For the augmented system of linear equations, Golub, Wu and Yuan recently studied an SOR-like method (BIT 41(2001)71-85). By further accelerating it with another parameter, in this paper we present a generalized SOR (GSOR) method for the augmented linear system. We prove its convergence under suitable restrictions on the iteration parameters, and determine its optimal iteration parameters and the corresponding optimal convergence factor. Theoretical analyses show that the GSOR method has faster asymptotic convergence rate than the SOR-like method. Also numerical results show that the GSOR method is more effective than the SOR-like method when they are applied to solve the augmented linear system. This GSOR method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration. Besides, we also demonstrate a complete version about the convergence theory of the SOR-like method.
Approximations to the solution of a large sparse symmetric system of equations are considered. The conjugate gradient and minimum residual approximations are studied without reference to their computation. Several different bases for the associated Krylov subspace are used, including the usual Lannos basis. The zeros of the iteration polynomial for the minimum residual approximation (harmonic Ritz values) are characterized in several ways and, in addition, attractive convergence properties are established. The connection of these harmonic Ritz values to Lehmann's optimal intervals for eigenvalues of the original matrix appears to be new.
Abstract. The Rayleigh Quotient Iteration (RQI) was developed for real symmetric matrices. Its rapid local convergence is due to the stationarity of the Rayleigh Quotient at an eigenvector. Its excellent global properties are due to the monotonie decrease in the norms of the residuals. These facts are established for normal matrices. Both properties fail for nonnormal matrices and no generalization of the iteration has recaptured both of them. We examine methods which employ either one or the other of them.1. History. In the 1870's John William Strutt (third Baron Rayleigh) began his mammoth work on the theory of sound. Basic to many of his studies were the small oscillations of a vibrating system about a position of equilibrium.In terms of suitable generalized coordinates, any set of small displacements (the state of the system) will be represented by a vector q with time derivative q. The potential energy V and the kinetic energy T at each instant t are represented by the two quadratic forms V = i (Mq, q) = \q*Mq, T = £ {Hq, q) = \q*Kq, where M and H are suitable symmetric positive definite matrices, constant in time, and q* denotes the transpose of q.Of principal interest are the smallest natural frequencies w and their corresponding natural modes which can be expressed in the generalized coordinates in the form exp(i«r)x where x is a constant vector (the mode shape) which satisfiesAmong other things, Lord Rayleigh showed that the frequency can be written in terms of the mode shape
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