scholarly journals A New Approach to Solve Perturbed Symmetric Eigenvalue Problems

2020 ◽  
Vol 14 ◽  
Keyword(s):  
Eigenvalue Problem ◽  
Dynamic Analysis ◽  
Eigenvalue Problems ◽  
Symmetric Polynomial ◽  
New Approach ◽  
Calculation Time ◽  
Polynomial Eigenvalue Problem ◽  
Large Size ◽  
Eigen Value ◽  
Symmetric Eigenvalue Problems

The objective of this study is to ef-ciently resolve a perturbed symmetric eigen-value problem, without resolving a completelynew eigenvalue problem. When the size of aninitial eigenvalue problem is large, its multipletimes solving for each set of perturbations can becomputationally expensive and undesired. Thistype of problems is frequently encountered inthe dynamic analysis of mechanical structures.This study deals with a perturbed symmetriceigenvalue problem. It propose to develop atechnique that transforms the perturbed sym-metric eigenvalue problem, of a large size, toa symmetric polynomial eigenvalue problem ofa much reduced size. To accomplish this, weonly need the introduced perturbations, the sym-metric positive-de nite matrices representing theunperturbed system and its rst eigensolutions.The originality lies in the structure of the ob-tained formulation, where the contribution of theunknown eignsolutions of the unperturbed sys-tem is included. The e ectiveness of the pro-posed method is illustrated with numerical tests.High quality results, compared to other existingmethods that use exact reanalysis, can be ob-tained in a reduced calculation time, even if theintroduced perturbations are very signi cant.

scholarly journals Linearization schemes for Hermite matrix polynomials

2015 ◽  
Author(s):  
Nikta Shayanfar ◽  
Heike Fassbender
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Matrix Polynomial ◽  
Multiple Input Multiple Output ◽  
Matrix Pencil ◽  
Least Square ◽  
Coupled Systems ◽  
Matrix Polynomials ◽  
Polynomial Eigenvalue Problem ◽  
The Matrix

The polynomial eigenvalue problem is to find the eigenpair of $(\lambda,x) \in \mathbb{C}\bigcup \{\infty\} \times \mathbb{C}^n \backslash \{0\}$ that satisfies $P(\lambda)x=0$, where $P(\lambda)=\sum_{i=0}^s P_i \lambda ^i$ is an $n\times n$ so-called matrix polynomial of degree $s$, where the coefficients $P_i, i=0,\cdots,s$, are $n\times n$ constant matrices, and $P_s$ is supposed to be nonzero. These eigenvalue problems arise from a variety of physical applications including acoustic structural coupled systems, fluid mechanics, multiple input multiple output systems in control theory, signal processing, and constrained least square problems. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Such methods convert the eigenvalue problem into a well-studied linear eigenvalue problem, and meanwhile, exploit and preserve the structure and properties of the original eigenvalue problem. The linearizations have been extensively studied with respect to the basis that the matrix polynomial is expressed in. If the matrix polynomial is expressed in a special basis, then it is desirable that its linearization be also expressed in the same basis. The reason is due to the fact that changing the given basis ought to be avoided \cite{H1}. The authors in \cite{ACL} have constructed linearization for different bases such as degree-graded ones (including monomial, Newton and Pochhammer basis), Bernstein and Lagrange basis. This contribution is concerned with polynomial eigenvalue problems in which the matrix polynomial is expressed in Hermite basis. In fact, Hermite basis is used for presenting matrix polynomials designed for matching a series of points and function derivatives at the prescribed nodes. In the literature, the linearizations of matrix polynomials of degree $s$, expressed in Hermite basis, consist of matrix pencils with $s+2$ blocks of size $n \times n$. In other words, additional eigenvalues at infinity had to be introduced, see e.g. \cite{CSAG}. In this research, we try to overcome this difficulty by reducing the size of linearization. The reduction scheme presented will gradually reduce the linearization to its minimal size making use of ideas from \cite{VMM1}. More precisely, for $n \times n$ matrix polynomials of degree $s$, we present linearizations of smaller size, consisting of $s+1$ and $s$ blocks of $n \times n$ matrices. The structure of the eigenvectors is also discussed.

scholarly journals High performance solution of skew-symmetric eigenvalue problems with applications in solving the Bethe-Salpeter eigenvalue problem

2020 ◽  
Vol 96 ◽  
pp. 102639
Author(s):  
Carolin Penke ◽  
Andreas Marek ◽  
Christian Vorwerk ◽  
Claudia Draxl ◽  
Peter Benner
Keyword(s):  
Eigenvalue Problem ◽  
High Performance ◽  
Eigenvalue Problems ◽  
Symmetric Eigenvalue Problems

A new approach for solving symmetric eigenvalue problems

1992 ◽  
Vol 3 (6) ◽  
pp. 671-679 ◽  
Author(s):  
C. Carey ◽  
H.-C. Chen ◽  
G. Golub ◽  
A. Sameh
Keyword(s):  
Eigenvalue Problems ◽  
New Approach ◽  
Symmetric Eigenvalue Problems

scholarly journals Homotopy continuation method for the numerical solutions of generalised symmetric eigenvalue problems

1991 ◽  
Vol 32 (4) ◽  
pp. 437-456 ◽  
Author(s):  
D. C. Dzeng ◽  
W. W. Lin
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Numerical Solutions ◽  
Continuation Method ◽  
Positive Semidefinite ◽  
Symmetric Matrices ◽  
Homotopy Continuation ◽  
Smooth Curves ◽  
Homotopy Continuation Method ◽  
Symmetric Eigenvalue Problems

AbstractWe consider a generalised symmetric eigenvalue problem Ax = λMx, where A and M are real n by n symmetric matrices such that M is positive semidefinite. The purpose of this paper is to develop an algorithm based on the homotopy methods in [9, 11] to compute all eigenpairs, or a specified number of eigenvalues, in any part of the spectrum of the eigenvalue problem Ax = λMx. We obtain a special Kronecker structure of the pencil A − λM, and give an algorithm to compute the number of eigenvalues in a prescribed interval. With this information, we can locate the lost eigenpair by using the homotopy algorithm when multiple arrivals occur. The homotopy maintains the structures of the matrices A and M (if any), and the homotopy curves are n disjoint smooth curves. This method can be used to find all/some isolated eigenpairs for large sparse A and M on SIMD machines.

A New Approach to the Rigid Finite Element Method in Modeling Spatial Slender Systems

2018 ◽  
Vol 18 (02) ◽  
pp. 1850017 ◽  
Author(s):  
Iwona Adamiec-Wójcik ◽  
Łukasz Drąg ◽  
Stanisław Wojciech
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Nonlinear Dynamic Analysis ◽  
New Approach ◽  
Static And Dynamic Analysis ◽  
Rigid Finite Element Method ◽  
Longitudinal Flexibility ◽  
Element Method

The static and dynamic analysis of slender systems, which in this paper comprise lines and flexible links of manipulators, requires large deformations to be taken into consideration. This paper presents a modification of the rigid finite element method which enables modeling of such systems to include bending, torsional and longitudinal flexibility. In the formulation used, the elements into which the link is divided have seven DOFs. These describe the position of a chosen point, the extension of the element, and its orientation by means of the Euler angles Z[Formula: see text]Y[Formula: see text]X[Formula: see text]. Elements are connected by means of geometrical constraint equations. A compact algorithm for formulating and integrating the equations of motion is given. Models and programs are verified by comparing the results to those obtained by analytical solution and those from the finite element method. Finally, they are used to solve a benchmark problem encountered in nonlinear dynamic analysis of multibody systems.

Transparent boundary conditions for discretized non-Hermitian eigenvalue problems

2021 ◽  
pp. 2150138
Author(s):  
Jonathan Heinz ◽  
Miroslav Kolesik
Keyword(s):  
Quantum Mechanics ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Complex Scaling ◽  
Transparent Boundary Conditions ◽  
Drastic Reduction ◽  
Optical Cavities ◽  
Energy Dependent ◽  
Lossy Waveguides

A method is presented for transparent, energy-dependent boundary conditions for open, non-Hermitian systems, and is illustrated on an example of Stark resonances in a single-particle quantum system. The approach provides an alternative to external complex scaling, and is applicable when asymptotic solutions can be characterized at large distances from the origin. Its main benefit consists in a drastic reduction of the dimesnionality of the underlying eigenvalue problem. Besides application to quantum mechanics, the method can be used in other contexts such as in systems involving unstable optical cavities and lossy waveguides.

Polynomial Eigenvalue Problem

2020 ◽  
pp. 195-210
Author(s):  
Federico Milano ◽  
Ioannis Dassios ◽  
Muyang Liu ◽  
Georgios Tzounas
Keyword(s):  
Eigenvalue Problem ◽  
Polynomial Eigenvalue Problem
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