Nonlinear eigenproblem

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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

where is a vector, and is a matrix-valued function of the number . The number is known as the (nonlinear) eigenvalue, the vector as the (nonlinear) eigenvector, and as the eigenpair. The matrix is singular at an eigenvalue .


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

Let , and let be a function that maps scalars to matrices. A scalar is called an eigenvalue, and a nonzero vector is called a right eigevector if . Moreover, a nonzero vector is called a left eigevector if , where the superscript denotes the Hermitian transpose. The definition of the eigenvalue is equivalent to , where denotes the determinant.[1]

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

In general, 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 such that . Otherwise it is said to be singular.[1][4]

Definition: An eigenvalue is said to have algebraic multiplicity if is the smallest integer such that the th derivative of with respect to , in is nonzero. In formulas that but for .[1][4]

Definition: The geometric multiplicity of an eigenvalue is the dimension of the nullspace of .[1][4]

Special cases[edit]

The following examples are special cases of the nonlinear eigenproblem.

  • The (ordinary) eigenvalue problem:
  • The generalized eigenvalue problem:
  • The quadratic eigenvalue problem:
  • The polynomial eigenvalue problem:
  • The rational eigenvalue problem: where are rational functions.
  • The delay eigenvalue problem: where are given scalars, known as delays.

Jordan chains[edit]

Definition: Let be an eigenpair. A tuple of vectors is called a Jordan chain if

for , where denotes the th derivative of with respect to and evaluated in . The vectors are called generalized eigenvectors, is called the length of the Jordan chain, and the maximal length a Jordan chain starting with is called the rank of .[1][4]

Theorem:[1] A tuple of vectors is a Jordan chain if and only if the function has a root in and the root is of multiplicity at least for , where the vector valued function is defined as

Mathematical software[edit]

  • 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 nonlinearity[edit]

Eigenvector nonlinearities is a related, but different, form of nonlinearity that is sometimes studied. In this case the function maps vectors to matrices, or sometimes hermitian matrices to hermitian matrices.[13][14]


  1. ^ a b c d e f g h Güttel, Stefan; Tisseur, Françoise (2017). "The nonlinear eigenvalue problem" (PDF). Acta Numerica. 26: 1–94. doi:10.1017/S0962492917000034. ISSN 0962-4929. S2CID 46749298.
  2. ^ Ruhe, Axel (1973). "Algorithms for the Nonlinear Eigenvalue Problem". SIAM Journal on Numerical Analysis. 10 (4): 674–689. doi:10.1137/0710059. ISSN 0036-1429. JSTOR 2156278.
  3. ^ Mehrmann, Volker; Voss, Heinrich (2004). "Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods". GAMM-Mitteilungen. 27 (2): 121–152. doi:10.1002/gamm.201490007. ISSN 1522-2608. S2CID 14493456.
  4. ^ a b c d e Voss, Heinrich (2014). "Nonlinear eigenvalue problems" (PDF). In Hogben, Leslie (ed.). Handbook of Linear Algebra (2 ed.). Boca Raton, FL: Chapman and Hall/CRC. ISBN 9781466507289.
  5. ^ Hernandez, Vicente; Roman, Jose E.; Vidal, Vicente (September 2005). "SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems". ACM Transactions on Mathematical Software. 31 (3): 351–362. doi:10.1145/1089014.1089019. S2CID 14305707.
  6. ^ Betcke, Timo; Higham, Nicholas J.; Mehrmann, Volker; Schröder, Christian; Tisseur, Françoise (February 2013). "NLEVP: A Collection of Nonlinear Eigenvalue Problems". ACM Transactions on Mathematical Software. 39 (2): 1–28. doi:10.1145/2427023.2427024. S2CID 4271705.
  7. ^ Polizzi, Eric (2020). "FEAST Eigenvalue Solver v4.0 User Guide". arXiv:2002.04807 [cs.MS].
  8. ^ Güttel, Stefan; Van Beeumen, Roel; Meerbergen, Karl; Michiels, Wim (1 January 2014). "NLEIGS: A Class of Fully Rational Krylov Methods for Nonlinear Eigenvalue Problems". SIAM Journal on Scientific Computing. 36 (6): A2842–A2864. doi:10.1137/130935045.
  9. ^ Van Beeumen, Roel; Meerbergen, Karl; Michiels, Wim (2015). "Compact rational Krylov methods for nonlinear eigenvalue problems". SIAM Journal on Matrix Analysis and Applications. 36 (2): 820–838. doi:10.1137/140976698. S2CID 18893623.
  10. ^ Lietaert, Pieter; Meerbergen, Karl; Pérez, Javier; Vandereycken, Bart (13 April 2022). "Automatic rational approximation and linearization of nonlinear eigenvalue problems". IMA Journal of Numerical Analysis. 42 (2): 1087–1115. doi:10.1093/imanum/draa098.
  11. ^ Berljafa, Mario; Steven, Elsworth; Güttel, Stefan (15 July 2020). "An overview of the example collection". index.m. Retrieved 31 May 2022.
  12. ^ Jarlebring, Elias; Bennedich, Max; Mele, Giampaolo; Ringh, Emil; Upadhyaya, Parikshit (23 November 2018). "NEP-PACK: A Julia package for nonlinear eigenproblems". arXiv:1811.09592 [math.NA].
  13. ^ Jarlebring, Elias; Kvaal, Simen; Michiels, Wim (2014-01-01). "An Inverse Iteration Method for Eigenvalue Problems with Eigenvector Nonlinearities". SIAM Journal on Scientific Computing. 36 (4): A1978–A2001. arXiv:1212.0417. doi:10.1137/130910014. ISSN 1064-8275. S2CID 16959079.
  14. ^ Upadhyaya, Parikshit; Jarlebring, Elias; Rubensson, Emanuel H. (2021). "A density matrix approach to the convergence of the self-consistent field iteration". Numerical Algebra, Control & Optimization. 11 (1): 99. doi:10.3934/naco.2020018. ISSN 2155-3297.

Further reading[edit]