### An integral method for solving nonlinear eigenvalue problems

Beyn W-J (2012)
Linear Algebra and its Applications 436(10): 3839-3863.

Zeitschriftenaufsatz | Veröffentlicht | Englisch

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Abstract / Bemerkung
We propose a numerical method for computing all eigenvalues (and the corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane. The method uses complex integrals of the resolvent operator, applied to at least k column vectors, where k is the number of eigenvalues inside the contour. The theorem of Keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k. No initial approximations of eigenvalues and eigenvectors are needed. The method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension. We also give an extension of the method to the case where k is larger than the matrix dimension. The quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours. Using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour. (C) 2011 Elsevier Inc. All rights reserved.
Stichworte
Numerical methods; Contour integrals; Nonlinear eigenvalue problems
Erscheinungsjahr
2012
Zeitschriftentitel
Linear Algebra and its Applications
Band
436
Ausgabe
10
Seite(n)
3839-3863
ISSN
0024-3795
Page URI
https://pub.uni-bielefeld.de/record/2501407

### Zitieren

Beyn W-J. An integral method for solving nonlinear eigenvalue problems. Linear Algebra and its Applications. 2012;436(10):3839-3863.
Beyn, W. - J. (2012). An integral method for solving nonlinear eigenvalue problems. Linear Algebra and its Applications, 436(10), 3839-3863. doi:10.1016/j.laa.2011.03.030
Beyn, W. - J. (2012). An integral method for solving nonlinear eigenvalue problems. Linear Algebra and its Applications 436, 3839-3863.
Beyn, W.-J., 2012. An integral method for solving nonlinear eigenvalue problems. Linear Algebra and its Applications, 436(10), p 3839-3863.
W.-J. Beyn, “An integral method for solving nonlinear eigenvalue problems”, Linear Algebra and its Applications, vol. 436, 2012, pp. 3839-3863.
Beyn, W.-J.: An integral method for solving nonlinear eigenvalue problems. Linear Algebra and its Applications. 436, 3839-3863 (2012).
Beyn, Wolf-Jürgen. “An integral method for solving nonlinear eigenvalue problems”. Linear Algebra and its Applications 436.10 (2012): 3839-3863.

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