# Nonlinearizing two-parameter eigenvalue problems

## TL;DR

This paper introduces a method to convert linear two-parameter eigenvalue problems into nonlinear eigenvalue problems by eliminating one equation, enabling the use of NEP techniques and simplifying certain structured problems.

## Contribution

The paper presents a novel transformation technique that converts two-parameter eigenvalue problems into NEPs, with theoretical characterization and computational strategies, including special cases and error analysis.

## Key findings

- Transformation is effective for problems with smaller eliminated equations.
- The method generalizes companion linearization for polynomial eigenvalue problems.
- Error analysis shows the approach is stable with backward stable solvers.

## Abstract

We investigate a technique to transform a linear two-parameter eigenvalue problem, into a nonlinear eigenvalue problem (NEP). The transformation stems from an elimination of one of the equations in the two-parameter eigenvalue problem, by considering it as a (standard) generalized eigenvalue problem. We characterize the equivalence between the original and the nonlinearized problem theoretically and show how to use the transformation computationally. Special cases of the transformation can be interpreted as a reversed companion linearization for polynomial eigenvalue problems, as well as a reversed (less known) linearization technique for certain algebraic eigenvalue problems with square-root terms. Moreover, by exploiting the structure of the NEP we present algorithm specializations for NEP methods, although the technique also allows general solution methods for NEPs to be directly applied. The nonlinearization is illustrated in examples and simulations, with focus on problems where the eliminated equation is of much smaller size than the other two-parameter eigenvalue equation. This situation arises naturally in domain decomposition techniques. A general error analysis is also carried out under the assumption that a backward stable eigenvalue solver method is used to solve the eliminated problem, leading to the conclusion that the error is benign in this situation.