Circulant and skew-circulant preconditioners for skew-hermitian type Toeplitz systems |
| |
Authors: | Raymond H Chan Xiao-Qing Jin |
| |
Institution: | (1) Department of Mathematics, University of Hong Kong, Hong Kong |
| |
Abstract: | We study the solutions of Toeplitz systemsA
n
x=b by the preconditioned conjugate gradient method. Then ×n matrixA
n
is of the forma
0
I+H
n
wherea
0 is a real number,I is the identity matrix andH
n
is a skew-Hermitian Toeplitz matrix. Such matrices often appear in solving discretized hyperbolic differential equations. The preconditioners we considered here are the circulant matrixC
n
and the skew-circulant matrixS
n
whereA
n
=1/2(C
n
+S
n
). The convergence rate of the iterative method depends on the distribution of the singular values of the matricesC
–1
n
An andS
–1
n
A
n
. For Toeplitz matricesA
n
with entries which are Fourier coefficients of functions in the Wiener class, we show the invertibility ofC
n
andS
n
and prove that the singular values ofC
–1
n
A
n
andS
–1
n
A
n
are clustered around 1 for largen. Hence, if the conjugate gradient method is applied to solve the preconditioned systems, we expect fast convergence. |
| |
Keywords: | 65F10 65F15 |
本文献已被 SpringerLink 等数据库收录! |
|