Eigenvalue problem for tridiagonal matrices arising in the scattering-theory analysis of disordered conductors

1992 ◽  
Vol 70 (4) ◽  
pp. 257-267 ◽  
Author(s):  
Josef A. Zuk
Keyword(s):  
Eigenvalue Problem ◽  
Scattering Theory ◽  
Perturbation Expansion ◽  
Approximate Solutions ◽  
Theory Analysis ◽  
Tridiagonal Matrices ◽  
Finite Matrix ◽  
Special Cases ◽  
Exactly Solvable ◽  
Qualitative Characteristics

The eigenvalue problem for a family of tridiagonal matrices arising in the scattering-theory analysis of the conductance in mesoscopic systems, and its fluctuations, is studied. The exactly solvable special cases are identified. For the general problem, qualitative characteristics of the spectrum are established, and approximate solutions for the eigenvalues are constructed. These comprise ones that are valid in the limit of large but finite matrix dimension, and those derived from a perturbation expansion around each of the exactly solvable cases.

scholarly journals Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 870
Author(s):  
Diego Caratelli ◽  
Paolo Emilio Ricci
Keyword(s):  
Functional Analysis ◽  
Computer Algebra ◽  
Toeplitz Matrices ◽  
Test Cases ◽  
Recursive Procedure ◽  
Tridiagonal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Matrix Eigenvalues ◽  
Complex Valued

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

Approximate Solutions

2020 ◽  
pp. 106-158
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Random Walk ◽  
Mean Field ◽  
Gaussian Model ◽  
Spherical Model ◽  
Approximate Solutions ◽  
Field Approximation ◽  
Approximation Schemes ◽  
Mean Field Approximation ◽  
Exactly Solvable ◽  
The Subject

Chapter 3 discusses the approximation schemes used to approach lattice statistical models that are not exactly solvable. In addition to the mean field approximation, it also considers the Bethe–Peierls approach to the Ising model. Moreover, there is a thorough discussion of the Gaussian model and its spherical version, both of which are two important systems with several points of interest. A chapter appendix provides a detailed analysis of the random walk on different lattices: apart from the importance of the subject on its own, it explains how the random walk is responsible for the critical properties of the spherical model.

scholarly journals Approximate Solutions of Nonlinear Partial Differential Equations by Modifiedq-Homotopy Analysis Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Shaheed N. Huseen ◽  
Said R. Grace
Keyword(s):  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Approximate Solutions ◽  
Power Series Solution ◽  
Analysis Method ◽  
Exactly Solvable ◽  
Homotopy Analysis

A modifiedq-homotopy analysis method (mq-HAM) was proposed for solvingnth-order nonlinear differential equations. This method improves the convergence of the series solution in thenHAM which was proposed in (see Hassan and El-Tawil 2011, 2012). The proposed method provides an approximate solution by rewriting thenth-order nonlinear differential equation in the form ofnfirst-order differential equations. The solution of thesendifferential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.

scholarly journals A perturbed algorithm for strongly nonlinear variational-like inclusions

2000 ◽  
Vol 62 (3) ◽  
pp. 417-426 ◽  
Author(s):  
C.-H. Lee ◽  
Q. H. Ansari ◽  
J.-C. Yao
Keyword(s):  
Exact Solution ◽  
General Setting ◽  
Approximate Solutions ◽  
Strongly Nonlinear ◽  
Special Cases ◽  
The One

In this paper, we define the concept of η- subdifferential in a more general setting than the one used by Yang and Craven in 1991. By using η-subdifferentiability, we suggest a perturbed algorithm for finding the approximate solutions of strongly nonlinear variational-like inclusions and prove that these approximate solutions converge to the exact solution. Several special cases are also discussed.

scholarly journals Eigenvibrations of a simply supported beam with elastically attached load

2018 ◽  
Vol 224 ◽  
pp. 04012 ◽  
Author(s):  
Anton A. Samsonov ◽  
Sergey I. Solov’ev ◽  
Pavel S. Solov’ev
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Eigenvalue Problem ◽  
Approximate Solutions ◽  
Uniform Grid ◽  
Simply Supported Beam ◽  
Simply Supported ◽  
Plates And Shells ◽  
The Finite Difference Method ◽  
Theoretical Results

The nonlinear differential eigenvalue problem describing eigenvibrations of a simply supported beam with elastically attached load is investigated. The existence of an increasing sequence of positive simple eigenvalues with limit point at infinity is established. To the sequence of eigenvalues, there corresponds a system of normalized eigenfunctions. To illustrate the obtained theoretical results, the initial problem is approximated by the finite difference method on a uniform grid. The accuracy of approximate solutions is studied. Investigations of the present paper can be generalized for the cases of more complicated and important problems on eigenvibrations of plates and shells with elastically attached loads.

Infinite Matrices, Ritz Theorem and Bound State Problems

1975 ◽  
Vol 30 (2) ◽  
pp. 256-261 ◽  
Author(s):  
A. K. Mitra
Keyword(s):  
Eigenvalue Problem ◽  
Convenient Method ◽  
Bound State ◽  
Wave Functions ◽  
Infinite Matrix ◽  
Variational Technique ◽  
Matrix Eigenvalue Problem ◽  
Schroedinger Operator ◽  
Matrix Eigenvalue ◽  
Finite Matrix

Abstract The straight forward application of the Ritz variational technique has been shown to be a very convenient method for obtaining numerically the first few discrete eigenvalues of the Schroedinger operator with certain special types of potentials. This method solves essentially the (finite) matrix eigenvalue problem obtained by truncating the infinite matrix representing the Schroedinger operator with respect to the Coulomb wave functions. The Ritz theorem justifies the validity of this truncation procedure.

scholarly journals Asymptotology—a cautionary tale

2002 ◽  
Vol 44 (1) ◽  
pp. 33-40 ◽  
Author(s):  
R. L. Dewar
Keyword(s):  
Eigenvalue Problem ◽  
Applied Mathematics ◽  
Theoretical Physics ◽  
Local Analysis ◽  
Cautionary Tale ◽  
Stokes Phenomenon ◽  
Solvable Problem ◽  
Exactly Solvable ◽  
Continuous Range

AbstractThe art of asymptotology is a powerful tool in applied mathematics and theoretical physics, but can lead to erroneous conclusions if misapplied. A seemingly paradoxical case is presented in which a local analysis of an exactly solvable problem appears to find solutions to an eigenvalue problem over a continuous range of the eigenvalue, whereas the spectrum is known to be discrete. The resolution of the paradox involves the Stokes phenomenon. The example illustrates two of Kruskal's Principles of Asymptotology.

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.

Dynamics of Coupled Fluid-Shells

1986 ◽  
Vol 108 (3) ◽  
pp. 339-347 ◽  
Author(s):  
M. K. Au-Yang
Keyword(s):  
Finite Element ◽  
Structural Analysis ◽  
Eigenvalue Problem ◽  
Cylindrical Shells ◽  
Computer Programs ◽  
Dynamic Problems ◽  
Special Cases ◽  
Analysis Computer

The hydrodynamic mass approach to the solution of dynamic problems in coupled fluid-cylindrical shells is reviewed; simplified equations for computing the hydrodynamic masses and for the subsequent solution of the eigenvalue problem are given in several commonly encountered special cases. Methods of incorporating the hydrodynamic mass concept into finite element structural analysis computer programs for the more general cases are discussed.

Sign in / Sign up

Export Citation Format

Share Document