Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1979"
Book•
01 Jan 1979
TL;DR: A submitted manuscript is the author's version of the article upon submission and before peer-review as discussed by the authors, and the final published version features the final layout of the paper including the volume, issue and page numbers.
Abstract: • A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.
177 citations
TL;DR: In this paper, an initial value method based on the use of certain compound matrices was presented for the treatment of eigenvalue problems for stiff ordinary differential equations, which appeared to overcome not only the parasitic growth problems associated with standard shooting methods but also the difficulties, due to the presence of singularities, of the Riccati method.
120 citations
01 Jan 1979
TL;DR: In this article, the Sperner Lemma proof of the Brouwer fixed point theorem has been used as a simplicial approximation of the map as is used in the simplicial methods developed by H. Scarf, B. C. Eaves, R. Saigal and others.
Abstract: Newton's method has probably long been the most frequently used method for solving systems of nonlinear equations but it is most useful when an approximate solution is available. Then it will generally provide a sequence of approximations which converge rapidly to a solution. Newton's method can also be tried by picking an "approximate" solution at random, but the approximate solution is likely to lie outside the domain of convergence. In comparison, numerous topological proofs proclaim the existence of solutions in situations in which local procedures, like Newton's method or the method of steepest descent, are of less value since topological methods by their nature have not provided us with good approximations of the solutions. In the past few years a number of surprising advances have been made in finding numerical procedures for solving such nonlinear systems of equations. Two closely related methods have become available. The simplicial methods developed by H. Scarf, B. C. Eaves, R. Saigal and others [1-7] was employed initially for finding the Brouwer fixed point. Their method is in essence based on using a simplicial approximation of the map as is used in the Sperner Lemma proof of the Brouwer Fixed Point Theorem. Kellogg, Li and Yorke developed an alternative approach [8,9] for the numerical solution of
114 citations
TL;DR: In this article, the homogenization of elliptic eigenvalue problems is studied, with a second order homogeneous Dirichlet problem as an example, where the structure of eigenvalues and eigenvectors is preserved.
Abstract: The aim of this paper is to study the homogenization of elliptic eigenvalue problems, with a second order homogeneous Dirichlet problem as an example. The main homogenization theorem states that the same operator which serves to homogenize the corresponding static problem works for the eigenvalue problem as well and that the structure of eigenvalues and eigenvectors is in some sense preserved. Formulae for first and second order correctors for eigenvalues are proposed and error estimates are obtained. These results are applied to the case of coefficients with a periodic structure and a simple numerical example is presented. Extensions to other types of boundary conditions and to higher order equations are indicated.
112 citations
TL;DR: In this paper, the eigenvalue problem Ax = λBx is shown to have a complete system of eigenvectors and that its eigenvalues are real.
98 citations
94 citations
TL;DR: In this paper, it was shown that the Rayleigh quotient can often give a good approximation to the dominant eigenvalue after a very few iterations, even when the order of the matrix is large.
Abstract: The power method for computing the dominant eigenvector of a positive definite matrix will converge slowly when the dominant eigenvalue is poorly separated from the next largest eigenvalue. In this note it is shown that in spite of this slow convergence, the Rayleigh quotient will often give a good approximation to the domi- nant eigenvalue after a very few iterations-even when the order of the matrix is large.
43 citations
TL;DR: In this article, the problem of finding scalars λ and vectors x = 0 for which Ax = λx has been studied, and the form of the matrix affects the way in which the scalars are found.
Abstract: Find scalars λ and vectors x = 0 for which Ax = λx The form of the matrix affects the way in which we solve this problem, and we also have variety as to what is to be found. • A symmetric and real (or Hermitian and complex). This is the most common case. In some cases we want only the eigenvalues (and perhaps only some of them); and in other cases, we also want the eigenvectors. There are special classes of such A, e.g. banded, positive definite, sparse, and others. • A non-symmetric, but with a diagonal Jordan canonical form. This means there is a nonsin-gular matrix P for which P −1 AP = D = diag[λ 1 , ..., λ n ]
26 citations
TL;DR: In this paper, the spectrum of the multigroup neutron transport operator A is studied for bounded spatial regions D which consist of a finite number of material subregions, and the authors provide simple conditions on the material cross sections which guarantee that A possesses eigenvalues in the finite plane.
Abstract: The spectrum of the multigroup neutron transport operator A is studied for bounded spatial regions D which consist of a finite number of material subregions. Our main results provide simple conditions on the material cross sections which guarantee that (1) A possesses eigenvalues in the finite plane; (2) A possesses a ’’leading’’ eigenvalue λ0 which is real, not less than the real part of any other eigenvalue, and to which there corresponds at least one nonnegative eigenfunction ψλ0; and (3) A possesses a ’’dominant’’ eigenvalue λ0 which is real, simple, greater than the real part of any other eigenvalue, and whose eigenfunction ψλ0 satisfies ψλ0⩾0 and ∫ψλ0d2Ω≳0. We give examples to illustrate the results and to show that a leading eigenvalue need not be simple, nor its eigenfunction(s) positive.
23 citations
TL;DR: In this article, it was shown that linear convergence to a local maximum of a certain potential is possible for a large class of such problems and a simple version of such algorithms that can be obtained.
Abstract: In recent years, a group of inverse iteration type algorithms have been developed for solving nonlinear elliptic eigenvalue problems in plasma physics [4]. Although these algorithms have been very successful in practice, no satisfactory theoretical justification of convergence has been available. The present paper fills this gap and proves for a large class of such problems and a simple version of such algorithms that linear convergence to a local maximum of a certain potential is obtained.
22 citations
01 Dec 1979
TL;DR: In this article, a solution to implicit model following using the theory of feedforward matrices is presented, which is reformulated and shown to be equivalent to eigenvalue eigenvector placement.
Abstract: In implicit model following one attempts to change the output dynamics of the plant using feed-back so as to equal the output dynamics of a desirable model. This paper presents a solution to implicit model following using the theory of feedforward matrices. The solution is reformulated and shown to be equivalent to eigenvalue eigenvector placement.
TL;DR: In this article, boundary value and eigenvalue problems of the Emden-Fowler equation (t α u′)′ + λt β ⨍(u) = 0, √ u = u γ and eu = u ǫ) were studied using the simple one parameter group properties.
Abstract: Two pragmatic boundary value and eigenvalue problems of the Emden-Fowler equation (t α u′)′ + λt β ⨍(u) = 0,⨍(u) = u γ and eu are studied using the simple one parameter group properties In all cases boundary value problems are converted into initial value problems using the property of the invariance group With ⨍(u) = u γ an eigenvalue problem is detailed and calculations presented
TL;DR: In this paper, the self-adjoint extensions of −d2/dx2−λx−n,n≧2, on [0, ∞] are described in terms of an inhomogeneous boundary value problem.
Abstract: The self-adjoint extensions of −d2/dx2−λx−n,n≧2, on [0, ∞) are described in terms of an inhomogeneous boundary value problem. The eigenvalue equation may be obtained in terms of the connection coefficients, connecting the solutions at zero and infinity.
TL;DR: In this paper, it was shown that the analysis of Morse of a non-Hermitian eigenvalue problem, connected with a string with non-rigid supports, leads to an erroneous expansion of the Green function into a set of related eigenfunctions.
Abstract: It is shown that the analysis of Morse of a non-Hermitian eigenvalue problem, connected with a string with non-rigid supports, leads to an erroneous expansion of the Green function into a set of related eigenfunctions. The correct expansion is derived.
TL;DR: In this article, the authors extended the work of J. Walter on regular eigenvalue problems with eigen value parameter in the boundary condition to the multiparameter setting and obtained completeness and expansion theorems.
TL;DR: The main purpose of as discussed by the authors is to find the eigenvalue spectrum of matrices appearing when numerical methods are used to solve the second and third boundary value problems for Laplace, Poisson, and Helmholtz equations.
01 Jan 1979
TL;DR: In this article, the best possible bounds for changes in the eigenvalues of a class of relative eigenvalue problems in finite-dimensional spaces are given. But these bounds are in terms of quadratic forms and are not applicable to some aspects of eigen value problems in infinite-dimensional space.
Abstract: This article gives some new bounds for changes in the eigenvalues of a class of relative eigenvalue problems in finite-dimensional spaces. These bounds are “best possible” in a certain sense, and improve those given recently by G. W. Stewart. Our formulation is in terms of quadratic forms and is applicable to some aspects of eigenvalue problems in infinite-dimensional spaces.
01 Jan 1979
TL;DR: In this paper, the convergence of an inverse iteration type algorithm for a certain class of nonlinear elliptic eigenvalue problems was discussed, but no satisfactory theoretical justification of convergence was known.
Abstract: In a recent paper [7] the convergence of an inverse iteration type algorithm for a certain class of nonlinear elliptic eigenvalue problems was discussed. Such algorithms have been used successfully in plasma physics [11], but no satisfactory theoretical justification of convergence was known. While in [7] only the nondiscretized case was discussed, here an analogous algorithm for nonlinear eigenvalue problems in ℝN will be treated. This algorithm is interesting in itself, but can also be interpreted as a suitably discretized version of the algorithm discussed in [7].
TL;DR: The theory of singular left-definite canonical eigenvalue problems treated by Nieβen and Schneider in this paper was generalized to arbitrary λ ∈ (ℂ\ℝ∪{0) and a natural analogue of the Titchmarsh-Kodaira formula was proved.
Abstract: The theory of singular left-definite canonical eigenvalue problems treated by Nieβen and Schneider in is generalized to arbitrary λ∈(ℂ\ℝ∪{0}. In this enlarged theory the Green's matrix of the problem is evaluated and a natural analogue of the Titchmarsh-Kodaira formula is proved. This formula permits the explicit computation of the spectral matrix playing the main role in the expansion theorems of this theory.
TL;DR: In this paper, the convergence of multi-step methods for solving eigenvalue problems for ordinary second-order differential equations is studied and convergence of both the eigenvalues and eigenfunctions is considered.
Abstract: METHODS of functional analysis are used to prove the convergence of multi-step methods for solving eigenvalue problems for ordinary second-order differential equations. The convergence of both the eigenvalues and the eigenfunctions is considered.
01 Jan 1979
TL;DR: In this article, it is shown how the linear system for a two-point boundary value problem can be reformulated in terms of associated nonlinear initial value systems, where the new dependent variables introduced can be interpreted as the unknown boundary conditions at one end of the interval.
Abstract: Analytical mathematical techniques abound for two-point boundary value problems which exploit the inherent advantages of linear systems. The success of the analytic approach has to a large extent been responsible for carrying these methods over into computational techniques, where linearity may no longer be an advantage. If large scale computer calculations are contemplated, there in fact may be computational advantages in reformulating a linear problem into a nonlinear counterpart. The aim of this paper is to illustrate just such a circumstance in the context of linear differential eigenvalue problems. By invoking matrix Riccati transformations, it is demonstrated how the linear system for a two point boundary value problem can be reformulated in terms of associated nonlinear initial value systems. The new dependent variables introduced can be interpreted as the unknown boundary conditions at one end of the interval. A straightforward algorithm can be used to solve directly for the eigenvalue of the original system. The algorithm will be outlined in the context of a simple beam problem. A parallel treatment of a well known difficult linear complex eigenvalue problem arising in the study of the stability of plane Poiseuille flow is then discussed. The method proves to be particularly well suited to this problem and reveals distinct advantages over traditional methods of solution in terms of (i) complexity and formulation, (ii) numerical analytic sophistication, and (iii) demands on computer time and storage.
01 Jan 1979
TL;DR: In this paper, the generalized eigenvalue problem x - Kx = μBx in a complex Banach space was considered, and the convergence of sequences of eigenvalues and eigenelements of the approximate eigen value problem to eigen values and eigenenelements was established.
Abstract: We consider the generalized eigenvalue problem x - Kx = μBx in a complex Banach space E . Here, K and B are bounded linear operators, B is compact, and 1 is not in the spectrum of K . If { E n : n = 1, 2,…} is a sequence of closed subspaces of E and P n : E → E n is a linear projection which maps E onto E n , then we consider the sequence of approximate eigenvalue problems { x n - P n Kx n = μP n Bx n in E n : n = 1, 2,…}. Assuming that ∥ K - P n K ∥ → 0 and t | B - P n B ∥ → 0 as n → ∞, we prove the convergence of sequences of eigenvalues and eigenelements of the approximate eigenvalue problem to eigenvalues and eigenelements of the original eigenvalue problem, and establish upper bounds for the errors. These error bounds are sharper than those given by Vainikko in Ref. 2 for the more general problem x = μTx in E , T linear and compact, and the sequence of approximate problems { x n = μT n x n in E n : n = 1, 2,…}, and do not involve the operator S n = T n - P n T ∥; E n .