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Nonlinear eigenproblem
Mathematical equation involving a matrix-valued function that is singular at the eigenvalue.

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 } .

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Definition

In the discipline of numerical linear algebra the following definition is typically used.1234

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 eigevector 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 eigevector 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.5

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.67

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} .89

Definition: The geometric multiplicity of an eigenvalue λ {\displaystyle \lambda } is the dimension of the nullspace of M ( λ ) {\displaystyle M(\lambda )} .1011

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}} .1213

Theorem:14 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 as χ 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.15
  • The NLEVP collection of nonlinear eigenvalue problems is a MATLAB package containing many nonlinear eigenvalue problems with various properties. 16
  • 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.17
  • The MATLAB toolbox NLEIGS contains an implementation of fully rational Krylov with a dynamically constructed rational interpolant.18
  • The MATLAB toolbox CORK contains an implementation of the compact rational Krylov algorithm that exploits the Kronecker structure of the linearization pencils.19
  • The MATLAB toolbox AAA-EIGS contains an implementation of CORK with rational approximation by set-valued AAA.20
  • 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. 21
  • The Julia package NEP-PACK contains many implementations of various numerical methods for nonlinear eigenvalue problems, as well as many benchmark problems.22
  • The review paper of Güttel & Tisseur23 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 {\displaystyle M} maps vectors to matrices, or sometimes hermitian matrices to hermitian matrices.2425

Further reading

References

  1. Güttel, Stefan; Tisseur, Françoise (2017). "The nonlinear eigenvalue problem" (PDF). Acta Numerica. 26: 1–94. doi:10.1017/S0962492917000034. ISSN 0962-4929. S2CID 46749298. /wiki/Fran%C3%A7oise_Tisseur ↩

  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. https://epubs.siam.org/doi/10.1137/0710059 ↩

  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. /wiki/Volker_Mehrmann ↩

  4. 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. 9781466507289 ↩

  5. Güttel, Stefan; Tisseur, Françoise (2017). "The nonlinear eigenvalue problem" (PDF). Acta Numerica. 26: 1–94. doi:10.1017/S0962492917000034. ISSN 0962-4929. S2CID 46749298. /wiki/Fran%C3%A7oise_Tisseur ↩

  6. Güttel, Stefan; Tisseur, Françoise (2017). "The nonlinear eigenvalue problem" (PDF). Acta Numerica. 26: 1–94. doi:10.1017/S0962492917000034. ISSN 0962-4929. S2CID 46749298. /wiki/Fran%C3%A7oise_Tisseur ↩

  7. 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. 9781466507289 ↩

  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. /wiki/Fran%C3%A7oise_Tisseur ↩

  9. 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. 9781466507289 ↩

  10. Güttel, Stefan; Tisseur, Françoise (2017). "The nonlinear eigenvalue problem" (PDF). Acta Numerica. 26: 1–94. doi:10.1017/S0962492917000034. ISSN 0962-4929. S2CID 46749298. /wiki/Fran%C3%A7oise_Tisseur ↩

  11. 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. 9781466507289 ↩

  12. Güttel, Stefan; Tisseur, Françoise (2017). "The nonlinear eigenvalue problem" (PDF). Acta Numerica. 26: 1–94. doi:10.1017/S0962492917000034. ISSN 0962-4929. S2CID 46749298. /wiki/Fran%C3%A7oise_Tisseur ↩

  13. 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. 9781466507289 ↩

  14. Güttel, Stefan; Tisseur, Françoise (2017). "The nonlinear eigenvalue problem" (PDF). Acta Numerica. 26: 1–94. doi:10.1017/S0962492917000034. ISSN 0962-4929. S2CID 46749298. /wiki/Fran%C3%A7oise_Tisseur ↩

  15. 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. /wiki/Doi_(identifier) ↩

  16. 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. /wiki/Doi_(identifier) ↩

  17. Polizzi, Eric (2020). "FEAST Eigenvalue Solver v4.0 User Guide". arXiv:2002.04807 [cs.MS]. /wiki/ArXiv_(identifier) ↩

  18. 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. /wiki/Bibcode_(identifier) ↩

  19. 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. https://lirias.kuleuven.be/handle/123456789/490706 ↩

  20. 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. /wiki/ArXiv_(identifier) ↩

  21. Berljafa, Mario; Steven, Elsworth; Güttel, Stefan (15 July 2020). "An overview of the example collection". index.m. Retrieved 31 May 2022. http://guettel.com/rktoolbox/examples/html/index.html#5 ↩

  22. 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]. /wiki/ArXiv_(identifier) ↩

  23. Güttel, Stefan; Tisseur, Françoise (2017). "The nonlinear eigenvalue problem" (PDF). Acta Numerica. 26: 1–94. doi:10.1017/S0962492917000034. ISSN 0962-4929. S2CID 46749298. /wiki/Fran%C3%A7oise_Tisseur ↩

  24. 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. https://epubs.siam.org/doi/10.1137/130910014 ↩

  25. 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. https://doi.org/10.3934%2Fnaco.2020018 ↩