Approximating the roots of a cubic equation in one variable through simultaneous geometric structures in the plane of complex numbers

This paper describes a numerical procedure, called cubic LC method, whose objective is to approximate the roots of a cubic polynomial equation in one complex variable by means of a set of simultaneous geometric structures in the complex plane; each geometric structure in this set consists of three components, interrelated through Vieta’s formulas: 1: A line L1 with inclination angle theta that seeks to contain one of the roots of the cubic equation. 2: A circumference C that depends on L1 and seeks to contain the product of the two other roots. 3: A semi-line LT, derived from L1, that also seeks to contain the product of the two other roots. The possible intersections between semi-line LT and circumference C provide information on how close line L1 is to one of the roots; this information is encoded into real proximity functions that depend on theta: the closer these proximity functions are to zero, the closer line L1 is to one of the roots. The estimated zeros of these proximity functions help us find approximations to the roots of the cubic equation. The cubic LC method is illustrated through a numerical example, whose results can be reproduced with the help of two R programs included in the supplementary material for this paper; these two R programs can also be used to analyze arbitrary cubic equations. This paper concludes by briefly outlining how to extend the cubic LC method to univariate polynomial equations of degree four or higher.