The solution of symmetric positive definite Toeplitz systems Ax = b by the preconditioned conjugate gradient (PCG) method was recently proposed by Strang and analyzed by R. Chan and Strang. The convergence rate of the PCG method depends heavily on the choice of preconditioners for the given Toeplitz matrices. In this paper, we present a general approach to the design of Toeplitz preconditioners based on the idea to approximate a partially characterized linear deconvolution with circular deconvolutions. All resulting preconditioners can therefore be inverted via various fast transform algorithms with O(N log N) operations. For a wide class of problems, the PCG method converges in a finite number of iterations independent of N so that the computational complexity for solving these Toeplitz systems is O(N log N) .
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.
A new preconditioner is proposed for the solution of an N x N Toeplitz system TNX = b, where TN can be symmetric indefinite or nonsymmetric, by preconditioned iterative methods. The preconditioner FN is obtained based on factorizing the generating function T(z) into the product of two terms corresponding, respectively, to minimum-phase causal and anticausal systems and therefore called the minimum-phase LU (MPLU) factorization preconditioner. Due to the minimum-phase property, IIFJ1II is bounded. For rational Toeplitz TN with generating function T(z) = A(z1)/B(z1) + C(z)/D(z), where A(z), B(z), C(z) and D(z) are polynomials of orders p , q , P2 and q , we show that the eigenvalues of FJ1 TN are repeated exactly at 1 except at most cF outliers, where cF depends on Pi ,qi , P2 , q2 and the number w of the roots of T(z) = A(z1)D(z) + B(z1)C(z) outside the unit circle. A preconditioner IN in circulant form generalized from the symmetric case is also presented for comparison.
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