The eigenvalue problem for infinite systems of linear equations

1977 ◽  
Vol 82 (2) ◽  
pp. 269-273 ◽  
Author(s):  
F. P. Sayer
Keyword(s):  
Eigenvalue Problem ◽  
Infinite System ◽  
Linear Equations ◽  
Theoretical Work ◽  
Rectangular Plates ◽  
System Of Linear Equations ◽  
Usual Procedure ◽  
Systems Of Linear Equations ◽  
Image Position

Given an infinite system of linear equationswhere the aij depend on a parameter λ, the eigenvalue problem is to determine values of λ for which xj (j = 1, 2, …) are not all zero. This problem (Taylor (3) and Vaughan (4)) can arise in the vibration of rectangular plates. Little theoretical work, however, appears to have been done concerning the existence and determination of the eigenvalues. The usual procedure (see (3) and (4)) is to consider a truncated or reduced system of N equations and find the values of λ for which the determinant of the N × N matrix [aij] vanishes. If a particular λ tends to a constant value as N is increased then this value is assumed to be an eigenvalue. The question therefore arises as to what happens if no limit exists. Can we assert that there are no eigenvalues? By constructing an appropriate example we show that the non-existence of a limit does not imply the non-existence of eigenvalues. In order to construct our example we first establish a result concerning the Legendre polynomials.

Numerical Linear Algebra

1966 ◽  
Vol 9 (05) ◽  
pp. 757-801 ◽  
Author(s):  
W. Kahan
Keyword(s):  
Eigenvalue Problem ◽  
Linear Algebra ◽  
Linear Equations ◽  
Numerical Linear Algebra ◽  
System Of Linear Equations ◽  
Image Position

The primordial problems of linear algebra are the solution of a system of linear equations and the solution of the eigenvalue problem for the eigenvalues λk, and corresponding eigenvectors of a given matrix A.

On the Stresses in a Notched Strip

1952 ◽  
Vol 19 (2) ◽  
pp. 141-146
Author(s):  
Chih-Bing Ling
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Numerical Results ◽  
Theoretical Solution ◽  
Experimental Results ◽  
System Of Linear Equations ◽  
Transverse Bending ◽  
Longitudinal Tension

Abstract In a previous paper by the author (1), a theoretical solution for a notched strip under longitudinal tension is given. The result demands the solution of an infinite system of linear equations. A considerable amount of labor is involved in solving such a system. It seems, however, that the labor can be diminished by adapting to the solution a process known as the promotion of rank. In this paper such a process is described and then applied to solve the problem of a notched strip under transverse bending. The solution of this problem seems also to be new. The numerical results obtained are compared graphically with the experimental results available.

A Diagonally Dominant Solution for the Cylinder End Problem

1981 ◽  
Vol 48 (4) ◽  
pp. 876-880 ◽  
Author(s):  
T. D. Gerhardt ◽  
Shun Cheng
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Infinite Cylinder ◽  
Algebraic Equations ◽  
Precise Calculation ◽  
Linear Algebraic Equations ◽  
Present Solution ◽  
Diagonally Dominant ◽  
Expansion Coefficients

An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

A System of Linear Equations with an Infinity of Unknowns

1924 ◽  
Vol 22 (3) ◽  
pp. 282-286
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Present Note ◽  
Linear Equations ◽  
The Other ◽  
System Of Linear Equations ◽  
Other Hand ◽  
Image Position ◽  
Monotonic Character

I have collected in the present note some theorems regarding the solution of a certain system of linear equations with an infinity of unknowns. The general form of the equations isthe numbers a1, a2, … c1, c2, … being given. Equations of this type are of course well known; but in studying them it is generally assumed that the series depend for convergence on the convergence-exponent of the sequences involved, e.g. that and are convergent. No assumptions of this kind are made here, and in fact the series need not be absolutely convergent. On the other hand rather special assumptions are made with regard to the monotonic character of the sequences an and cn.

scholarly journals The New Approach to Analysis of Thin Isotropic Symmetrical Plates

2020 ◽  
Vol 10 (17) ◽  
pp. 5931
Author(s):  
Mykhaylo Delyavskyy ◽  
Krystian Rosiński
Keyword(s):  
Present Method ◽  
Automatic Differentiation ◽  
Linear Equations ◽  
High Accuracy ◽  
Rectangular Plates ◽  
System Of Linear Equations ◽  
New Approach ◽  
Summation Convention ◽  
Analytical Formulae ◽  
Einstein Summation Convention

A new approach to solve plate constructions using combined analytical and numerical methods has been developed in this paper. It is based on an exact solution of an equilibrium equation. The proposed mathematical model is implemented as a computer program in which known analytical formulae are rewritten as wrapper functions of two arguments. Partial derivatives are calculated using automatic differentiation. A solution of a system of linear equations is substituted to these functions and evaluated using the Einstein summation convention. The calculated results are presented and compared to other analytical and numerical ones. The boundary conditions are satisfied with high accuracy. The effectiveness of the present method is illustrated by examples of rectangular plates. The model can be extended with the ability to solve plates of any shape.

scholarly journals The computation of stationary distributions of Markov chains through perturbations

1991 ◽  
Vol 4 (1) ◽  
pp. 29-46 ◽  
Author(s):  
Jeffery J. Hunter
Keyword(s):  
Stochastic Matrix ◽  
Linear Equations ◽  
Probability Vector ◽  
Doubly Stochastic ◽  
Rank One ◽  
Systems Of Linear Equations ◽  
Stationary Probability Vector ◽  
Irreducible Markov Chain ◽  
Algorithmic Procedure

An algorithmic procedure for the determination of the stationary distribution of a finite, m-state, irreducible Markov chain, that does not require the use of methods for solving systems of linear equations, is presented. The technique is based upon a succession of m, rank one, perturbations of the trivial doubly stochastic matrix whose known steady state vector is updated at each stage to yield the required stationary probability vector.

THE ELECTROSTATIC FIELD OF TWO EQUAL CIRCULAR CO-AXIAL CONDUCTING DISKS

1949 ◽  
Vol 2 (4) ◽  
pp. 428-451 ◽  
Author(s):  
E. R. LOVE
Keyword(s):  
Integral Equation ◽  
Explicit Solution ◽  
Existence And Uniqueness ◽  
Infinite System ◽  
Linear Equations ◽  
Neumann Series ◽  
Standard Theory ◽  
System Of Linear Equations ◽  
Electrostatic Problem ◽  
Spheroidal Harmonics

Abstract In the earliest discussion of this problem Nicholson (1) expressed the potential as a series of spheroidal harmonics with coefficients satisfying an infinite system of linear equations, and gave a formula for an explicit solution; but this formula appears to be meaningless and its derivation to contain serious errors. In the present paper, starting tentatively from Nicholson's infinite system of linear equations, a much simpler, though still implicit, specification of the potential is developed; this involves a Fredholm integral equation the existence and uniqueness of whose solution are deducible from standard theory. The specification so obtained for the potential is shown rigorously to satisfy the differential equation and boundary conditions of the electrostatic problem. The Neumann series of the integral equation is shown to converge to its solution, so that the potential, and other aspects of the field, can be explicitly formulated and thus computed. The errors in Nicholson's process of solving his system of equations are exhibited in detail, and it is concluded that attempts to carry through that process without error cannot lead to an explicit solution.

scholarly journals Application of the Concept of Linear Equation Systems in Balancing Chemical Reaction Equations

2020 ◽  
Vol 1 (4) ◽  
pp. 130-135
Author(s):  
Dwindi Agryanti Johar
Keyword(s):  
Chemical Reactions ◽  
Chemical Reaction ◽  
Mathematical Problem ◽  
Linear Equations ◽  
System Of Linear Equations ◽  
Reaction Coefficient ◽  
Chemical Equations ◽  
Systems Of Linear Equations ◽  
Chemical Equation ◽  
Reaction Equations

This study discusses the equalization of chemical reactions using a system of linear equations with the Gaussian and Gauss-Jordan elimination. The results show that there is a contradiction in the existing methods for balancing chemical reactions. This study also aims to criticize several studies that say that the equalization of the reaction coefficient can use a system of linear equations. In this paper, the chemical equations were balanced by representing the chemical equation into systems of linear equations. Particularly, the Gauss and Gauss-Jordan elimination methods were used to solve the mathematical problem with this method, it was possible to handle any chemical reaction with given reactants and products.

Analyzing Unstable Systems of Linear Equations

2000 ◽  
Vol 93 (5) ◽  
pp. 388-390
Author(s):  
Harris S. Shultz
Keyword(s):  
High School ◽  
Linear Equations ◽  
Mathematical Concept ◽  
School Mathematics ◽  
Problem Situation ◽  
High School Mathematics ◽  
System Of Linear Equations ◽  
Unstable Systems ◽  
Evaluation Standards ◽  
Systems Of Linear Equations

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989, 146) states, “Students who are able to apply and translate among different representations of the same problem situation or of the same mathematical concept will have at once a powerful, flexible set of tools for solving problems and a deeper appreciation of the consistency and beauty of mathematics”. The Standards also states, “The connections between algebra and geometry are among the most important in high school mathematics” (1989, 147). This article shows how the phenomenon of instability in the solution of a system of linear equations can be analyzed both algebraically and geometrically.

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