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.

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.

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.

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 $\ell =0,1,2,\dots ,k-1$.

Definition: The geometric multiplicity of an eigenvalue $\lambda$ is the dimension of the nullspace of $M(\lambda )$.

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 $x_{0}$.

Theorem: 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.
• The NLEVP collection of nonlinear eigenvalue problems is a MATLAB package containing many nonlinear eigenvalue problems with various properties.
• 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.
• The MATLAB toolbox NLEIGS contains an implementation of fully rational Krylov with a dynamically constructed rational interpolant.
• The MATLAB toolbox CORK contains an implementation of the compact rational Krylov algorithm that exploits the Kronecker structure of the linearization pencils.
• The MATLAB toolbox AAA-EIGS contains an implementation of CORK with rational approximation by set-valued AAA.
• 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.
• The Julia package NEP-PACK contains many implementations of various numerical methods for nonlinear eigenvalue problems, as well as many benchmark problems.
• The review paper of Güttel & Tisseur 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.

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)