7 research outputs found
Convergence Analysis of Extended LOBPCG for Computing Extreme Eigenvalues
This paper is concerned with the convergence analysis of an extended
variation of the locally optimal preconditioned conjugate gradient method
(LOBPCG) for the extreme eigenvalue of a Hermitian matrix polynomial which
admits some extended form of Rayleigh quotient. This work is a generalization
of the analysis by Ovtchinnikov (SIAM J. Numer. Anal., 46(5):2567-2592, 2008).
As instances, the algorithms for definite matrix pairs and hyperbolic quadratic
matrix polynomials are shown to be globally convergent and to have an
asymptotically local convergence rate. Also, numerical examples are given to
illustrate the convergence.Comment: 21 pages, 2 figure
GMRES ON A TRIDIAGONAL TOEPLITZ LINEAR SYSTEM
The Generalized Minimal Residual method (GMRES) is often used to solve a nonsymmetric linear system Ax = b. But its convergence analysis is a rather difficult task in general. A commonly used approach is to diagonalize A = XΛX-1 and then separate the study of GMRES convergence behavior into optimizing the condition number of X and a polynomial minimization problem over As spectrum. This artificial separation could greatly overestimate GMRES residuals and likely yields error bounds that are too far from the actual ones. On the other hand, considering the effects of both As spectrum and the conditioning of X at the same time poses a difficult challenge, perhaps impossible to deal with in general but only possible for certain particular linear systems. This thesis will do so for a (nonsymmetric) tridiagonal Toeplitz system. Sharp error bounds on and sometimes exact expressions for residuals are obtained. These expressions and/or bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first kind or the second kind