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

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

Definition

In the discipline of numerical linear algebra the following definition is typically used.

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

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

In general, M ( λ ) {\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 z ∈ Ω {\displaystyle z\in \Omega } such that det ( M ( z ) ) ≠ 0 {\displaystyle \det(M(z))\neq 0}. Otherwise it is said to be singular.

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

Definition: The geometric multiplicity of an eigenvalue λ {\displaystyle \lambda } is the dimension of the nullspace of M ( λ ) {\displaystyle M(\lambda )}.

Special cases

The following examples are special cases of the nonlinear eigenproblem.

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

Jordan chains

Definition: Let ( λ 0 , x 0 ) {\displaystyle (\lambda _{0},x_{0})} be an eigenpair. A tuple of vectors ( x 0 , x 1 , … , x r − 1 ) ∈ C n × C n × ⋯ × C n {\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∑ k = 0 ℓ M ( k ) ( λ 0 ) x ℓ − k = 0 , {\displaystyle \sum _{k=0}^{\ell }M^{(k)}(\lambda _{0})x_{\ell -k}=0,}for ℓ = 0 , 1 , … , r − 1 {\displaystyle \ell =0,1,\dots ,r-1}, where M ( k ) ( λ 0 ) {\displaystyle M^{(k)}(\lambda _{0})} denotes the k {\displaystyle k}th derivative of M {\displaystyle M} with respect to λ {\displaystyle \lambda } and evaluated in λ = λ 0 {\displaystyle \lambda =\lambda _{0}}. The vectors x 0 , x 1 , … , x r − 1 {\displaystyle x_{0},x_{1},\dots ,x_{r-1}} are called generalized eigenvectors, r {\displaystyle r} is called the length of the Jordan chain, and the maximal length a Jordan chain starting with x 0 {\displaystyle x_{0}} is called the rank of x 0 {\displaystyle x_{0}}.

Theorem: A tuple of vectors ( x 0 , x 1 , … , x r − 1 ) ∈ C n × C n × ⋯ × C n {\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 M ( λ ) χ ℓ ( λ ) {\displaystyle M(\lambda )\chi _{\ell }(\lambda )} has a root in λ = λ 0 {\displaystyle \lambda =\lambda _{0}} and the root is of multiplicity at least ℓ {\displaystyle \ell } for ℓ = 0 , 1 , … , r − 1 {\displaystyle \ell =0,1,\dots ,r-1}, where the vector valued function χ ℓ ( λ ) {\displaystyle \chi _{\ell }(\lambda )} is defined asχ ℓ ( λ ) = ∑ k = 0 ℓ x k ( λ − λ 0 ) k . {\displaystyle \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 is a MATLAB package containing many nonlinear eigenvalue problems with various properties.
The 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 contains an implementation of fully rational Krylov with a dynamically constructed rational interpolant.
The MATLAB toolbox contains an implementation of the compact rational Krylov algorithm that exploits the Kronecker structure of the linearization pencils.
The MATLAB toolbox contains an implementation of CORK with rational approximation by set-valued AAA.
The MATLAB toolbox (Rational Krylov Toolbox) contains implementations of the rational Krylov method for nonlinear eigenvalue problems as well as features for rational approximation.
The Julia package 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