Total positivity and accurate computations related to $q$-Abel polynomials

TL;DR

This paper explores the total positivity of $q$-Abel polynomial bases, providing factorizations and accuracy results that enable high-precision solutions to algebraic problems involving ill-conditioned matrices.

Contribution

It characterizes the total positivity of the $q$-Abel basis change matrix and extends high-accuracy Vandermonde results to negative nodes, improving numerical solutions.

Findings

Total positivity of the $q$-Abel basis change matrix is established.

High relative accuracy is achieved for algebraic problems with $q$-Abel polynomials.

Numerical tests confirm the theoretical advantages over standard methods.

Abstract

The attainment of accurate numerical solutions of ill-conditioned linear algebraic problems involving totally positive matrices has been gathering considerable attention among researchers over the last years. In parallel, the interest of $q$-calculus has been steadily growing in the literature. In this work the $q$-analogue of the Abel polynomial basis is studied. The total positivity of the matrix of change of basis between monomial and $q$-Abel bases is characterized, providing its bidiagonal factorization. Moreover, well-known high relative accuracy results of Vandermonde matrices corresponding to increasing positive nodes are extended to the decreasing negative case. This further allows to solve with high relative accuracy several algebraic problems concerning collocation, Wronskian and Gramian matrices of $q$-Abel polynomials. Finally, a series of numerical tests support the presented theoretical results and illustrate the goodness of the method where standard approaches fail to deliver accurate solutions.