On computing the zeros of a class of Sobolev orthogonal polynomials

TL;DR

The paper addresses the zeros of a Sobolev-orthogonal hypergeometric polynomial sequence $${}_2F_2(-n,1;\\alpha+1,\\kappa+1;x)$$ by transforming the associated four-term recurrence into a generalized eigenvalue problem and further into a comrade-matrix eigenproblem. The zeros of $${}\mathfrak{L}_n(x)$$ are shown to be the eigenvalues of a symmetric tridiagonal matrix with a rank-one update, enabling the use of a structure-preserving QR algorithm to compute all zeros in $O(n^2)$ time and $O(n)$ memory. The authors compare unsymmetric QR, a standard comrade-matrix QR, and a fast structure-preserving QR method across double and multiple precision, illustrating the fast method’s efficiency and stability within practical parameter ranges while highlighting conditioning issues for larger $\alpha$ and $\kappa$. This work provides a practically efficient tool to study the asymptotic distribution of zeros of these polynomials, aiding theoretical investigations into their spectral behavior.

Abstract

A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre polynomials. The theoretical study of the asymptotic distribution of the spectrum of these polynomials is an active research topic. In this article we do not contribute to the theory, but provide a practical method to contribute to further and better understanding of the asymptotic behavior. The polynomials under consideration fit into the class of Sobolev orthogonal polynomials, satisfying a four--term recurrence relation. This allows computing the roots via a generalized eigenvalue problem. After condition enhancing similarity transformations, the problem is transformed into the computation of the eigenvalues of a comrade matrix, which is a symmetric tridiagonal modified by a rank--one matrix. The eigenvalues are then retrieved by relying on an existing structured rank based fast algorithm. Numerical examples are reported studying the accuracy, stability and conforming the efficiency for various parameter settings of the proposed approach.

Figures (8)

• Figure 1: Comparison of timings in seconds between the classical order $n^3$ QR algorithm and the fast $n^2$ structured one.
• Figure 2: Maximum absolute error on the eigenvalues for the non-structured QR algorithm on the nonsymmetrized matrix $X$.
• Figure 3: Comparison of the absolute error of the three algorithms and two additional curves representing upper bounds for both nonsymmetric and symmetric solvers. Note that the legend holds for all subfigures, but the y-axis scale differs significantly.
• Figure 4: High precision and double precision plot of part of the spectrum, to illustrate the ill-conditioning. The eigenvalues computed in double precision deviate from the correct ones.
• Figure 5: Separation lines in the $\alpha$$\kappa$-plane to distinguish two regions. Below left the eigenvalues are real, crossing the line results in complex eigenvalues.

Theorems & Definitions (3)

• Remark 1
• Theorem 1
• proof