Accurate algorithms for Bessel matrices

TL;DR

This paper proves strict total positivity of Bessel polynomial collocation matrices, introduces an accurate bidiagonal factorization method for high-precision computations, and demonstrates these results with numerical examples.

Contribution

It establishes strict total positivity for Bessel polynomial collocation matrices and develops an accurate bidiagonal factorization method for eigenvalues, singular values, and inverses.

Findings

All minors of Bessel polynomial collocation matrices are positive.

The bidiagonal factorization enables high-accuracy computations.

Numerical examples confirm theoretical results.

Abstract

In this paper, we prove that any collocation matrix of Bessel polynomials at positive points is strictly totally positive, that is, all its minors are positive. Moreover, an accurate method to construct the bidiagonal factorization of these matrices is obtained and used to compute with high relative accuracy the eigenvalues, singular values and inverses. Similar results for the collocation matrices for the reverse Bessel polynomials are also obtained. Numerical examples illustrating the theoretical results are included.