摘要:

AbstractWe consider the problem of reconstructing Jacobi matrices and real symmetric arrow matrices from two eigenpairs. Algorithms for solving these inverse problems are presented. We show that there are reasonable conditions under which this reconstruction is always possible. Moreover, it is seen that in certain cases reconstruction can proceed with little or no cancellation. The algorithm is particularly elegant for the tridiagonal matrix associated with a bidiagonal singular value decomposition.

ISSN:1070-5325    

DOI:10.1002/nla.1680020302

出版商:John Wiley&Sons, Ltd    

年代:1995

数据来源: WILEY

摘要:

AbstractMany researchers are now working on computing the product of a matrix function and a vector, using approximations in a Krylov subspace. We review our results on the analysis of one implementation of that approach for symmetric matrices, which we call the spectral lanczos decomposition method (SLDM).We have proved a general convergence estimate, relating SLDM error bounds to those obtained through approximation of the matrix function by a part of its Chebyshev series. Thus, we arrived at effective estimates for matrix functions arising when solving parabolic, hyperbolic and elliptic partial differential equations. We concentrate on the parabolic case, where we obtain estimates that indicate superconvergence of SLDM. For this case a combination of SLDM and splitting methods is also considered and some numerical results are presented.We implement our general estimates to obtain convergence bounds of Lanczos approximations to eigenvalues in the internal part of the spectrum. Unlike Kaniel‐Saad estimates, our estimates are independent of the set of eigenvalues between the required one and the nearest spectrum bound.We consider an extension of our general estimate to the case of the simple Lanczos method (without reorthogonalization) in finite computer arithmetic which shows that for a moderate dimension of the Krylov subspace the results, proved for the exact arithmetic, are stable up to roundof

ISSN:1070-5325    

DOI:10.1002/nla.1680020303

出版商:John Wiley&Sons, Ltd    

年代:1995

数据来源: WILEY

摘要:

AbstractWith the growing demands from disciplinary and interdisciplinary fields of science and engineering for the numerical solution of the nonsymmetric eigenvalue problem, competitive new techniques have been developed for solving the problem. In this paper we examine the state of the art of the algorithmic techniques and the software scene for the problem. Some current developments are also outlined.

ISSN:1070-5325    

DOI:10.1002/nla.1680020304

出版商:John Wiley&Sons, Ltd    

年代:1995

数据来源: WILEY

摘要:

AbstractAn optimization method is developed based on ellipsoidal trust regions that are defined by conic functions. It provides a powerful unifying theory from which can be derived a variety of interesting and potentially useful optimization algorithms, in particular, conjugate‐gradient‐like algorithms for nonlinear minimization and Karmarkar‐like interior‐point algorithms for linear prog

ISSN:1070-5325    

DOI:10.1002/nla.1680020305

出版商:John Wiley&Sons, Ltd    

年代:1995

数据来源: WILEY

摘要:

AbstractWe show that a certain matrix norm ratio studied by Parlett has a supermum that isO(\documentclass{article}\pagestyle{empty}\begin{document}$\mathop \[\sqrt n \] $\end{document}) when the chosen norm is the Frobenius norm, while it isO(logn) for the 2‐norm. This ratio arises in Parlett's analysis of the Cholesky decomposition of annbynmatri

ISSN:1070-5325    

DOI:10.1002/nla.1680020306

出版商:John Wiley&Sons, Ltd    

年代:1995

数据来源: WILEY

摘要:

AbstractThe Rayleigh quotient iteration method finds an eigenvector and the corresponding eigenvalue of a symmetric matrix. This is a fundamental problem in science and engineering. Parlett and Kahan have shown, in 1968, that for almost any initial vector in the unit sphere, the Rayleigh quotient iteration method converges to some eigenvector. In this paper, the regions of the unit sphere which include all possible initial vectors converging to a specific eigenvector are studied. The generalized eigenvalue problemAx= λBxis considered. It is shown that the regions do not change when the matrix is shifted or multiplied by a scalar. These regions are completely characterized in the three‐dimensional case. It is shown that, in this case, the area of the region of convergence corresponding to the interior eigenvalue is larger than the area of those corresponding to any extreme one. This can be interpreted, with the appropriate choice of probability distribution, as: the probability of converging to an eigenvector corresponding to the interior eigenvalue is larger than the probability of converging to an eigenvector corresponding to any extreme eigenvalue. It is conjectured that the same is true for matrices of any order. Experiments in higher dimensions are presented which conform with the conjectu

ISSN:1070-5325    

DOI:10.1002/nla.1680020307

出版商:John Wiley&Sons, Ltd    

年代:1995

数据来源: WILEY

摘要:

AbstractLetAbe anm×nmatrix,bbe anm‐vector, and x̃ be a purported solution to the problem of minimizing ‖b—Ax‖2. We consider the following open problem: find the smallest perturbationEofAsuch that the vector x̃ exactly minimizes ‖b— (A+E)x‖2. This problem is completely solved whenEis measured in the Frobenius norm. When using the spectral norm ofE, upper and lower bounds are given, and the optimum is found under ce

ISSN:1070-5325    

DOI:10.1002/nla.1680020308

出版商:John Wiley&Sons, Ltd    

年代:1995

数据来源: WILEY

摘要:

AbstractThis paper describes a divide‐and‐conquer strategy for solving block Hessenberg systems. For dense matrices the method is as efficient as Gaussian elimination; however, because it works almost entirely with the original blocks, it is much more efficient for sparse matrices or matrices whose blocks can be generated on the fly. For Toeplitz matrices, the algorithm can be combined with the fast Fourier transf

ISSN:1070-5325    

DOI:10.1002/nla.1680020309

出版商:John Wiley&Sons, Ltd    

年代:1995

数据来源: WILEY

摘要:

AbstractFor many years techniques from numerical analysis have been applied fruitfully to the study of dynamical systems. In this paper it is shown that the theory of dynamical systems may be applied to certain computational problems. In particular the question of global convergence of various QR algorithms can be reduced to the study of certain vector iterations derived from Schur forms of matrices. The technique is illustrated in determining the convergence behavior of normal Hessenberg matrices under the Francis and multishift QR iterations.

ISSN:1070-5325    

DOI:10.1002/nla.1680020310

出版商:John Wiley&Sons, Ltd    

年代:1995

数据来源: WILEY

ISSN:1070-5325    

DOI:10.1002/nla.1680020311

出版商:John Wiley&Sons, Ltd    

年代:1995

数据来源: WILEY