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Uniform Asymptotic approximation and numerical evaluation of the Reverse Generalized Bessel Polynomial zeros

T. M. Dunster, Amparo Gil, Diego Ruiz-Antolin, Javier Segura

TL;DR

This work addresses the problem of locating all zeros of the reverse generalized Bessel polynomials $\theta_n(z;a)$ in the large-$n$ regime with $a$ growing at most linearly with $n$. It develops a Liouville-Green (WKB) framework and Airy-type uniform asymptotics to derive a uniform expansion for the zeros, $t_m(u,a)\sim u\sum_{s\ge0} \tau_{m,s}(\alpha)/u^{2s}$, with explicit corrections up to $\tau_{m,4}$, and shows these approximations achieve machine-precision accuracy for moderate to large $n$. The authors then introduce a Taylor-series based numerical algorithm that uses these asymptotic zeros as starting values to efficiently compute all remaining complex zeros via iterative refinement, avoiding expensive matrix methods. The resulting approach provides a fast, robust, and scalable method for computing the zeros across broad parameter ranges, with practical implications for applications in applied mathematics and related fields.

Abstract

Uniform asymptotic expansions are derived for the zeros of the reverse generalized Bessel polynomials of large degree $n$ and real parameter $a$. It is assumed that $-Δ_{1} n+\frac{3}{2} \leq a \leq Δ_{2} n$ for fixed arbitrary $Δ_{1} \in (0,1)$ and bounded positive $Δ_{2}$. For this parameter range at most one of the zeros is real, with the rest being complex conjugates. The new expansions are uniformly valid for all the zeros, and are shown to be highly accurate for moderate or large values of $n$. They are consequently used as initial values in a very efficient numerical algorithm designed to obtain the remaining complex zeros using Taylor series.

Uniform Asymptotic approximation and numerical evaluation of the Reverse Generalized Bessel Polynomial zeros

TL;DR

This work addresses the problem of locating all zeros of the reverse generalized Bessel polynomials in the large- regime with growing at most linearly with . It develops a Liouville-Green (WKB) framework and Airy-type uniform asymptotics to derive a uniform expansion for the zeros, , with explicit corrections up to , and shows these approximations achieve machine-precision accuracy for moderate to large . The authors then introduce a Taylor-series based numerical algorithm that uses these asymptotic zeros as starting values to efficiently compute all remaining complex zeros via iterative refinement, avoiding expensive matrix methods. The resulting approach provides a fast, robust, and scalable method for computing the zeros across broad parameter ranges, with practical implications for applications in applied mathematics and related fields.

Abstract

Uniform asymptotic expansions are derived for the zeros of the reverse generalized Bessel polynomials of large degree and real parameter . It is assumed that for fixed arbitrary and bounded positive . For this parameter range at most one of the zeros is real, with the rest being complex conjugates. The new expansions are uniformly valid for all the zeros, and are shown to be highly accurate for moderate or large values of . They are consequently used as initial values in a very efficient numerical algorithm designed to obtain the remaining complex zeros using Taylor series.
Paper Structure (5 sections, 1 theorem, 77 equations, 5 figures, 3 tables)

This paper contains 5 sections, 1 theorem, 77 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Liouville-Green coefficients
  3. Uniform asymptotic expansions for the zeros
  4. Numerical examples
  5. A numerical algorithm to compute the zeros of theta_n(z;a)

Key Result

Lemma 2.1

Each $(z-z_{1})^{1/2}\{\mathrm{E}_{2s+1}(\alpha,\phi)+d_{2s+1}(\alpha)\}$ ($s=0,1,2,\ldots$), regarded as a function of $z$, is meromorphic at $z=z_{1}$.

Figures (5)

  • Figure 1: Relative errors as a function of $a$ (with $n$ fixed at $15$ and $m$ at $3$).
  • Figure 1: Zeros in the second quadrant obtained by the numerical algorithm for $n=30$ and $a=1.2, \, 30.7$.
  • Figure 2: Plot of the function $F(w)$ for $a=1.01$, $m=10$ and $n=30$.
  • Figure 2: Zeros in the second quadrant obtained by the numerical algorithm for $n=500$ and $a=1.2, \, 30.7$.
  • Figure 3: Relative errors of the computed zeros obtained using the numerical algorithm.

Theorems & Definitions (1)

  • Lemma 2.1