Accurate computations with Said-Ball-Vandermonde matrices

TL;DR

This paper introduces an efficient, accurate algorithm for computing the bidiagonal decomposition of Said-Ball-Vandermonde matrices, which are strictly totally positive within (0,1), enabling improved numerical stability and performance.

Contribution

It presents a novel fast algorithm for bidiagonal decomposition of Said-Ball-Vandermonde matrices, leveraging their total positivity for enhanced numerical accuracy.

Findings

Algorithm is fast and guarantees high relative accuracy.

Numerical experiments demonstrate good behavior and stability.

Applicable to matrices with nodes in (0,1).

Abstract

A generalization of the Vandermonde matrices which arise when the power basis is replaced by the Said-Ball basis is considered. When the nodes are inside the interval (0,1), then those matrices are strictly totally positive. An algorithm for computing the bidiagonal decomposition of those Said-Ball-Vandermonde matrices is presented, which allows to use known algorithms for totally positive matrices represented by their bidiagonal decomposition. The algorithm is shown to be fast and to guarantee high relative accuracy. Some numerical experiments which illustrate the good behaviour of the algorithm are included.