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.

Filtering algorithm for eigenvalue bounds of fuzzy symmetric matrices

2016 ◽  
Vol 33 (3) ◽  
Author(s):  
Nisha Rani Mahato ◽  
Snehashish Chakraverty
Keyword(s):  
Eigenvalue Problem ◽  
Material Properties ◽  
Fuzzy Number ◽  
Design Methodology ◽  
Eigenvalue Problems ◽  
Triangular Fuzzy Number ◽  
Real Eigenvalue ◽  
Eigenvalue Bounds ◽  
Filtering Algorithm ◽  
Structural Problems

Purpose The solution of dynamic problems of structures using finite element method leads to generalised eigenvalue problem. In general, if the material properties are crisp (exact) then we get crisp eigenvalue problem. But in actual practice, instead of crisp material properties we may have only bounds of values as a result of errors in measurements, observations and calculations or it may be due to maintenance induced error etc. Such bounds of values may be considered in terms of interval or fuzzy numbers. The purpose of this paper is to develop a fuzzy filtering procedure for finding real eigenvalue bounds of different structural problems. Design/methodology/approach The proposed fuzzy filtering algorithm has been developed in terms of fuzzy number to solve the fuzzy eigenvalue problem. The initial bounds of fuzzy eigenvalues are filtered to obtain precise eigenvalue bounds which are depicted by fuzzy (Triangular Fuzzy Number) plots using α-cut. Findings Previously, bounds of eigenvalues of interval matrices have been investigated by few authors. But when the structural problem consists of fuzzy material properties, then the interval eigenvalue bounds may be obtained for each interval of the fuzzy number. The proposed algorithm has been applied for standard fuzzy eigenvalue problems which may be extended to generalised fuzzy eigenvalue problems for obtaining filtered fuzzy bounds. Originality/value The developed fuzzy filtering method is found to be efficient for different structural dynamics problems with fuzzy material properties.

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×.

A Numerical Approach to the Stability of Rotor-Bearing Systems

1982 ◽  
Vol 104 (2) ◽  
pp. 356-363 ◽  
Author(s):  
K. Athre ◽  
J. Kurian ◽  
K. N. Gupta ◽  
R. D. Garg
Keyword(s):  
Eigenvalue Problem ◽  
Characteristic Polynomial ◽  
Eigenvalue Problems ◽  
Numerical Technique ◽  
Numerical Approach ◽  
Generalized Eigenvalue ◽  
Generalized Eigenvalue Problems ◽  
Rotor Bearing ◽  
The Stability ◽  
Bearing System

The stability characteristics of a rotor-bearing system which indicate the threshold of instability are generally obtained by applying the Routh-Hurwitz criterion to the characteristic polynomial. Usually the characteristic polynomial is obtained analytically from the characteristic determinant. In the case of the generalized eigenvalue problems, this is practically impossible. To study the stability characteristics of a floating bush bearing, the characteristic polynomial is constructed from the generalized eigenvalue problem using a recently developed numerical technique. Results obtained through this computer package are compared with those already available in the literature.

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 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.

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