On the Spectrum of a Family of Preconditioned Block Toeplitz Matrices

Abstract

Research on preconditioning Toeplitz matrices with circulant matrices has been active recently. The preconditioning technique can be easily generalized to block Toeplitz matrices.That is, for a block Toeplitz matrix T consisting of N N blocks with M M elements per block, a block circulant matrix R is used with the same block structure as its preconditioner. In this research, the spectral clustering property of the preconditioned matrix R-1T with T generated by two-dimensional rational functions T(z,,zy) of order (p:r,q:,pu,qv) is examined. It is shown that the eigenvalues of R-1T are clustered around unity except at most O(M/u + N"/) outliers, where max(p, q) and max(p, qy). Furthermore, if T is separable, the outliers are clustered together such that R-1T has at most (2/x + 1)(2 + 1) asymptotic distinct eigenvalues. The superior convergence behavior of the preconditioned conjugate gradient (PCG) method over the conjugate gradient (CG) method is explained by a smaller condition number and a better clustering property of the spectrum of the preconditioned matrix R-1T.

Preconditioning strategies for asymptotically ill-conditioned block Toeplitz systems

“…Now we compare our preconditioning technique with the block circulant preconditioners of [19,6] and we show that the better theoretical result is confirmed in numerical experiments. Moreover, we remark that the papers [19,6] focus their attention on the case of asymptotically well-conditioned block Toeplitz systems: actually, only very few systems considered in the numerical examples of|19, 6] are ill-conditioned. Let us consider Example 5 in [19], whose block Toeplitz matrix An, m is characterized by m = n and by the two-dimensional mask Masks of this nature anse in the discretization of constant-coefficient elliptic lade.…”

“…Very favourable results can be reached when a preconditioner Pn, m is applied: the best of all is, of course, to obtain a constant condition number of p-1/2A •--1/2 We observe that the preconditioners proposed by Di Benedetto n,m "*n,m~n,m • [9], Ku and Kuo [19], and R. H. Chan and Jin [6] do not ensure a condition number bounded by a constant independent of n and m. By using the results on the eigenvalues of Theorems 2.1-2.4 we devise a new preconditioner Pn, m with the following features: 1) P.,m is a band block Toeplitz matrix having band Toeplitz blocks.…”

“…In this case in [19,6,9], by using a preconditioner in the two-level circulant algebra [8] and in the r..., class [2], respectively, it is shown that good clustering properties of the preconditioned matrix are achieved. In the case of a ---0, Theorem 2.2 indicates that the class of matrices A.,m(f) can be illconditioned for large values of n and m and therefore the techniques proposed in [ 19,6,9] do not ensure a constant condition number for the preconditioned matrix.…”

“…Moreover, we remark that the papers [19,6] focus their attention on the case of asymptotically well-conditioned block Toeplitz systems: actually, only very few systems considered in the numerical examples of|19, 6] are ill-conditioned. Let us consider Example 5 in [19], whose block Toeplitz matrix An, m is characterized by m = n and by the two-dimensional mask Masks of this nature anse in the discretization of constant-coefficient elliptic lade. The preconditioning block circulant matrix Cn.n [19] is singular for this problem.…”

A Preconditioned MINRES Method for Nonsymmetric Toeplitz Matrices

“…We can define a preconditioner G n = J n ⊗ K m for B n , where J n ∈ R n×n is a circulant preconditioner for S n and K m ∈ R m×m is a circulant preconditioner for K m and, since the eigenvalues of a Kronecker product are products of the eigenvalues of the constituent matrices, the results of previous sections extend to these separable problems. The fact that the effective preconditioners for single Toeplitz problems can be used in Kronecker products was discussed by, for example, Benedetto, Estatico and Serra-Capizzano [14] and Ku and Kuo [29]. 6.…”