scholarly journals A Prediction-Correction Dynamic Method for Large-Scale Generalized Eigenvalue Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Xin-long Luo ◽  
Jia-ru Lin ◽  
Wei-ling Wu
Keyword(s):  
Dynamical System ◽  
Eigenvalue Problem ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Correction Method ◽  
Differential Algebraic Equations ◽  
Euler Method ◽  
Generalized Eigenvalue Problem ◽  
Algebraic Equations ◽  
Generalized Eigenvalue

This paper gives a new prediction-correction method based on the dynamical system of differential-algebraic equations for the smallest generalized eigenvalue problem. First, the smallest generalized eigenvalue problem is converted into an equivalent-constrained optimization problem. Second, according to the Karush-Kuhn-Tucker conditions of this special equality-constrained problem, a special continuous dynamical system of differential-algebraic equations is obtained. Third, based on the implicit Euler method and an analogous trust-region technique, a prediction-correction method is constructed to follow this system of differential-algebraic equations to compute its steady-state solution. Consequently, the smallest generalized eigenvalue of the original problem is obtained. The local superlinear convergence property for this new algorithm is also established. Finally, in comparison with other methods, some promising numerical experiments are presented.

Inverse multi-parameter eigenvalue problems for matrices

1985 ◽  
Vol 100 (1-2) ◽  
pp. 29-38 ◽  
Author(s):  
Patrick J. Browne ◽  
B. D. Sleeman
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Classical Problem ◽  
Diagonal Matrix ◽  
Generalized Eigenvalue Problem ◽  
Quadratic Eigenvalue Problem ◽  
Generalized Eigenvalue ◽  
The One ◽  
Quadratic Eigenvalue ◽  
Two Parameter

SynopsisWe study the possibility of perturbing a matrix A by a diagonal matrix so that an eigenvalue problem with leading matrix A has specifiedeigenvalues when A is replaced by A+D. The particular cases presented are the one-parameter generalized eigenvalue problem (A× = λB μ×, a two-parameter eigenvalue problem (A + λB + μC)× = 0, a linked system ofsuch two-parameter problems and a quadratic eigenvalue problem (A + λB + λ2C)× = 0. The work extends results of Hadeler for the classical problem A× = λ×.

scholarly journals On the Generalized Eigenvalue Problem for the Rossby Wave Vertical Velocity in the Presence of Mean Flow and Topography

2012 ◽  
Vol 42 (6) ◽  
pp. 1045-1050 ◽  
Author(s):  
Rémi Tailleux
Keyword(s):  
Eigenvalue Problem ◽  
Rossby Wave ◽  
Vertical Velocity ◽  
Vertical Structure ◽  
Nonlinear Term ◽  
Eigenvalue Problems ◽  
Equations Of Motion ◽  
Generalized Eigenvalue Problem ◽  
Mean Flow ◽  
Generalized Eigenvalue

Abstract In a series of papers, Killworth and Blundell have proposed to study the effects of a background mean flow and topography on Rossby wave propagation by means of a generalized eigenvalue problem formulated in terms of the vertical velocity, obtained from a linearization of the primitive equations of motion. However, it has been known for a number of years that this eigenvalue problem contains an error, which Killworth was prevented from correcting himself by his unfortunate passing and whose correction is therefore taken up in this note. Here, the author shows in the context of quasigeostrophic (QG) theory that the error can ultimately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a nonlinear one (the eigenvalue appears both in the numerator and denominator), unlike that for the pressure. The reason that this nonlinear term is lacking in the Killworth and Blundell theory comes from neglecting the depth dependence of a depth-dependent term. This nonlinear term is shown on idealized examples to alter significantly the Rossby wave dispersion relation in the high-wavenumber regime but is otherwise irrelevant in the long-wave limit, in which case the eigenvalue problems for the vertical velocity and pressure are both linear. In the general dispersive case, however, one should first solve the generalized eigenvalue problem for the pressure vertical structure and, if needed, diagnose the vertical velocity vertical structure from the latter.

scholarly journals Local defect-correction method based on multilevel discretization for Steklov eigenvalue problem

2021 ◽  
Author(s):  
Fei Xu ◽  
Liu Chen ◽  
Qiumei Huang
Keyword(s):  
Eigenvalue Problem ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Correction Method ◽  
Small Scale ◽  
Local Defect ◽  
Defect Correction ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Steklov Eigenvalue Problems

In this paper, we propose a local defect-correction method for solving the Steklov eigenvalue problem arising from the scalar second order positive definite partial differential equations based on the multilevel discretization. The objective is to avoid solving large-scale equations especially the large-scale Steklov eigenvalue problem whose computational cost increases exponentially. The proposed algorithm transforms the Steklov eigenvalue problem into a series of linear boundary value problems, which are defined in a multigrid space sequence, and a series of small-scale Steklov eigenvalue problems in a coarse correction space. Furthermore, we use the local defect-correction technique to divide the large-scale boundary value problems into small-scale subproblems. Through our proposed algorithm, we avoid solving large-scale Steklov eigenvalue problems. As a result, our proposed algorithm demonstrates significantly improved the solving efficiency. Additionally, we conduct numerical experiments and a rigorous theoretical analysis to verify the effectiveness of our proposed approach.

Filtering Algorithm for Real Eigenvalue Bounds of Interval and Fuzzy Generalized Eigenvalue Problems

2016 ◽  
Vol 2 (4) ◽  
Author(s):  
Nisha Rani Mahato ◽  
S. Chakraverty
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Triangular Fuzzy Number ◽  
Generalized Eigenvalue Problem ◽  
Real Eigenvalue ◽  
Eigenvalue Bounds ◽  
Generalized Eigenvalue ◽  
Generalized Eigenvalue Problems ◽  
Precise Bound ◽  
Fuzzy Parameter

This paper deals with an interval and fuzzy generalized eigenvalue problem involving uncertain parameters. Based on a sufficient regularity condition for intervals, an interval filtering eigenvalue procedure for generalized eigenvalue problems with interval parameters is proposed, which iteratively eliminates the parts that do not contain an eigenvalue and thus reduces the initial eigenvalue bound to a precise bound. The same iterative procedure has been proposed for generalized fuzzy eigenvalue problems. In general, the solution of dynamic problems of structures using the finite element method (FEM) leads to a generalized eigenvalue problem. Based on the proposed procedures, various structural examples with an interval and fuzzy parameter such as triangular fuzzy number (TFN) are investigated to show the efficiency of the algorithms stated. Finally, fuzzy filtered eigenvalue bounds are depicted by fuzzy plots using the α-cut.

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