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

${\displaystyle M(\lambda )x=0,}$

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

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

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

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

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

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

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

The following examples are special cases of the nonlinear eigenproblem.

• The (ordinary) eigenvalue problem: ${\displaystyle M(\lambda )=A-\lambda I.}$
• The generalized eigenvalue problem: ${\displaystyle M(\lambda )=A-\lambda B.}$
• The quadratic eigenvalue problem: ${\displaystyle M(\lambda )=A_0+\lambda A_1+{\lambda }^2{A}_2.}$
• The polynomial eigenvalue problem: ${\displaystyle M(\lambda )={\displaystyle \sum _{i=0}^m}{\lambda }^i{A}_i.}$
• The rational eigenvalue problem: ${\displaystyle M(\lambda )={\displaystyle \sum _{i=0}^{m_1}}{A}_i{\lambda }^i+{\displaystyle \sum _{i=1}^{m_2}}{B}_i{r}_i(\lambda ),}$ where ${\displaystyle r_i(\lambda )}$ are rational functions.
• The delay eigenvalue problem: ${\displaystyle M(\lambda )=-I\lambda +A_0+{\displaystyle \sum _{i=1}^m}{A}_i{e}^{-{\tau }_i\lambda },}$ where ${\displaystyle {\tau }_1,{\tau }_2,\dots ,{\tau }_m}$ are given scalars, known as delays.

Definition: Let ${\displaystyle ({\lambda }_0,x_0)}$ be an eigenpair. A tuple of vectors ${\displaystyle (x_0,x_1,\dots ,x_{r-1})\in {\mathbb{ℂ}}^n\times {\mathbb{ℂ}}^n\times \dots \times {\mathbb{ℂ}}^n}$ is called a Jordan chain if$${\displaystyle {\displaystyle \sum _{k=0}^{\ell }}{M}^{(k)}({\lambda }_0)x_{\ell -k}=0,}$$for ${\displaystyle \ell =0,1,\dots ,r-1}$, where ${\displaystyle M^{(k)}({\lambda }_0)}$ denotes the ${\displaystyle k}$th derivative of ${\displaystyle M}$ with respect to ${\displaystyle \lambda }$ and evaluated in ${\displaystyle \lambda ={\lambda }_0}$. The vectors ${\displaystyle x_0,x_1,\dots ,x_{r-1}}$ are called generalized eigenvectors, ${\displaystyle r}$ is called the length of the Jordan chain, and the maximal length a Jordan chain starting with ${\displaystyle x_0}$ is called the rank of ${\displaystyle x_0}$.^[1][4]

Theorem:^[1] A tuple of vectors ${\displaystyle (x_0,x_1,\dots ,x_{r-1})\in {\mathbb{ℂ}}^n\times {\mathbb{ℂ}}^n\times \dots \times {\mathbb{ℂ}}^n}$ is a Jordan chain if and only if the function ${\displaystyle M(\lambda ){\chi }_{\ell }(\lambda )}$ has a root in ${\displaystyle \lambda ={\lambda }_0}$ and the root is of multiplicity at least ${\displaystyle \ell }$ for ${\displaystyle \ell =0,1,\dots ,r-1}$, where the vector valued function ${\displaystyle {\chi }_{\ell }(\lambda )}$ is defined as$${\displaystyle {\chi }_{\ell }(\lambda )={\displaystyle \sum _{k=0}^{\ell }}{x}_k(\lambda -{\lambda }_0{)}^k.}$$

• 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 nonlinearities is a related, but different, form of nonlinearity that is sometimes studied. In this case the function ${\displaystyle M}$ maps vectors to matrices, or sometimes hermitian matrices to hermitian matrices.^[13][14]

• 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)

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Nonlinear eigenproblem. Read more