# Nonlinear Eigenvalue Problems: Theory and Numerics

Lead Research Organisation:
University of Manchester

Department Name: Mathematics

### Abstract

Nonlinear eigenvalue problems arise in a wide variety of science and engineering applications, such as the dynamic analysis of mechanical systems (where the eigenvalues represent vibrational frequencies), the linear stability of flows in fluid mechanics, the stability analysis of time-delay systems, and electronic band structure calculations for photonic crystals. These problems present many mathematical challenges. For some there is a lack of underlying theory. For others numerical methods struggle to provide any accuracy or to solve very large problems in a reasonable time.The trend towards extreme designs (such as in micro-electromechanical (MEMS) devices and superjumbo jets) means that these nonlinear eigenproblems are often poorly conditioned (hence difficult to solve accurately) while also having algebraic structure that should be exploited in a numerical method in order to ensure physical meaning of the computed results. As a specific example, in a project at TU Berlin modelling the sound and vibration levels in European high-speed trains it was found that standard finite element packages provided no correct figures in the computed solutions until linear algebra techniques of the type to be developed in this project were brought into play in the underlying quadratic (degree 2) eigenvalue problem (see the cover article in SIAM News, Nov. 2004).With the help of the funded research team I will develop theory and methods that enable the solution of new classes of emerging eigenproblems (e.g., rational) and more efficient and more accurate solution of existing problems. The project will exploit the new concept of structure preserving transformations for matrix polynomials and a new linearization-based approach for rational eigenproblems. For the general nonlinear eigenproblem, we will to devise good approximations to the nonlinear parts by rational or polynomial functions that will then be handled with techniques for the latter problems.The work will have significant impact through the provision of algorithms and software, either open source or distributed through numerical libraries, that enables efficient computer solution of these problems.

### Planned Impact

The numerical solution of algebraic eigenvalue problems, both linear and nonlinear, is a key technology underpinning many areas of computational science and engineering, including acoustics, aeronautics, control theory, fluid mechanics, population modelling, quantum physics, robotics, and structural engineering. In all these areas, the need for fast and numerically reliable solution of eigenvalue problems arises. The problems can be large, so that time to solution can be unacceptably long, and they can be very ill conditioned, making it difficult to obtain accurate solutions. The research in this proposal will provide new theory, algorithms and software that will allow: -- Solution of problems that could not previously be solved reliably, or for which there is no library software available. This includes many rational and nonlinear eigenvalue problems. -- Faster and more meaningful solution of problems for which available techniques do not fully respect the problem structure. This includes the quadratic eigenvalue problem. These advances will facilitate improved design of commercial products and design of more efficient and environmentally friendly transport. This project will help to train a new generation of research students and postdoctoral researchers in numerical algorithms and scientific computing. This will help build UK capacity in this important area, providing highly qualified researchers not only for academia but also for other employment sectors including industry and government. An important output of the research facilitated by the Fellowship is software. Library products such as LAPACK and NAG are widely used across diverse industries and application areas, including in finance, pharmaceuticals, visualization, earth sciences, and engineering, and on a wide range of machines. The general public will benefit from increased understanding of the research carried out within the project. The PI will prepare a talk aimed at a general audience. General aspects of computational mathematics will also be disseminated through articles for non-specialist publications. In summary, the activities of the Fellowship will benefit diverse people, organizations and institutions, ranging from academia, through industry and commerce, to individuals and the general public, within the UK and internationally. The timescales for the impact range from short to long. Rapid impact can be expected when, for example, algorithmic improvements can be quickly incorporated into new or improved library codes. We hope to commercialize the outputs of this Fellowship into NAG software through a Knowledge Transfer Partnership with NAG, to the benefit of the UK economy. Open source software will also be produced, documented and made freely available to all.

## People |
## ORCID iD |

Francoise Tisseur (Principal Investigator) |

### Publications

Al-Ammari M
(2012)

*Hermitian matrix polynomials with real eigenvalues of definite type. Part I: Classification*in Linear Algebra and its Applications
Al-Ammari M
(2012)

*Standard triples of structured matrix polynomials*in Linear Algebra and its Applications
Arslan B
(2019)

*The Structured Condition Number of a Differentiable Map between Matrix Manifolds, with Applications*in SIAM Journal on Matrix Analysis and Applications
Betcke T
(2013)

*NLEVP A Collection of Nonlinear Eigenvalue Problems*in ACM Transactions on Mathematical Software
Chen H
(2017)

*Improving the numerical stability of the Sakurai-Sugiura method for quadratic eigenvalue problems*in JSIAM Letters
De Terán F
(2014)

*Flanders' theorem for many matrices under commutativity assumptions*in Linear Algebra and its Applications
Guterman A
(2018)

*Linear isomorphisms preserving Green's relations for matrices over anti-negative semifields*in Linear Algebra and its Applications
Güttel S
(2017)

*The nonlinear eigenvalue problem*in Acta Numerica
Güttel S
(2016)

*Scaled and Squared Subdiagonal Padé Approximation for the Matrix Exponential*in SIAM Journal on Matrix Analysis and Applications
Hammarling S
(2013)

*An algorithm for the complete solution of quadratic eigenvalue problems*in ACM Transactions on Mathematical SoftwareDescription | Existence of reduced forms for matrix polynomials and development of algorithms to compute them. Have also shown that max-plus algebra is useful to numerical algebra as it allows "cheap" order of magnitude approximations of roots of scalar polynomials, moduli of eigenvalues and singular values. |

Exploitation Route | Our findings will make their way into more numerically reliable software for nonlinear eigenvalue computation. |

Sectors | Aerospace, Defence and Marine,Chemicals,Construction,Digital/Communication/Information Technologies (including Software),Electronics,Energy,Other |

URL | http://www.maths.manchester.ac.uk/~ftisseur/papers/ |

Description | They have been used to develop numerically reliable software in several libraries. |

First Year Of Impact | 2014 |

Sector | Aerospace, Defence and Marine,Chemicals,Construction,Digital/Communication/Information Technologies (including Software),Electronics,Energy,Other |

Impact Types | Economic |