Convergence Analysis of a Nonlinear Eigensolver Based on Rational Approximation of the Resolvent

Jul 16, 2026

Given a holomorphic matrix-valued function, eigenvalues emerge as the poles of a sketched resolvent. After constructing a high-quality rational approximation to this sketched resolvent, the poles of the rational approximant typically cluster near the true eigenvalues, enabling a flexible route for linear and nonlinear eigenvalue problems. The paper studies convergence of this pole-based eigensolver and introduces block probing and zooming-in to improve eigenvalue accuracy, with proven backward/forward stability and supporting numerical experiments.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages24

Reading Time1 h 0 min

Introduction

Nonlinear eigenvalue problems and resolvent-pole approach

Rational approximation via AAA

Limitations of naive polefinding

Convergence strategy improvements

Zooming-in

Block probing

Stability of polefinding

arXiv :2607 . 10377v1 [math .NA] 11 Jul 2026 CONVERGENCE ANALYSIS OF A NONLINEAR EIGENSOLVER BASED ON RATIONAL APPROXIMATION OF THE RESOLVENT NIAN SHAO˚ AND YUJI NAKATSUKASA: Abstract. Given a holomorphic matrix-valued function, the poles of its sketched resolvent are generically its eigenvalues. Once a good rational approximation of the sketched resolvent is obtained, the poles of this rational approximation typically lie close to those eigenvalues, thus providing a flexible framework for solving both linear and nonlinear eigenvalue problems. However, the accuracy of the computed eigenvalues is limited and remains poorly understood. This paper analyzes the convergence of this approach and demonstrates the effectiveness of two techniques to improve accuracy: block probing and zooming in. We also establish the backward and forward stability of polefinding for a barycentric rational form via a generalized eigenproblem. Numerical experiments demonstrate the sharpness of our theoretical results. Key words. Nonlinear eigenvalue problem, rational approximation, pole/root finding. AMS subject classifications. 65H17, 41A20, 65F15 1. Introduction. This work is concerned with nonlinear eigenvalue problems (NEPs) [19, 28] of the form (1.1) Tpλqv “ 0, λ P D and v P Cnzt0u. Throughout this paper, we assume that D Ă C is a nonempty domain enclosed by a Jordan curve, and T is an n ˆ n holomorphic matrix-valued function on some domain that contains the closure of D. We also assume that the NEP (1.1) is regular, that is, that det Tpz0 q ‰ 0 for some z0 P D. In this paper, we consider an approach that computes eigenvalues through the poles of rational approximations of the resolvent T ´1 . To our knowledge, such an approach was first proposed in [1] for symmetric eigenvalue problems, and later developed in [7] for the evaluation of resonances. A general discussion can be found in [33] . Let ΩL and ΩR be two n ˆ b probing matrices, such as Gaussian random matrices. Our approach starts with the following surrogate function for the resolvent: Fpzq :“ ΩHLTpzq´1ΩR , z P D. For generic ΩL and ΩR , the poles of the surrogate function F are exact eigenvalues of T. Let R be a good rational approximation of F in D. Then we expect that the eigenvalues of T are approximated by the poles of R. Once we obtain an approximate eigenvalue λ, the approximate eigenvector can be computed by SVD of Tpλq or residual inverse iteration [34] . The main ingredient of our approach is computing a rational approximation. The AAA (adaptive Antoulas-Anderson) algorithm [32] provides an efficient tool for rational approximation, and has been employed in a number of applications [33] . The AAA algorithm basically computes a barycentric representation of the rational approximation by a greedy strategy. For a matrix-valued function, the set-valued AAA method [27] computes a matrix-valued rational approximation in a similar approach. Unlike existing contour integral-based approaches [5, 14 , 24 , 35 , 38], which interpret ˚ Institute of Mathematics, EPF Lausanne, 1015 Lausanne, Switzerland ([nian.shao@epfl.ch](nian.shao@epfl.ch)) : Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK (nakat[sukasa@maths.ox.ac.uk](sukasa@maths.ox.ac.uk)). 2 NIAN SHAO AND YUJI NAKATSUKASA the quadrature rule as a rational approximation of the indicator function of D, our method directly approximates the resolvent itself and extracts the eigenvalues from its poles. Moreover, our method differs from the nonlinear eigensolvers based on rational approximation proposed in [17, 18 , 20 , 37], which approximate surrogate functions of T instead of its resolvent T ´1 . Even though the (set-valued) AAA can compute a near-best rational approximation effectively, the accuracy of the (nonlinear) eigensolver is not always satisfactory. Consider a toy example where Tpzq “ diagt0 .1 ,..., 0.9u ´ zI9 and ΩL and ΩR are two complex Gaussian random vectors. We uniformly sa

FAQs

How are eigenvalues obtained in this method?

Eigenvalues are extracted from the poles of rational approximations of the resolvent sketched from a holomorphic matrix-valued function.

Why is eigenvalue accuracy limited in the naive approach?

Even with well-separated eigenvalues, numerical examples show that the poles computed from a basic rational approximation can have incorrect digits, especially for interior eigenvalues.

What techniques does the paper propose to improve accuracy?

The paper demonstrates two strategies: zooming in (recursively subdividing the domain) and block probing (using probing matrices instead of vectors).

Jul 16, 2026

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