# NONLINEAR EIGENVALUE PROBLEMS: THEORY AND ALGORITHMS

• Gian Maria Negri Porzio

Student thesis: Phd

### Abstract

In this thesis we focus on the theoretical and computational aspects of nonlinear eigenvalue problems (NEPs), which arise in several fields of computational science and engineering, such as fluid dynamics, optics, and structural engineering. In the last twenty years several researchers devoted their time in studying efficient and precise ways to solve NEPs, which cemented their importance in numerical linear algebra. The most successful algorithms developed towards this goal are either based on contour integrals, or on rational approximations and linearizations. The first part of the thesis is dedicated to contour integral algorithms. In this framework, one computes specific integrals of a holomorphic function G(z) over the contour of a target region and exploits results of complex analysis to retrieve the eigenvalues of G(z) inside it. Our main contribution consists in having expanded the theory to include meromorphic functions, i.e., functions with poles inside the target region. We showed that under some circumstances, these algorithms can mistake a pole for an eigenvalue, but these situations are easily recognised and the main results from the holomorphic case can be extended. Furthermore, we proposed several heuristics to automatically choose the parameters that are needed to precisely retrieve the eigenpairs. In the second part of the thesis, we focus on rational approximations. Our goal was developing algorithms that construct robust, i.e., reliable for a given tolerance and scaling independent, rational approximations for functions given in split form or in black-box form. In the first case, we proposed a variant of the set-valued AAA, named weighted AAA, which guarantees to return an approximation with the chosen accuracy. In the second one, we built a two-phase approach, where an initial step performed by the surrogate AAA is followed by a cyclic Leja-Bagby refinement. We concluded the section with numerous numerical experiments based on the NLEVP library, comparing contour integral and rational approximation algorithms. The third and final part of the thesis is about tropical linear algebra. Our research on this topic started has a way to set the parameters of the aforementioned contour integral algorithms: in order to do that, we extended the theory of tropical roots from tropical polynomials to tropical Laurent series. Unlike in the polynomial case, a tropical Laurent series may have infinite tropical roots, but they are still in bijection with the slopes of the associated Newton polygon and they still provide annuli of exclusion for the eigenvalues of the Laurent series.
Date of Award 31 Dec 2021 English The University of Manchester Nicholas Higham (Supervisor) & Francoise Tisseur (Supervisor)

### Keywords

• Eigenvalue Problems
• Tropical Algebra
• Tropical Roots
• NLEVP
• Numerical Analysis
• Eigenvalue
• Rational Approximations
• Nonlinear eigenvalue problems
• Numerical Linear Algebra
• Contour Integrals
• Eigenpair

'