A practical algorithm to design fast and optimal band-Toeplitz preconditioners for hermitian toeplitz systems

Abstract

In this paper we are concerned with the iterative solution ofn×n Hermitian Toeplitz systems by means of preconditioned conjugate gradient (PCG) methods. In many applications [9] such as signal processing [24], differential equations [39], linear prediction of stationary processes [18], the related Toeplitz systems have the formAn(f)x=b where the symbolf, the generating function, is anL1 function and the entries ofAn(f) along thek-th diagnonal coincide with thek-th Fourier coefficient off. When the essential range of the generating function has a convex hull containing zero, the matricesAn(f) are asymptotically ill-conditioned [21, 33, 28] and circulant or Hartley preconditioners do not work [15]. For this difficult case the only optimal preconditioners in the sense of [3, 29] are found in the τ algebra [15, 35] and especially in the band Toeplitz matrix class [7, 16]. In particular the band Toeplitz preconditioning strategy has been shown to be the most flexible one since it allows one to treat the nonnegative case [7, 16, 11, 31], the nondefinite one [27, 30, 34, 26]. On the other hand, the main criticism to this approach is surely the assumption that we must know the position and the order of the zeros off: in some applicative fields this is a feasible assumption, in other applications it is merely a theoretical possibility. Therefore, we discuss an economical technique in order to discover the sign off, the position of the possible zeros of the generating function and to evaluate approximately the order of these zeros. Finally, we exhibit some numerical experiments which confirm the effectiveness of the proposed idea.