Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1971"

An eigenvalue algorithm for skew-symmetric matrices

Abstract: A Jacobi-like algorithm is presented for the skew-symmetric eigenvalue problem. The process constructs iteratively, with elementary orthogonal transformations, a sequence of matrices which converges to the so-called Murnaghan form of the intial matrix.

Parallel Computation of Eigenvalues of Real Matrices

Abstract: : The paper describes the implementation of three standard matrix eigenvalue computation methods on an array machine with high efficiency. A brief description of the ILLIAC 4 computer is provided as background material. Three major sections follow--the first two describe Jacobi and Householder algorithms for real symmetric matrices, and the third describes the QR algorithm for real nonsymmetric matrices.

Note on the computation of the maximal eigenvalue of a non-negative irreducible matrix

Abstract: We prove the convergence of Wielandt's method in the computation of the maximal eigenvalue and eigenvector of a non-negative irreducible matrix.

An Algorithm for the Generalized Matrix Eigenvalue Problem Ax = Lambda Bx

Abstract: : A new method, called the QZ algorithm, is presented for the solution of the matrix eigenvalue problem Ax = lambda Bx with general square matrices A and B. Particular attention is paid to the degeneracies which result when B is singular. No inversions of B or its submatrices are used. The algorithm is a generalization of the QR algorithm, and reduces to it when B = I. A FORTRAN program and some illustrative examples are included.

On the inverse eigenvalue problem for matrices and related problems for difference and differential equations

Abstract: The problem of estimating parameters α1,...,αk of the matrix valued function M(λ,α) given eigenvalue data λ1,...,λp, p ≥ k, is considered. Two algorithms are presented. The first reduces the estimation problem to an unconstrained minimisation and contains as special cases methods suggested by other authors. The second reduces the problem to one of minimisation subject to equality constraints. Examples are given to show that the behaviour of the solutions can be involved so that the application of numerical methods is probably of necessity tentative. The results of some numerical experiments are summarised.

Transversality in Nonlinear Eigenvalue Problems

Abstract: Publisher Summary This chapter describes the transversality in nonlinear eigenvalue problems. The use of transversality or general position arguments together with traditional techniques for studying nonlinear eigenvalue problems enables one to obtain new results on the geometric structure and multiplicity of solutions of nonlinear eigenvalue problems. The chapter explains how transversality can be used to obtain a continuum of solutions for an eigenvalue problem involving a cone preserving map. For that, odd multiplicity is not needed, but the cone preservation is used to obtain an unbounded continuum of solutions of an equation u = λ F (u) , where F is a compact map that preserves the cone of non-negative functions in an L 2 space and has a derivative at zero with a single eigenvalue λ 0 corresponding to eigenvectors in the cone. In this case, the continuum emanates from the point (λ 0 , 0) . The result is applied to a simple example involving a pair of linked differential equations.

A variational approach to a class of nonlinear eigenvalue problems.

Abstract: Letf be a real-valued differentiable function defined on the real reflexive Banach space X. The problem of minimizing f over a subset of X is investigated under the following mild monotonicity assumption on the derivative f' off: if { u, } is a sequence in X converging weakly to some u EX, then lim sup (fu, uny -u) >0. The eigenvalue problem f'u =Xg'u for some X GRi, with g' being the derivative of a further function g, is then reduced to that first question. Introduction. We consider the following variational PROBLEM (*). To minimize a real-valued function f over a subset C oJ the real reflexive Banach space X. If the function f is differentiable, a solution of Problem (*) with C = X yields a solution of the equation f'u = 0. Iff is differentiable and C=M0(g) = {uEX:g(u) =c} is a level set of a further differentiable function g, a solution of Problem (*) is (under additional restrictions) a solution of the eigenvalue problemf'u =Xg'u for some real X. For functionsf which can be represented in the formf(u) ='P(u, u), with 4b a semiconvex mapping of XXX into the reals, Problem (*) is treated in Browder [I], where the abstract results are also applied to multiple integral functionals. In the recent note [6] Browder attacks Problem (*) for differentiable functions f and convex sets C, making use of properties of the derivativef' rather than of those of f. In the application to multiple integral functionals, his new results are, however, weaker than those he got in [1]. Further the convexity assumption on the set C is rather restrictive; eigenvalue problems cannot be studied by this method. In the present note we investigate Problem (*) for differentiable functions with a related method which does not demand the convexity of C. Compared with [6] we require more continuity properties on the derivative of the function f, but can considerably weaken the monotonicity assumptions on f'. Presented to the Society, August 17, 1970; received by the editors August 12, 1970 and, in revised form, October 26, 1970. AMS 1969 subject classifications. Primary 4780, 4610.

On the Bifurcation of Solutions of the Nonlinear Eigenvalue Problem $Lu + \lambda b(x)u = g(x,u)$

Abstract: Under certain circumstances the nonlinear eigenvalue problem $Lu + \lambda b(x)u = g(x,u)$ has two positive solutions for a continuous range of values of $\lambda $. The detailed nature of these conditions is analyzed by a method originally proposed by A. Hammerstein ; the number of solutions for eigenvalues in the neighborhood of a special eigenvalue is found to depend on certain properties of the nonlinear function $g(x,u)$ and its derivatives. A method of obtaining bounds on this special eigenvalue is presented. Since this problem describes physical phenomena, a stability analysis of the related time-dependent problem is performed in order to determine which solution is stable for small perturbations. An iterative method of solution is proposed which yields both solutions; error estimates are provided. An illustrative example from nonlinear nuclear reactor statics concludes the analysis.

Eigenvalue Moments for Ensembles of Random Matrices

Abstract: A straightforward method is given for calculating the moments of the joint eigenvalue distribution corresponding to an ensemble of random Hermitian matrices. The method enables one to calculate the averages using the matrix element distribution. Thus an explicit expression for the joint eigenvalue distribution is not required.

Variational solution of certain nonlinear eigenvalue problems

Abstract: s of Australasian Ph.D. theses Variational solution of certain nonlinear eigenvalue problems Alan L. Andrew The \"nonlinear eigenvalue problems\" studied in this thesis are nonlinear in the eigenvalue but generally involve linear operators. One such problem, a standard formulation of the astrophysical problem of non-radial oscillations of stars, has been solved numerically by the Ritz method by several workers. However the problem is not covered by the classical theory of the Ritz method. This thesis examines a class of nonlinear eigenvalue problems which contains the astrophysical problem as a special case. Various sufficient conditions are established for Ritz approximate (R.a.) solutions (solutions obtained by the Ritz method) of some of this class of problems to possess certain properties of the exact solutions and for R.a. solutions to converge to exact solutions. In several important respects these R.a. solutions are shown to be more sensitive to changes in coordinate functions than is the case for related linear eigenvalue problems. Theoretical results proved here are supplemented by results of carefully designed numerical experiments. The problems studied here are converted into a form linear in the eigenvalue (but not by the classical method involving product spaces). R.a. solutions obtained from the linear and the nonlinear formulations are compared. New results are also proved concerning the use of the Ritz and several other numerical methods for linear eigenvalue problems in Hilbert space. Fuller summaries of this work are contained in the author's papers [1 5]. However the thesis contains much material not in these papers, Received 2*+ February 1971. Thesis submitted to La Trobe University, July 1970. Degree approved, February 1971Supervisor: Dr A.R. Jones. 137

Allmat: a tss/36o fortran iv subroutine for eigenvalues and eigenvectors of a

Abstract: number of iterations necessary for each eigenvalue and eigenvector. The program, ALLMAT, uses the com- INTRODUCTION Many areas of physics, mathematics, statistics, and engineering require the eigen- values and eigenvectors of square matrices. times called the algebraic eigenvalue problem, holds a place that is