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

[math]\displaystyle{ M (\lambda) x = 0 , }[/math]

where [math]\displaystyle{ x\neq0 }[/math] is a vector, and [math]\displaystyle{ M }[/math] is a matrix-valued function of the number [math]\displaystyle{ \lambda }[/math]. The number [math]\displaystyle{ \lambda }[/math] is known as the (nonlinear) eigenvalue, the vector [math]\displaystyle{ x }[/math] as the (nonlinear) eigenvector, and [math]\displaystyle{ (\lambda,x) }[/math] as the eigenpair. The matrix [math]\displaystyle{ M (\lambda) }[/math] is singular at an eigenvalue [math]\displaystyle{ \lambda }[/math].

## 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 [math]\displaystyle{ \Omega \subseteq \Complex }[/math], and let [math]\displaystyle{ M : \Omega \rightarrow \Complex^{n\times n} }[/math] be a function that maps scalars to matrices. A scalar [math]\displaystyle{ \lambda \in \Complex }[/math] is called an eigenvalue, and a nonzero vector [math]\displaystyle{ x \in \Complex^n }[/math]is called a right eigevector if [math]\displaystyle{ M (\lambda) x = 0 }[/math]. Moreover, a nonzero vector [math]\displaystyle{ y \in \Complex^n }[/math]is called a left eigevector if [math]\displaystyle{ y^H M (\lambda) = 0^H }[/math], where the superscript [math]\displaystyle{ ^H }[/math] denotes the Hermitian transpose. The definition of the eigenvalue is equivalent to [math]\displaystyle{ \det(M (\lambda)) = 0 }[/math], where [math]\displaystyle{ \det() }[/math] denotes the determinant.[1]

The function [math]\displaystyle{ M }[/math] is usually required to be a holomorphic function of [math]\displaystyle{ \lambda }[/math] (in some domain [math]\displaystyle{ \Omega }[/math]).

In general, [math]\displaystyle{ M (\lambda) }[/math] 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 [math]\displaystyle{ z\in\Omega }[/math] such that [math]\displaystyle{ \det(M (z)) \neq 0 }[/math]. Otherwise it is said to be singular.[1][4]

Definition: An eigenvalue [math]\displaystyle{ \lambda }[/math] is said to have algebraic multiplicity [math]\displaystyle{ k }[/math] if [math]\displaystyle{ k }[/math] is the smallest integer such that the [math]\displaystyle{ k }[/math]th derivative of [math]\displaystyle{ \det(M (z)) }[/math] with respect to [math]\displaystyle{ z }[/math], in [math]\displaystyle{ \lambda }[/math] is nonzero. In formulas that [math]\displaystyle{ \left.\frac{d^k \det(M (z))}{d z^k} \right|_{z=\lambda} \neq 0 }[/math] but [math]\displaystyle{ \left.\frac{d^\ell \det(M (z))}{d z^\ell} \right|_{z=\lambda} = 0 }[/math] for [math]\displaystyle{ \ell=0,1,2,\dots, k-1 }[/math].[1][4]

Definition: The geometric multiplicity of an eigenvalue [math]\displaystyle{ \lambda }[/math] is the dimension of the nullspace of [math]\displaystyle{ M (\lambda) }[/math].[1][4]

## Special cases

The following examples are special cases of the nonlinear eigenproblem.

* The (ordinary) eigenvalue problem: [math]\displaystyle{ M (\lambda) = A-\lambda I. }[/math]
* The generalized eigenvalue problem: [math]\displaystyle{ M (\lambda) = A-\lambda B. }[/math]
* The quadratic eigenvalue problem: [math]\displaystyle{ M (\lambda) = A_0 + \lambda A_1 + \lambda^2 A_2. }[/math]
* The polynomial eigenvalue problem: [math]\displaystyle{ M (\lambda) = \sum_{i=0}^m \lambda^i A_i. }[/math]
* The rational eigenvalue problem: [math]\displaystyle{ M (\lambda) = \sum_{i=0}^{m_1} A_i \lambda^i + \sum_{i=1}^{m_2} B_i r_i(\lambda), }[/math] where [math]\displaystyle{ r_i(\lambda) }[/math] are rational functions.
* The delay eigenvalue problem: [math]\displaystyle{ M (\lambda) = -I\lambda + A_0 +\sum_{i=1}^m A_i e^{-\tau_i \lambda}, }[/math] where [math]\displaystyle{ \tau_1,\tau_2,\dots,\tau_m }[/math] are given scalars, known as delays.

## Jordan chains

Definition: Let [math]\displaystyle{ (\lambda_0,x_0) }[/math] be an eigenpair. A tuple of vectors [math]\displaystyle{ (x_0,x_1,\dots, x_{r-1})\in\Complex^n\times\Complex^n\times\dots\times\Complex^n }[/math] is called a Jordan chain if[math]\displaystyle{ \sum_{k=0}^{\ell} M^{(k)} (\lambda_0) x_{\ell - k} = 0 , }[/math]for [math]\displaystyle{ \ell = 0,1,\dots , r-1 }[/math], where [math]\displaystyle{ M^{(k)}(\lambda_0) }[/math] denotes the [math]\displaystyle{ k }[/math]th derivative of [math]\displaystyle{ M }[/math] with respect to [math]\displaystyle{ \lambda }[/math] and evaluated in [math]\displaystyle{ \lambda=\lambda_0 }[/math]. The vectors [math]\displaystyle{ x_0,x_1,\dots, x_{r-1} }[/math] are called generalized eigenvectors, [math]\displaystyle{ r }[/math] is called the length of the Jordan chain, and the maximal length a Jordan chain starting with [math]\displaystyle{ x_0 }[/math] is called the rank of [math]\displaystyle{ x_0 }[/math].[1][4]

Theorem:[1] A tuple of vectors [math]\displaystyle{ (x_0,x_1,\dots, x_{r-1})\in\Complex^n\times\Complex^n\times\dots\times\Complex^n }[/math] is a Jordan chain if and only if the function [math]\displaystyle{ M(\lambda) \chi_\ell (\lambda) }[/math] has a root in [math]\displaystyle{ \lambda=\lambda_0 }[/math] and the root is of multiplicity at least [math]\displaystyle{ \ell }[/math] for [math]\displaystyle{ \ell=0,1,\dots,r-1 }[/math], where the vector valued function [math]\displaystyle{ \chi_\ell (\lambda) }[/math] is defined as[math]\displaystyle{ \chi_\ell(\lambda) = \sum_{k=0}^\ell x_k (\lambda-\lambda_0)^k. }[/math]

## 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 [math]\displaystyle{ M }[/math] 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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