orders of magnitude. Validation of the method is assessed by casting a cubic equation of state as a polynomial in terms of the compressibility factor and the reduced molar volume for propylene at temperature and pressure conditions where Cardano’s formula and iterative root finders fail.