Nonlinear eigenproblem

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(\lambda )x=0,

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

Definition

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

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

The function $M$ is usually required to be a holomorphic function of $\lambda$ (in some domain $\Omega$ ).

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

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

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

Special cases

The following examples are special cases of the nonlinear eigenproblem.

The (ordinary) eigenvalue problem: $M(\lambda )=A-\lambda I.$
The generalized eigenvalue problem: $M(\lambda )=A-\lambda B.$
The quadratic eigenvalue problem: $M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.$
The polynomial eigenvalue problem: $M(\lambda )=\sum _{i=0}^{m}\lambda ^{i}A_{i}.$
The rational eigenvalue problem: $M(\lambda )=\sum _{i=0}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ where $r_{i}(\lambda )$ are rational functions.
The delay eigenvalue problem: $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ where $\tau _{1},\tau _{2},\dots ,\tau _{m}$ are given scalars, known as delays.

Jordan chains

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

Theorem:^[1] A tuple of vectors $(x_{0},x_{1},\dots ,x_{r-1})\in \mathbb {C} ^{n}\times \mathbb {C} ^{n}\times \dots \times \mathbb {C} ^{n}$ is a Jordan chain if and only if the function $M(\lambda )\chi _{\ell }(\lambda )$ has a root in $\lambda =\lambda _{0}$ and the root is of multiplicity at least $\ell$ for $\ell =0,1,\dots ,r-1$ , where the vector valued function $\chi _{\ell }(\lambda )$ is defined as $\chi _{\ell }(\lambda )=\sum _{k=0}^{\ell }x_{k}(\lambda -\lambda _{0})^{k}.$

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

^ ^a ^b ^c ^d ^e ^f ^g ^h Güttel, Stefan; Tisseur, Françoise (2017). "The nonlinear eigenvalue problem" (PDF). Acta Numerica. 26: 1–94. doi:10.1017/S0962492917000034. ISSN 0962-4929. S2CID 46749298.
^ Ruhe, Axel (1973). "Algorithms for the Nonlinear Eigenvalue Problem". SIAM Journal on Numerical Analysis. 10 (4): 674–689. Bibcode:1973SJNA...10..674R. doi:10.1137/0710059. ISSN 0036-1429. JSTOR 2156278.
^ Mehrmann, Volker; Voss, Heinrich (2004). "Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods". GAMM-Mitteilungen. 27 (2): 121–152. doi:10.1002/gamm.201490007. ISSN 1522-2608. S2CID 14493456.
^ ^a ^b ^c ^d ^e Voss, Heinrich (2014). "Nonlinear eigenvalue problems" (PDF). In Hogben, Leslie (ed.). Handbook of Linear Algebra (2 ed.). Boca Raton, FL: Chapman and Hall/CRC. ISBN 9781466507289.
^ 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. S2CID 14305707.
^ 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. S2CID 4271705.
^ Polizzi, Eric (2020). "FEAST Eigenvalue Solver v4.0 User Guide". arXiv:2002.04807 [cs.MS].
^ 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. Bibcode:2014SJSC...36A2842G. doi:10.1137/130935045.
^ 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. S2CID 18893623.
^ 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. arXiv:1801.08622. doi:10.1093/imanum/draa098.
^ Berljafa, Mario; Steven, Elsworth; Güttel, Stefan (15 July 2020). "An overview of the example collection". index.m. Retrieved 31 May 2022.
^ 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].
^ 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. arXiv:1212.0417. Bibcode:2014SJSC...36A1978J. doi:10.1137/130910014. ISSN 1064-8275. S2CID 16959079.
^ 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. arXiv:1809.02183. doi:10.3934/naco.2020018. ISSN 2155-3297.