The solution of ill-conditioned symmetric toeplitz systems via two-grid and wavelet methods

Abstract

In this paper, we present a generalized multigrid method combined with wavelet filters for solving ill-conditioned symmetric Toeplitz systems T n x = b, where T n ϵ R n× n is generated by nonnegative functions with zeros. First, we propose the construction of general Cohen, Daubechies, and Feauveau (CDF) 9/7 biorthogonal wavelet systems, so that a new class of compactly supported biorthogonal wavelet systems GCDF are achieved with specified vanishing moments for scaling functions. In order to solve ill-conditioned Toeplitz systems by using the two-grid method (TGM), we use the constructed GCDF wavelets to get prolongation and restriction operators. As a result, the proposed TGM is proved by numerical experiment to be more efficient than the classic TGM, especially when T n is seriously ill-conditioned. For the vanishing moments being N, experimental tests illustrate that the TGM with damped-Jacobi smoother converges when the generating function has zeros of order less than or equal to 2N ( N ≤ 8). Besides, we prove in theory that the proposed method converges for Toeplitz systems that are generated by functions with zeros of order less than or equal to four.