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Accurate algorithms for Bessel matrices

Jorge Delgado, Héctor Orera, Juan Manuel Peña

TL;DR

The paper establishes that collocation matrices of Bessel and reverse Bessel polynomials at positive nodes are strictly totally positive, enabling their bidiagonal factorization to be computed with high relative accuracy. This structure allows reliable, high-accuracy computation of eigenvalues, singular values, inverses, and linear system solutions via Koev's HRA algorithms. Numerical experiments demonstrate markedly improved accuracy over standard methods, particularly for small spectral values, and extend to both Bessel and reverse Bessel matrices across moderate sizes. The results provide a practical framework for accurate algebraic operations on these structured matrices and connect total positivity with Bessel function theory.

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.

Accurate algorithms for Bessel matrices

TL;DR

The paper establishes that collocation matrices of Bessel and reverse Bessel polynomials at positive nodes are strictly totally positive, enabling their bidiagonal factorization to be computed with high relative accuracy. This structure allows reliable, high-accuracy computation of eigenvalues, singular values, inverses, and linear system solutions via Koev's HRA algorithms. Numerical experiments demonstrate markedly improved accuracy over standard methods, particularly for small spectral values, and extend to both Bessel and reverse Bessel matrices across moderate sizes. The results provide a practical framework for accurate algebraic operations on these structured matrices and connect total positivity with Bessel function theory.

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.
Paper Structure (5 sections, 8 theorems, 22 equations, 4 figures, 5 tables, 2 algorithms)

This paper contains 5 sections, 8 theorems, 22 equations, 4 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Bessel polynomials and matrices
  4. Reverse Bessel polynomials and matrices
  5. Numerical experiments

Key Result

theorem 1

Let $A$ be a nonsingular matrix. Then $A$ is TP if and only if there are no row exchanges in the NE of $A$ and $U^T$ and the pivots of both NE are nonnegative.

Figures (4)

  • Figure 1: Relative error for the minimal eigenvalue and singular value of $M_n$
  • Figure 2: Relative errors for $(M_n)^{-1}$
  • Figure 3: Relative error for the minimal eigenvalue and singular value of $M_n^r$
  • Figure 4: Relative errors for $(M_n^r)^{-1}$

Theorems & Definitions (15)

  • theorem 1
  • theorem 2
  • remark 1
  • theorem 3
  • proof
  • definition 1
  • theorem 4
  • proof
  • corollary 1
  • theorem 5
  • ...and 5 more