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 $$x\neq0$$ 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 $$\Omega \subseteq \Complex$$, and let $$M : \Omega \rightarrow \Complex^{n\times n}$$ be a function that maps scalars to matrices. A scalar $$\lambda \in \Complex $$ is called an eigenvalue, and a nonzero vector $$x \in \Complex^n $$is called a right eigevector if $$M (\lambda) x = 0$$. Moreover, a nonzero vector $$y \in \Complex^n $$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 $$\left.\frac{d^k \det(M (z))}{d z^k} \right|_{z=\lambda} \neq 0$$ but  $$\left.\frac{d^\ell \det(M (z))}{d z^\ell} \right|_{z=\lambda} = 0$$  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 $$(x_0,x_1,\dots, x_{r-1})\in\Complex^n\times\Complex^n\times\dots\times\Complex^n$$ 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 $$(x_0,x_1,\dots, x_{r-1})\in\Complex^n\times\Complex^n\times\dots\times\Complex^n$$ 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.