Enhancing cubic polynomial solutions: A comprehensive analysis and iterative refinement strategy in response to the versatility of cubic equations of state

Abstract

The study presents a comprehensive evaluation of ten distinct methods for solving cubic equations, encompassing both analytical, numerical, and linear algebra-based approaches. Among these, an analytical technique involving the transformation of cubic equations into Chebyshev polynomials emerged as the most efficient. To mitigate issues arising from numerical round-off errors, the study proposes an iterative refinement strategy. Furthermore, the research introduces a novel criterion grounded in Vieta's formulas to identify and isolate areas where round-off errors may occur. This criterion aids in the application of the proposed iterative refinement technique. The novel solution method, which involves the conversion of cubic equations into Chebyshev polynomials and subsequent error-checking and iterative refinement, exhibits significant advantages. Firstly, it proves efficient, reducing computational time by 15–20 % in comparison to the well-established Cardano analytical method. It also demonstrates robustness in effortlessly identifying regions of potential error and offers a method for iterative improvement when needed. Additionally, its reliability is underscored by its independence from the specific cubic equation of state under consideration, making it well-suited for the intensive solution of a wide range of cubic equations.