Division Free Inversion of Certain Class of Vandermonde Matrices

The study of matrices and determinants has a long history.The subject has seen great progress with the advent of computers.In the present era of building models of machine learning the use of matrices of large sizes has become imminent.Solving a system of equations using matrices has been a paradigm shift in the theory of equations and fits perfectly well for solving them with computers.One of the challenges that still persist is when the matrix is ill-conditioned (either very large or small numbers as matrix elements).As a result, one might encounter instabilities in inversion, either in the form of elemental inaccuracy or the final inverse itself.This challenge paved the way for inventing novel algorithms and numerical approaches to handle the calculation of these ill-conditioned matrices.In this article we propose a unique method for Vandermonde matrices.Here we convert a class of analytically solvable system of equations to a problem that in principle can be numerically friendly and accurate.We show that the method of conversion avoids division completely and is well behaved to handle ill-conditioned matrices.We also provide simple programs that can be used as a model to use on large dimensional matrices and show how the errors can be minimized with the proposed approach.