# An algorithm for the complete solution of the quartic eigenvalue problem

## TL;DR

This paper introduces a new numerical algorithm for solving the quartic eigenvalue problem completely, including all eigenvalues and eigenvectors, using quadratification and preprocessing techniques to improve accuracy and efficiency.

## Contribution

The paper presents a novel method combining quadratification and preprocessing to solve the quartic eigenvalue problem fully, outperforming existing methods in accuracy and robustness.

## Key findings

- Numerical examples demonstrate the method's superior accuracy.
- Backward error analysis confirms the method's reliability.
- Preprocessing effectively deflates zero and infinite eigenvalues.

## Abstract

Quartic eigenvalue problem $(\lambda^4 A + \lambda^3 B + \lambda^2C + \lambda D + E)x = \mathbf{0}$ naturally arises e.g. when solving the Orr-Sommerfeld equation in the analysis of the stability of the {Poiseuille} flow, in theoretical analysis and experimental design of locally resonant phononic plates, modeling a robot with electric motors in the joints, calibration of catadioptric vision system, or e.g. computation of the guided and leaky modes of a planar waveguide. This paper proposes a new numerical method for the full solution (all eigenvalues and all left and right eigenvectors) that is based on quadratification, i.e. reduction of the quartic problem to a spectraly equivalent quadratic eigenvalue problem, and on a careful preprocessing to identify and deflate zero and infinite eigenvalues before the linearized quadratification is forwarded to the QZ algorithm. Numerical examples and backward error analysis confirm that the proposed algorithm is superior to the available methods.