Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1978"

Numerical procedures for solving nonsymmetric eigenvalue problems associated with optical resonators.

Abstract: The Prony method is extended to handle the nonsymmetric algebraic eigenvalue problem and improved to search automatically for the number of dominant eigenvalues. A simple iterative algorithm is given to compute the associated eigenvectors. Resolution studies using the QR method are made in order to determine the accuracy of the matrix approximation. Numerical results are given for both simple well defined resonators and more complex advanced designs containing multiple propagation geometries and misaligned mirrors.

Perturbation Theory for the Generalized Eigenvalue Problem

Abstract: Publisher Summary This chapter discusses the perturbation theory for the generalized eigenvalue problem. The problems in which have nearly common null spaces give rise to unstable eigenvalues that can affect otherwise stable eigenvalues. This places an unfortunate limitation on a priori about the generalized eigenvalue problem, and in general, one shall have to restrict the size of the perturbations so that none of the eigenvalues can move too much. It should be possible to prove strong conditional theorems about individual eigenvalues under the assumption that the perturbations leave them well separated from their neighbors. Incidentally, ill-conditioned problems of this kind can easily be generated in practice. It is found that if A and B are Rayleigh–Ritz approximations to two operators that have been obtained by using a nearly degenerate basis, then A and B will have an approximate null vector whose components are the coefficients of a linear combination of the basis that approximates zero. There remains the anomaly that the supposedly well-conditioned eigenvalue is badly perturbed by the perturbation in A.

The additive inverse eigenvalue problem and topological degree

Abstract: A short proof using topological degree is given of the additive inverse eigenvalue problem: The diagonal elements of any square complex matrix can be altered so as to cause the altered matrix to have any prescribed set of eigenvalues. The (finite-dimensional) additive inverse eigenvalue problem is the following. Given an n by n complex matrix M and a set of n complex numbers 41, . . . , (n possibly with repetitions, does there exist a diagonal matrix D such that the eigenvalues of M + D are precisely 41, . . ., ,n? The answer is yes. In fact generically in 41, . . . , 4n, there exist n! such D, and hence there are always between 1 and n! solutions D. This problem was solved by S. Friedland after some others had obtained various partial solutions. He has given two proofs. The first [2] is completely algebraic and the second [4] uses some powerful results from algebraic geometry. The purpose of this note is to give a short proof using topological degree. Degree theory is a standard tool for many analysts and the present proof may feel closer to home. Also, the fact that the degree is nonzero allows the use of methods developed by Kellogg, Li, and Yorke [5], [6] and Hirsch and Smale (unpublished) to find the solutions numerically. See [1, Theorem 4.2]. It should also be mentioned that Friedland has solved the multiplicative inverse eigenvalue problem (where one tries to specify the eigenvalues of MD instead of M + D), also using degree theory [3]. An elegant general reference for degree theory is John Milnor's book [7]. We turn to the proof. Let tn + a,tn-i + * + an be the monic polynomial with roots 41, . .. , cn. We want the characteristic polynomial of M + D to bep. ConsiderfM: Cn -> Cn defined by fM (D) = (a1, . . . , an) if det(tI (M + D)) = tn + a Itn-1 + + anIf M = 0, the zero matrix, and D has entries dl, d2, ..., dn, then Received by the editors October 15, 1976. AMS (MOS) subject classifications (1970). Primary 15A18; Secondary 15-04. 'Partially supported by an NSF contract and a University of Maryland Faculty Research Grant. ? American Mathematical Society 1978 5 This content downloaded from 157.55.39.111 on Wed, 03 Aug 2016 05:16:58 UTC All use subject to http://about.jstor.org/terms

A Note on the Eigenvalues of Hermitian Matrices

Abstract: Two simple relations are derived that connect the eigenvalues of a Hermitian matrix with those of the submatrix obtained by deleting a row and the corresponding column. The relations, which readily establish the interlacing of these two sets of eigenvalues, are used to obtain an upper bound for the largest eigenvalue and a lower bound for the smallest eigenvalue of a Hermitian matrix.

Bifurcation from a Multiple Eigenvalue and Stability of Solutions

Abstract: A study is made of the branching of solutions of nonlinear operators in the case of a general multiple eigenvalue. An analysis is made of the structure of the nontrivial branches, and the stability of the bifurcating solutions is discussed. Some generalizations of the well-known result pertaining to a simple eigenvalue are obtained.

An imbedding method for matrix eigenvalue problems

Abstract: Determination of the (possibly complex) eigenvalues of a general square matrix is reduced to integration in the complex plane of ordinary differential equations subject to known initial conditions. Some formulae from the theory of functions of a complex variable play a crucial role. The method is illustrated with results of some numerical experiments.

Eigenvalue sensitivity method for optimal stabilisation of linear systems

Abstract: Using properties of eigenvalue sensitivity matrices a procedure is presented for designing state feedback controllers that give the closed loop system a prescribed set of preassigned eigenvalues, minimising a quadratic cost functional.

An inverse eigenvalue problem for random matrices

Abstract: The mean and variance of the top eigenvalue of a discrete version of the operator $ - abla ^2 + q$ are shown to be sufficient to determine the mean of the random vector q.

Numerical solution of a non-linear self-consistent eigenvalue problem☆

Abstract: THE POSSIBILITY of the numerical solution of the system of non-linear differential equations arising in the self-consistent problem of the interaction of a particle with a quantum field, on the basis of continuous analog of Newton's method, is investigated.

On the Asymptotic Behavior of the Perron Eigenvalue

Abstract: A real n×n matrix A = (aij) is said to be an ml-matrix if aij ≥ 0 for all i ≠ j. Such matrices arise in the study of stability of multiple markets in mathematical economics [1], [3] as well as in the theory of finite Markov processes [5]. As can be seen through the Perron-Frobenius theorem, any ml-matrix has a real eigenvalue which is called the Perron eigenvalue of A, so that if λ is any other eigenvalue of A then . Applications for this eigenvalue can be found in the above areas. In this paper, the asymptotic behavior of the Perron eigenvalue is studied. In particular, if A is an n×n ml-matrix and an n× nonnegative matrix, bounds are found for . Further, the asymptotic behavior of approaches infinity, is studied.

Stability and Eigenvalue Monotonicity of Linear Systems

Abstract: The matrices that occur most frequently in system modeling share with their submatrices, referred to as principal majors, a number of restrictive properties. In modeling, this situation arises naturally. Often, important properties of the system must be retained when the degrees of freedom are reduced.