The LC Method: A parallelizable numerical method for approximating the roots of single-variable polynomials

TL;DR

The paper introduces the LC method, a parallelizable geometric approach using lines and circles in the complex plane to approximate polynomial roots efficiently, demonstrated with numerical examples and accessible R code.

Contribution

It presents the LC method, a novel geometric and parallelizable technique for approximating roots of single-variable polynomials, with implementation details and numerical demonstrations.

Findings

Effective approximation of roots for quadratic, cubic, and quartic polynomials.

Parallel processing enables simultaneous construction of geometric structures.

Accessible R programs facilitate reproduction of results on personal computers.

Abstract

The LC method described in this work seeks to approximate the roots of polynomial equations in one variable. This book allows you to explore the LC method, which uses geometric structures of Lines L and Circumferences C in the plane of complex numbers, based on polynomial coefficients. These structures depend on the inclination angle of a line with fixed point that seeks to contain one of the roots; they are associated with an error measure that indicates the degree of proximity to that root, without knowing a priori its location. Using a computer with parallel processing capabilities, it is feasible to construct several of these geometric structures at the same time, varying the inclination angle of the lines with fixed point, in order to obtain an error measure map, with which it is possible to identify, approximately, the location of all polynomial roots. To show how the LC method works, this book includes numerical examples for quadratic, cubic, and quartic polynomials, and also for polynomials of degree greater than or equal to 5; this book also includes R programs that allow you to reproduce the results of the examples on a typical personal computer; these R programs use vectorization of operations instead of loops, which can be seen as a basic and accessible form of parallel processing. This book, in the end, invites us to explore beyond the basic ideas and concepts described here, motivating the development of a more efficient and complete computational implementation of the LC method.