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

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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 .

Definition

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 eigenvector if . Moreover, a nonzero vector is called a left eigenvector 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

The following examples are special cases of the nonlinear eigenproblem.

Jordan chains

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

Eigenvector nonlinearity

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]

References

  1. 1 2 3 4 5 6 7 8 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. Bibcode:1973SJNA...10..674R. 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. 1 2 3 4 5 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. Bibcode:2014SJSC...36A2842G. 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. arXiv: 1801.08622 . 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 . Bibcode:2014SJSC...36A1978J. 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. arXiv: 1809.02183 . doi: 10.3934/naco.2020018 . ISSN   2155-3297.

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