Nonlinear eigenproblem

Short description: Mathematical equation involving a matrix-valued function that is singular at the eigenvalue.

In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form

M(λ)x=0,

where x≠0 is a vector, and M is a matrix-valued function of the number λ. The number λ is known as the (nonlinear) eigenvalue, the vector x as the (nonlinear) eigenvector, and (λ,x) as the eigenpair. The matrix M(λ) is singular at an eigenvalue λ.

## Contents

* 1 Definition
* 2 Special cases
* 3 Jordan chains
* 4 Mathematical software
* 5 Eigenvector nonlinearity
* 6 References
* 7 Further reading

## Definition

In the discipline of numerical linear algebra the following definition is typically used.[1][2][3][4]

Let Ω⊆ℂ, and let M:Ω->ℂn×n be a function that maps scalars to matrices. A scalar λ∈ℂ is called an eigenvalue, and a nonzero vector x∈ℂn is called a right eigenvector if M(λ)x=0. Moreover, a nonzero vector y∈ℂn is called a left eigenvector if yHM(λ)=0H, where the superscript  H denotes the Hermitian transpose. The definition of the eigenvalue is equivalent to det⁡(M(λ))=0, where det⁡() denotes the determinant.[1]

The function M is usually required to be a holomorphic function of λ (in some domain Ω).

In general, M(λ) could be a linear map, but most commonly it is a finite-dimensional, usually square, matrix.

Definition: The problem is said to be regular if there exists a z∈Ω such that det⁡(M(z))≠0. Otherwise it is said to be singular.[1][4]

Definition: An eigenvalue λ is said to have algebraic multiplicity k if k is the smallest integer such that the kth derivative of det⁡(M(z)) with respect to z, in λ is nonzero. In formulas that dkdet⁡(M(z))dzk|z=λ≠0 but dℓdet⁡(M(z))dzℓ|z=λ=0 for ℓ=0,1,2,…,k−1.[1][4]

Definition: The geometric multiplicity of an eigenvalue λ is the dimension of the nullspace of M(λ).[1][4]

## Special cases

The following examples are special cases of the nonlinear eigenproblem.

* The (ordinary) eigenvalue problem: M(λ)=A−λI.
* The generalized eigenvalue problem: M(λ)=A−λB.
* The quadratic eigenvalue problem: M(λ)=A0+λA1+λ2A2.
* The polynomial eigenvalue problem: M(λ)=∑i=0mλiAi.
* The rational eigenvalue problem: M(λ)=∑i=0m1Aiλi+∑i=1m2Biri(λ), where ri(λ) are rational functions.
* The delay eigenvalue problem: M(λ)=−Iλ+A0+∑i=1mAie−τiλ, where τ1,τ2,…,τm are given scalars, known as delays.

## Jordan chains

Definition: Let (λ0,x0) be an eigenpair. A tuple of vectors (x0,x1,…,xr−1)∈ℂn×ℂn×…×ℂn is called a Jordan chain if∑k=0ℓM(k)(λ0)xℓ−k=0,for ℓ=0,1,…,r−1, where M(k)(λ0) denotes the kth derivative of M with respect to λ and evaluated in λ=λ0. The vectors x0,x1,…,xr−1 are called generalized eigenvectors, r is called the length of the Jordan chain, and the maximal length a Jordan chain starting with x0 is called the rank of x0.[1][4]

Theorem:[1] A tuple of vectors (x0,x1,…,xr−1)∈ℂn×ℂn×…×ℂn is a Jordan chain if and only if the function M(λ)χℓ(λ) has a root in λ=λ0 and the root is of multiplicity at least ℓ for ℓ=0,1,…,r−1, where the vector valued function χℓ(λ) is defined asχℓ(λ)=∑k=0ℓxk(λ−λ0)k.

## Mathematical software

* The eigenvalue solver package SLEPc contains C-implementations of many numerical methods for nonlinear eigenvalue problems.[5]
* The NLEVP collection of nonlinear eigenvalue problems is a MATLAB package containing many nonlinear eigenvalue problems with various properties. [6]
* The FEAST eigenvalue solver is a software package for standard eigenvalue problems as well as nonlinear eigenvalue problems, designed from density-matrix representation in quantum mechanics combined with contour integration techniques.[7]
* The MATLAB toolbox NLEIGS contains an implementation of fully rational Krylov with a dynamically constructed rational interpolant.[8]
* The MATLAB toolbox CORK contains an implementation of the compact rational Krylov algorithm that exploits the Kronecker structure of the linearization pencils.[9]
* The MATLAB toolbox AAA-EIGS contains an implementation of CORK with rational approximation by set-valued AAA.[10]
* The MATLAB toolbox RKToolbox (Rational Krylov Toolbox) contains implementations of the rational Krylov method for nonlinear eigenvalue problems as well as features for rational approximation. [11]
* The Julia package NEP-PACK contains many implementations of various numerical methods for nonlinear eigenvalue problems, as well as many benchmark problems.[12]
* The review paper of Güttel & Tisseur[1] contains MATLAB code snippets implementing basic Newton-type methods and contour integration methods for nonlinear eigenproblems.

## Eigenvector nonlinearity

Eigenvector nonlinearities is a related, but different, form of nonlinearity that is sometimes studied. In this case the function M maps vectors to matrices, or sometimes hermitian matrices to hermitian matrices.[13][14]

## References

1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Güttel, Stefan; Tisseur, Françoise (2017). "The nonlinear eigenvalue problem" (in en). Acta Numerica 26: 1–94. doi:10.1017/S0962492917000034. ISSN 0962-4929. http://eprints.maths.manchester.ac.uk/2538/1/main.pdf.
2. ↑ Ruhe, Axel (1973). "Algorithms for the Nonlinear Eigenvalue Problem". SIAM Journal on Numerical Analysis 10 (4): 674–689. doi:10.1137/0710059. ISSN 0036-1429. Bibcode: 1973SJNA...10..674R. https://epubs.siam.org/doi/10.1137/0710059.
3. ↑ Mehrmann, Volker; Voss, Heinrich (2004). "Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods" (in en). GAMM-Mitteilungen 27 (2): 121–152. doi:10.1002/gamm.201490007. ISSN 1522-2608. https://onlinelibrary.wiley.com/doi/abs/10.1002/gamm.201490007.
4. ↑ 4.0 4.1 4.2 4.3 4.4 Voss, Heinrich (2014). "Nonlinear eigenvalue problems". in Hogben, Leslie. Handbook of Linear Algebra (2 ed.). Boca Raton, FL: Chapman and Hall/CRC. ISBN 9781466507289. https://www.mat.tuhh.de/forschung/rep/rep174.pdf.
5. ↑ Hernandez, Vicente; Roman, Jose E.; Vidal, Vicente (September 2005). "SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems". ACM Transactions on Mathematical Software 31 (3): 351–362. doi:10.1145/1089014.1089019.
6. ↑ Betcke, Timo; Higham, Nicholas J.; Mehrmann, Volker; Schröder, Christian; Tisseur, Françoise (February 2013). "NLEVP: A Collection of Nonlinear Eigenvalue Problems". ACM Transactions on Mathematical Software 39 (2): 1–28. doi:10.1145/2427023.2427024.
7. ↑ Polizzi, Eric (2020). "FEAST Eigenvalue Solver v4.0 User Guide". arXiv:2002.04807 [cs.MS].
8. ↑ Güttel, Stefan; Van Beeumen, Roel; Meerbergen, Karl; Michiels, Wim (1 January 2014). "NLEIGS: A Class of Fully Rational Krylov Methods for Nonlinear Eigenvalue Problems". SIAM Journal on Scientific Computing 36 (6): A2842–A2864. doi:10.1137/130935045. Bibcode: 2014SJSC...36A2842G.
9. ↑ Van Beeumen, Roel; Meerbergen, Karl; Michiels, Wim (2015). "Compact rational Krylov methods for nonlinear eigenvalue problems". SIAM Journal on Matrix Analysis and Applications 36 (2): 820–838. doi:10.1137/140976698. https://lirias.kuleuven.be/handle/123456789/490706.
10. ↑ Lietaert, Pieter; Meerbergen, Karl; Pérez, Javier; Vandereycken, Bart (13 April 2022). "Automatic rational approximation and linearization of nonlinear eigenvalue problems". IMA Journal of Numerical Analysis 42 (2): 1087–1115. doi:10.1093/imanum/draa098.
11. ↑ Berljafa, Mario; Steven, Elsworth; Güttel, Stefan (15 July 2020). "An overview of the example collection". http://guettel.com/rktoolbox/examples/html/index.html#5.
12. ↑ Jarlebring, Elias; Bennedich, Max; Mele, Giampaolo; Ringh, Emil; Upadhyaya, Parikshit (23 November 2018). "NEP-PACK: A Julia package for nonlinear eigenproblems". arXiv:1811.09592 [math.NA].
13. ↑ Jarlebring, Elias; Kvaal, Simen; Michiels, Wim (2014-01-01). "An Inverse Iteration Method for Eigenvalue Problems with Eigenvector Nonlinearities". SIAM Journal on Scientific Computing 36 (4): A1978–A2001. doi:10.1137/130910014. ISSN 1064-8275. Bibcode: 2014SJSC...36A1978J. https://epubs.siam.org/doi/10.1137/130910014.
14. ↑ Upadhyaya, Parikshit; Jarlebring, Elias; Rubensson, Emanuel H. (2021). "A density matrix approach to the convergence of the self-consistent field iteration". Numerical Algebra, Control & Optimization 11 (1): 99. doi:10.3934/naco.2020018. ISSN 2155-3297.

## Further reading

* Françoise Tisseur and Karl Meerbergen, "The quadratic eigenvalue problem," SIAM Review 43 (2), 235–286 (2001) (link).
* Gene H. Golub and Henk A. van der Vorst, "Eigenvalue computation in the 20th century," Journal of Computational and Applied Mathematics 123, 35–65 (2000).
* Philippe Guillaume, "Nonlinear eigenproblems," SIAM Journal on Matrix Analysis and Applications 20 (3), 575–595 (1999) (link).
* Cedric Effenberger, "Robust solution methods fornonlinear eigenvalue problems", PhD thesis EPFL (2013) (link)
* Roel Van Beeumen, "Rational Krylov methods fornonlinear eigenvalue problems", PhD thesis KU Leuven (2015) (link)

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