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

is a vector, and is a matrix-valued function is known as the (nonlinear) eigenvalue as the (nonlinear) eigenvector as the eigenpair.

Definition

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

is called an eigenvalue is called a right eigenvector is called a left eigenvector denotes the Hermitian transpose denotes the determinant.

The function is usually required to be a holomorphic function (in some domain).

could be a linear map, but most commonly it is a finite-dimensional, usually square, matrix.

Definition: The problem is said to be regular. Otherwise it is said to be singular.

Definition: is said to have algebraic multiplicity is the smallest integerth derivative.

Definition: The geometric multiplicity is the dimension of the nullspace.

Special cases

The following examples are special cases of the nonlinear eigenproblem.

• The (ordinary) eigenvalue problem

• The generalized eigenvalue problem

• The quadratic eigenvalue problem

• are rational functions.

• The delay eigenvalue problem are given scalars, known as delays.

Jordan chains

Definition: be an eigenpair. A tuple is called a Jordan chainth derivative are called generalized eigenvectors is called the length is called the rank.

Theorem: is a Jordan chain if and only if has a root and the root is of multiplicity

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

maps vectors to matrices, or sometimes hermitian matrices to hermitian matrices.

References

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)