scholarly journals A Method of Indefinite Krylov Subspace for Eigenvalue Problem

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
M. Aliyari ◽  
M. Ghasemi Kamalvand
Keyword(s):  
Eigenvalue Problem ◽  
Krylov Subspace ◽  
Eigenvalue Problems ◽  
Lanczos Method ◽  
Hermitian Matrices ◽  
Better Than

We describe an indefinite state of Arnoldi’s method for solving the eigenvalues problems. In the following, we scrutinize the indefinite state of Lanczos’ method for solving the eigenvalue problems and we show that this method for the J-Hermitian matrices works much better than Arnoldi’s method.

Generalizing of Numerically Solving Methods of Eigenvalue Problems to Asymmetrical, Damping Included Case

2001 ◽  
Author(s):  
Shahram Rezaei
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Lanczos Method ◽  
Subspace Method ◽  
Damping Matrix ◽  
Stiffness Matrices

Abstract In this paper, “Subspace” method is generalized to asymmetrical case. In the new algorithm described here, “Lanczos” method is used to find the first subspace and to solve the eigenvalue problem resulted in generalized subspace method. To solve the standard eigenvalue problem developed by “Lanczos” method “Jacoby” method is used. If eigenvalue problem includes damping matrix, that will be imported in new defined mass and stiffness matrices.

scholarly journals Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians

2013 ◽  
Vol 149 (9) ◽  
pp. 1569-1582 ◽  
Author(s):  
David Anderson ◽  
Edward Richmond ◽  
Alexander Yong
Keyword(s):  
Eigenvalue Problem ◽  
Recent Work ◽  
Equivariant Cohomology ◽  
Eigenvalue Problems ◽  
Schubert Calculus ◽  
Hermitian Matrices ◽  
Common Features ◽  
Saturation Theorem ◽  
The Common

AbstractThe saturation theorem of Knutson and Tao concerns the nonvanishing of Littlewood–Richardson coefficients. In combination with work of Klyachko, it implies Horn’s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization to majorized sums of Hermitian matrices, due to S. Friedland. We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland’s problem, via equivariant cohomology of Grassmannians. In particular, we prove a saturation theorem for this setting. Our arguments employ the aforementioned work together with recent work of H. Thomas and A. Yong.

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.

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 Separation Principle for Gyroscopic Conservative Systems

1997 ◽  
Vol 119 (1) ◽  
pp. 110-119 ◽  
Author(s):  
L. Meirovitch
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Separation Theorem ◽  
Separation Principle ◽  
Numerical Illustration ◽  
Closed Form Solutions ◽  
Conservative Systems ◽  
Algebraic Eigenvalue Problem

Closed-form solutions to differential eigenvalue problems associated with natural conservative systems, albeit self-adjoint, can be obtained in only a limited number of cases. Approximate solutions generally require spatial discretization, which amounts to approximating the differential eigenvalue problem by an algebraic eigenvalue problem. If the discretization process is carried out by the Rayleigh-Ritz method in conjunction with the variational approach, then the approximate eigenvalues can be characterized by means of the Courant and Fischer maximin theorem and the separation theorem. The latter theorem can be used to demonstrate the convergence of the approximate eigenvalues thus derived to the actual eigenvalues. This paper develops a maximin theorem and a separation theorem for discretized gyroscopic conservative systems, and provides a numerical illustration.

scholarly journals Natural Vibrations of Repetitive Structures

2000 ◽  
Vol 16 (2) ◽  
pp. 85-95 ◽  
Author(s):  
Dajun Wang ◽  
C.-C. Wang
Keyword(s):  
Eigenvalue Problem ◽  
Natural Vibration ◽  
Eigenvalue Problems ◽  
Vibration Frequencies ◽  
Natural Vibrations

ABSTRACTNatural vibration frequencies and modes of repetitive structures, including symmetric, periodic, linking structures, are considered in this work. By using the repetition of the identical parts, we reduce the eigenvalue problem of the structure to a set of eigenvalue problems of lower dimensions associated with the parts. Special forms and properties of the modes of natural vibrations are observed.

Analysis of vibration levels of large structural system with recursive component mode synthesis method: Theory and convergence

2006 ◽  
Vol 220 (9) ◽  
pp. 1339-1345 ◽  
Author(s):  
C W Kim
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Large Scale ◽  
Synthesis Method ◽  
Lanczos Method ◽  
Component Mode Synthesis ◽  
Structural System ◽  
Accurate Analysis ◽  
Satisfactory Performance ◽  
And Performance

The component mode synthesis (CMS) method has been extensively used in industries. However, industry finite-element (FE) models need a more efficient CMS method for satisfactory performance since the size of FE models needs to be increased for a more accurate analysis. Recently, the recursive component mode synthesis (RCMS) method was introduced to solve large-scale eigenvalue problem efficiently. This article focuses on the convergence of the RCMS method with respect to different parameters, and evaluates the accuracy and performance compared with the Lanczos method.

Positive solutions of nth-order impulsive eigenvalue problems with an advanced argument

2017 ◽  
Vol 10 (04) ◽  
pp. 1750072
Author(s):  
Gaoli Lu ◽  
Meiqiang Feng
Keyword(s):  
Eigenvalue Problem ◽  
Positive Solutions ◽  
Eigenvalue Problems ◽  
Hölder’S Inequality ◽  
Transformation Technique ◽  
Hölder's Inequality ◽  
Advanced Argument

In this paper, we study a [Formula: see text]th-order impulsive eigenvalue problem with an advanced argument. We shall establish several criteria for the optimal intervals of the parameter [Formula: see text] so as to ensure existence of single or many positive solutions. Our methods are based on transformation technique, Hölder’s inequality and the eigenvalue theory.

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