Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1974"
104 citations
45 citations
TL;DR: The higher eigenvalues can be computed successively by minimizing the Rayleigh quotient of a modified eigenvalue problem based on a deflation process and some important computational advantages with regard to the amount of work are found.
36 citations
TL;DR: In this paper, a general technique for solving a large class of well-posed inverse eigenvalue problems is illustrated by means of a specific example, which consists in reconstructing the coefficients p(x and q(x) associated with the eigen value problem d4)- (pu’)’ +qu = Lu from a knowledge of three spectra.
Abstract: Summary A general technique for solving a large class of well-posed inverse eigenvalue problems is illustrated by means of a specific example. The example considered consists in reconstructing the coefficients p(x) and q(x) associated with the eigenvalue problem d4)- (pu’)’ +qu = Lu from a knowledge of three spectra.
34 citations
TL;DR: In this paper, a gradient method for the minimization of the functional F(x)=?Ax?(?Ax, Bx?/?Bx, bx?)Bx?2 is developed.
Abstract: LetA andB ben×m matrices. A gradient method for the minimization of the functionalF(x)=?Ax?(?Ax, Bx?/?Bx, Bx?)Bx?2 is developed. The minima ofF are the eigenvectors of the eigenproblemAx=?Bx. The concept of a non-defective eigenvalue for this generalized eigenvalue problem is developed. It is then shown that geometric convergence is attained for non-defective eigenvalues. A convergence rate analysis is given where it is shown that the rapidity of convergence of the gradient method to an eigenvalue ? depends on the degree of non-defectiveness of ? and the singular values ofA??B.
31 citations
TL;DR: An iterative algorithm for computing the smallest eigenvalue and its corresponding eigenvector, based on the successive overrelaxation splitting of the matrices, is developed, and its global convergence is proved.
Abstract: The eigenvalue problem Ax = XBx, where A and B are large and sparse symmetric matrices, is considered. An iterative algorithm for computing the smallest eigenvalue and its corresponding eigenvector, based on the successive overrelaxation splitting of the matrices, is developed, and its global convergence is proved. An ex- pression for the optimal overrelaxation factor is found in the case where A and B are two-cyclic (property A). Further, it is shown that this SOR algorithm is the first order approximation to the coordinate relaxation algorithm, which implies that the same overrelaxation can be applied to this latter algorithm. Several numerical tests are reported. It is found that the SOR method is more effective than coordinate relaxation. If the separation of the eigenvalues is not too bad, the SOR algorithm has a fast rate of convergence, while, for problems with more severe clustering, the c-g or Lanczos algorithms should be preferred.
30 citations
01 Jan 1974
TL;DR: Several iterative eigenvalue algorithms applicable to large sparse symmetric matrices, of so high an order that the matrix A cannot be stored in the memory of the computer, while it is easy to compute y = Ax for a given vector x.
Abstract: We compare several iterative eigenvalue algorithms applicable to large sparse symmetric matrices, of so high an order that the matrix A cannot be stored in the memory of the computer, while it is easy to compute y = Ax for a given vector x.
20 citations
TL;DR: In this paper, five algorithms for the reduction of the molecular orbital equations to the standard eigenvalue form are compared. But the results of these algorithms are limited to two, three and four functions approaching coalescence, and it is shown that the elements of the density matrix corresponding to the coalescing functions are almost entirely determined by the coalescence.
17 citations
Proceedings Article•
01 Jan 1974
TL;DR: This paper surveys techniques for calculating eigenvalues and eigenvectors of very large matrices of verylarge matrices.
Abstract: This paper surveys techniques for calculating eigenvalues and eigenvectors of very large matrices.
13 citations
TL;DR: In this article, the feedback eigenvalue control problem of distributed-parameter systems subject to boundary inputs is solved and results are derived for both single-eigenvalue and multi-Eigenvalue assignment.
Abstract: The feedback eigenvalue control problem of distributed-parameter systems subject to boundary inputs is solved and results are derived for both single-eigenvalue and multi-eigenvalue assignment. An illustrative example is included.
8 citations
01 Jan 1974
TL;DR: This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library.
Abstract: Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library
TL;DR: In this article, an expression for the upper and lower bounds of the largest eigenvalue of the one-velocity transport equation in terms of the Rayleigh quotient was derived using the variational method.
Abstract: Making use of the variational method, an expression is derived for the upper and lower bounds of the largest eigenvalue of the one-velocity transport equation, in terms of the Rayleigh quotient. It is found that the error contained in the eigenvalue thus obtained increases or decreases in keeping with the error inherent in the trial function used for expressing the neutron flux distribution. The first approximation for the eigenvalue and the extrapolation distances of finite cylindrical reactors are determined by using the asymptotic flux shape as trial function. The second and third approximations for the eigenvalue are derived by supplementing the asymptotic function with additional orthogonal terms. It is proposed to combine the eigenvalue determined by the variational method with the second approximation of the flux obtained by applying the integral transport operator to the asymptotic flux. Evidence is presented to prove the convergence of an iterative procedure devised for successively applying the ...
TL;DR: In this paper, a centered finite-difference approximation to the non-selfadjoint Sturm-Liouville eigenvalue problem is proposed, where the eigenvectors Wp of the M X M-matrix (Ax = 1 /(M + 1) mesh size) are bounded in the maximum norm independent of M if they are normalized so that 1WpI2 = 1.
Abstract: This paper is concerned with a centered finite-difference approximation to to the nonselfadjoint Sturm-Liouville eigenvalue problem Ltu] =[a(x)ux]x b(x)ux + c(x)u = Xu, 0 < x < 1, u(O) = u(i) = 0. It is shown that the eigenvectors Wp of the M X M-matrix (Ax = 1 /(M + 1) mesh size), which approximates L, are bounded in the maximum norm independent of M if they are normalized so that 1WpI2 = 1.
Book•
01 Jan 1974
TL;DR: In this paper, it was shown that for a general class of linear eigenvalue problems, the error in these approximations is at least O(h(II)]d+1) where h(II) is the maximal subinterval length.
Abstract: Approximations to the eigenvalues ofmth-order linear eigenvalue problems are determined by using a collocation method with piecewise-polynomial functions of degreem+d possessingm continuous derivatives as basis functions. It is shown that for a general class of problems the error in these approximations is at leastO([h(II)]
d+1) whereh(II) is the maximal subinterval length. The question of stability of the discretized problems is also considered. It is not assumed that the eigenvalue problems are in any sense self-adjoint.
TL;DR: In this article, the authors derived semi-analytical solutions of the Schrodinger equation for a particle moving in the electrostatic field of two other particles a fixed distance apart, such that the resulting matrix eigenvalue equations contain real symmetric band matrices.
Abstract: Semi-analytical solutions of the Schrodinger equation for a particle moving in the electrostatic field of two other particles a fixed distance apart, are derived in such a way that the resulting matrix eigenvalue equations contain real symmetric band matrices. Numerical techniques appropriate to the solution of the two simultaneous matrix eigenvalue equations are described; in particular the bisection method is used to determine precisely the significant truncation order of the matrices for a given numerical precision.
TL;DR: In this article, an eigenvalue problem in the Hilbert space with nonlinearities in the eigen value parameter was constructed, and the parameters of which were used to approximate those of the original problem.
Abstract: Given an eigenvalue problem in the Hilbert spaceL 2[a,b] with nonlinearities in the eigenvalue parameter. We construct an eigenvalue problem in ?n, the eigenvalues of which are used to approximate those of the original problem. With additional conditions we give bounds for the difference in the case when the eigenvalue parameter appears in quadratic form.
TL;DR: In this article, the upper bound of the spectrum of a quasilinear eigenvalue problem subject to a positivity requirement is derived, as a function of the surface heat-transfer coefficient.
TL;DR: In this paper, simple optimality conditions are derived which must be satisfied by the eigenvalue spectra of single-input closed-loop linear control systems, which is the same as the optimality condition in this paper.
Abstract: Simple optimality conditions are derived which must be satisfied by the eigenvalue spectra of single-input closed-loop linear control systems.
TL;DR: Characteristic polynomials are generated for a class of eigenvalue problems using the singularity condition previously encountered in the solution of corresponding forced system using the Leverrier-Faddeev method.
Abstract: Characteristic polynomials are here generated for a class of eigenvalue problems using the singularity condition previously encountered in the solution of corresponding forced system. The process is efficient and requires n2/2 operations in contrast to n3 operations required by techniques such as the Leverrier-Faddeev method.
01 Jan 1974
Dissertation•
01 Jan 1974
01 Jul 1974
TL;DR: An iterative method for the solution of the algebraic eigenvalue problem for Hermitian matrices is described in this paper, where a microfilm copy of the original document is used.
Abstract: An iterative method for the solution of the algebraic eigenvalue problem for Hermitian matrices " (1974). INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1.The sign or "target" for pagss apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper ICI I iiaiiu uuiiici ui a laiyc diicci aiiu lu ouiiuiiuc piiuiuiiiu iiuiii leii lu right in equal sections with a small overlap. If necessary, sectioning is continued again-beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver rvto\/ rwe\areit\ o* orl#4t^înnol WF 1^1 I WKWGI (V • • • MF V W & WWW »W# WFTVII^W WY #*# I LI# the Order Department, giving the catalog number, title, author and specific pages you wish reproduced.
01 Dec 1974
TL;DR: Backward error analysis of the Householder transformation for the generalized eigenvalue problem is presented in this paper, and bounds for the norm of the resulting perturbation matrices are compared with existing bounds when known.
Abstract: Backward error analyses of the application of Householder transformations to both the standard and the generalized eigenvalue problems are presented. The analysis for the standard eigenvalue problem determines the error from the application of an exact similarity transformation, and the analysis for the generalized eigenvalue problem determines the error from the application of an exact equivalence transformation. Bounds for the norms of the resulting perturbation matrices are presented and compared with existing bounds when known.