Nonlinear eigenproblem

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title: "Nonlinear eigenproblem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["linear-algebra"] description: "Type of equation involving matrix-valued functions" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Nonlinear_eigenproblem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Type of equation involving matrix-valued functions ::

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 eigenvector if M (\lambda) x = 0. Moreover, a nonzero vector y \in \Complex^n is called a left eigenvector 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 kth 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 kth 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.

References

References

1. (2017). "The nonlinear eigenvalue problem". Acta Numerica.
2. Ruhe, Axel. (1973). "Algorithms for the Nonlinear Eigenvalue Problem". SIAM Journal on Numerical Analysis.
3. (2004). "Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods". GAMM-Mitteilungen.
4. Voss, Heinrich. (2014). "Handbook of Linear Algebra". Chapman and Hall/CRC.
5. (September 2005). "SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems". ACM Transactions on Mathematical Software.
6. (February 2013). "NLEVP: A Collection of Nonlinear Eigenvalue Problems". ACM Transactions on Mathematical Software.
7. Polizzi, Eric. (2020). "FEAST Eigenvalue Solver v4.0 User Guide".
8. (1 January 2014). "NLEIGS: A Class of Fully Rational Krylov Methods for Nonlinear Eigenvalue Problems". SIAM Journal on Scientific Computing.
9. (2015). "Compact rational Krylov methods for nonlinear eigenvalue problems". SIAM Journal on Matrix Analysis and Applications.
10. (13 April 2022). "Automatic rational approximation and linearization of nonlinear eigenvalue problems". IMA Journal of Numerical Analysis.
11. (15 July 2020). "An overview of the example collection".
12. (23 November 2018). "NEP-PACK: A Julia package for nonlinear eigenproblems".
13. (2014-01-01). "An Inverse Iteration Method for Eigenvalue Problems with Eigenvector Nonlinearities". SIAM Journal on Scientific Computing.
14. (2021). "A density matrix approach to the convergence of the self-consistent field iteration". Numerical Algebra, Control & Optimization.

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