Circulant Preconditioners for Toeplitz Matrices with Piecewise Continuous Generating Functions Man-chung Yeung and Raymond
نویسندگان
چکیده
We consider the solution of «-by-« Toeplitz systems T„x = b by preconditioned conjugate gradient methods. The preconditioner Cn is the T. Chan circulant preconditioner, which is defined to be the circulant matrix that minimizes \\B„ T„\\f over all circulant matrices B„ . For Toeplitz matrices generated by positive In -periodic continuous functions, we have shown earlier that the spectrum of the preconditioned system C„ ' Tn is clustered around 1 and hence the convergence rate of the preconditioned system is superlinear. However, in this paper, we show that if instead the generating function is only piecewise continuous, then for all e sufficiently small, there are 0(log«) eigenvalues of Cñ ' T„ that lie outside the interval ( 1 — e, 1 + e ). In particular, the spectrum of Cñ ' Tn cannot be clustered around 1. Numerical examples are given to verify that the convergence rate of the method is no longer superlinear in general.
منابع مشابه
Circulant Preconditioners Constructed From Kernels
We consider circulant preconditioners for Hermitian Toeplitz systems from the view point of function theory. We show that some well-known circulant preconditioners can be derived from convoluting the generating function f of the Toeplitz matrix with famous kernels like the Dirichlet and the Fej er kernels. Several circulant precondition-ers are then constructed using this approach. Finally, we ...
متن کاملToeplitz-Circulant Preconditioners for Toeplitz Systems and their Applications to Queueing Networks with Batch Arrivals
The preconditioned conjugate gradient method is employed to solve Toeplitz systems T n x = b where the generating functions of the n-by-n Toeplitz matrices T n are functions with zeros. In this case, circulant preconditioners are known to give poor convergence, whereas band-Toeplitz preconditioners only ooer linear convergence and can only handle real-valued functions with zeros of even orders....
متن کاملThe best circulant preconditioners for Hermitian Toeplitz systems II: The multiple-zero case
In 10, 14], circulant-type preconditioners have been proposed for ill-conditioned Her-mitian Toeplitz systems that are generated by nonnegative continuous functions with a zero of even order. The proposed circulant preconditioners can be constructed without requiring explicit knowledge of the generating functions. It was shown that the spectra of the preconditioned matrices are uniformly bounde...
متن کاملPreconditioners for Nondeenite Hermitian Toeplitz Systems 1
This paper is concerned with the construction of circulant preconditioners for Toeplitz systems arising from a piecewise continuous generating function with sign changes. If the generating function is given, we prove that for any " > 0, only O(log N) eigenvalues of our preconditioned Toeplitz systems of size N N are not contained in ?1?"; ?1+"] 1?"; 1+"]. The result can be modiied for trigonome...
متن کاملFast Band-Toeplitz Preconditioners for Hermitian Toeplitz Systems
We consider the solutions of Hermitian Toeplitz systems where the Toeplitz matrices are generated by nonnegative functions f. The preconditioned conjugate gradient method with well-known circulant preconditioners fails in the case when f has zeros. In this paper, we employ Toeplitz matrices of xed band-width as preconditioners. Their generating functions g are trigonometric poly-nomials of xed ...
متن کامل