History of confluent Vandermonde matrices and inverting them algorithms

TL;DR

This paper reviews the history and algorithms for inverting confluent Vandermonde matrices, providing a practical quadratic-time numerical method suitable for large root multiplicities.

Contribution

It offers a comprehensive historical overview and introduces a non-symbolic, implementable quadratic-time algorithm for inverting confluent Vandermonde matrices.

Findings

Provides a history of confluent Vandermonde matrices since 1891.

Surveys existing algorithms for matrix inversion.

Introduces a practical numerical inversion algorithm.

Abstract

The author was encouraged to write this review by numerous enquiries from researchers all over the world, who needed a ready-to-use algorithm for the inversion of confluent Vandermonde matrices which works in quadratic time for any values of the parameters allowed by the definition, including the case of large root multiplicities of the characteristic polynomial. Article gives the history of the title special matrix since 1891 and surveys algorithms for solving linear systems with the title class matrix and inverting it. In particular, it presents, also by example, a numerical algorithm which does not use symbolic computations and is ready to be implemented in a general-purpose programming language or in a specific mathematical package.