Circulant Preconditioners for Hermitian Toeplitz Systems
نویسنده
چکیده
We study the solutions of Hermitian positive deenite Toeplitz systems Ax = b by the preconditioned conjugate gradient method for three families of circulant preconditioners C. The convergence rates of these iterative methods depend on the spectrum of C ?1 A. For a Toeplitz matrix A with entries which are Fourier coeecients of a positive function f in the Wiener class, we establish the invertiblity of C, and that the spectrum of the preconditioned matrix C ?1 A clusters around one. We prove that if f is (l + 1)-times diierentiable, with l > 0, then the error after 2q conjugate gradient steps will decrease like ((q ? 1)!) ?2l. We also show that if C copies the central diagonals of A, then C minimizes jjC ? Ajj 1 and jjC ? Ajj 1 .
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 کاملCirculant/Skewcirculant Matrices as Preconditioners for Hermitian Toeplitz Systems
We study the solutions of Hermitian positive definite Toeplitz systems Tnx = b by the preconditioned conjugate gradient method. For preconditioner An the convergence rate is known to be governed by the distribution of the eigenvalues of the preconditioned matrix A−1 n Tn . New properties of the circulant preconditioners introduced by Strang, R. Chan, T. Chan, Szegö/Grenander and Tyrtyshnikov ar...
متن کاملGeneralized circulant Strang-type preconditioners
SUMMARY Strang's proposal to use a circulant preconditioner for linear systems of equations with a Hermitian positive definite Toeplitz matrix has given rise to considerable research on circulant preconditioners. This paper presents an {e iϕ }-circulant Strang-type preconditioner.
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 کاملThe Best Circulant Preconditioners for Hermitian Toeplitz Systems
In this paper, we propose a new family of circulant preconditioners for ill-conditioned Hermitian Toeplitz systems Ax = b. The preconditioners are constructed by con-volving the generating function f of A with the generalized Jackson kernels. For an n-by-n Toeplitz matrix A, the construction of the preconditioners only requires the entries of A and does not require the explicit knowledge of f. ...
متن کاملCirculant Preconditioners for Ill-Conditioned Hermitian Toeplitz Matrices
In this paper, we propose a new family of circulant preconditioners for solving ill-conditioned Hermitian Toeplitz systems Ax = b. The eigenval-ues of the preconditioners are given by the convolution products of the generating function f of A with some summation kernels. When f is a nonnegative 2-periodic continuous function deened on ?; ] with a zero of order 2p, we show that the circulant pre...
متن کامل