Iterative Projection Methods for Large-Scale Nonlinear Eigenvalue Problems

2010 ◽  
Vol 1 ◽  
pp. 187-214 ◽  
Author(s):  
H. Voss
Keyword(s):  
Large Scale ◽  
Eigenvalue Problems ◽  
Projection Methods ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Eigenvalue

scholarly journals Computation of pseudospectral abscissa for large-scale nonlinear eigenvalue problems

2017 ◽  
pp. drw065 ◽  
Author(s):  
Karl Meerbergen ◽  
Wim Michiels ◽  
Roel Van Beeumen ◽  
Emre Mengi
Keyword(s):  
Large Scale ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Eigenvalue

RESTARTING PROJECTION METHODS FOR RATIONAL EIGENPROBLEMS ARISING IN FLUID‐SOLID VIBRATIONS

2008 ◽  
Vol 13 (2) ◽  
pp. 171-182 ◽  
Author(s):  
Marta M. Betcke ◽  
Heinrich Voss
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Projection Methods ◽  
Nonlinear Eigenvalue Problems ◽  
Solid Structure ◽  
Minmax Characterization ◽  
Nonlinear Eigenvalue

For nonlinear eigenvalue problems T(λ)x = 0 satisfying a minmax characterization of its eigenvalues iterative projection methods combined with safeguarded iteration are suitable for computing all eigenvalues in a given interval. Such methods hit their limitations if a large number of eigenvalues is required. In this paper we discuss restart procedures which are able to cope with this problem, and we evaluate them for a rational eigenvalue problem governing vibrations of a fluid‐solid structure.

scholarly journals Automatic rational approximation and linearization of nonlinear eigenvalue problems

2021 ◽  
Author(s):  
Pieter Lietaert ◽  
Karl Meerbergen ◽  
Javier Pérez ◽  
Bart Vandereycken
Keyword(s):  
Rational Approximation ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Space Representation ◽  
Rational Approximations ◽  
Krylov Methods ◽  
Nonlinear Eigenvalue Problems ◽  
State Space Representation ◽  
Rational Polynomial ◽  
Nonlinear Eigenvalue

Abstract We present a method for solving nonlinear eigenvalue problems (NEPs) using rational approximation. The method uses the Antoulas–Anderson algorithm (AAA) of Nakatsukasa, Sète and Trefethen to approximate the NEP via a rational eigenvalue problem. A set-valued variant of the AAA algorithm is also presented for building low-degree rational approximations of NEPs with a large number of nonlinear functions. The rational approximation is embedded in the state-space representation of a rational polynomial by Su and Bai. This procedure perfectly fits the framework of the compact rational Krylov methods (CORK and TS-CORK), allowing solve large-scale NEPs to be efficiently solved. One advantage of our method, compared to related techniques such as NLEIGS and infinite Arnoldi, is that it automatically selects the poles and zeros of the rational approximations. Numerical examples show that the presented framework is competitive with NLEIGS and usually produces smaller linearizations with the same accuracy but with less effort for the user.

scholarly journals Erratum to: “Computation of pseudospectral abscissa for large-scale nonlinear eigenvalue problems”

2017 ◽  
Vol 38 (3) ◽  
pp. 1598-1598
Author(s):  
Karl Meerbergen ◽  
Emre Mengi ◽  
Wim Michiels ◽  
Roel Van Beeumen
Keyword(s):  
Large Scale ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Eigenvalue

scholarly journals NONLINEAR RAYLEIGH-RITZ ITERATIVE METHOD FOR SOLVING LARGE SCALE NONLINEAR EIGENVALUE PROBLEMS

2010 ◽  
Vol 14 (3A) ◽  
pp. 869-883 ◽  
Author(s):  
Ben-Shan Liao ◽  
Zhaojun Bai ◽  
Lie-Quan Lee ◽  
Kwok Ko
Keyword(s):  
Iterative Method ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Eigenvalue
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