Abstract
In this paper, we consider the solution ofn-by-n symmetric positive definite Toeplitz systemsT n x=b by the preconditioned conjugate gradient (PCG) method. The preconditionerM n is defined to be the minimizer of ‖T n −B n ‖ F over allB n εH n whereH n is the Hartley algebra. We show that if the generating functionf ofT n is a positive 2π-periodic continuous even function, then the spectrum of the preconditioned systemM −1 n T n will be clustered around 1. Thus, if the PCG method is applied to solve the preconditioned system, the convergence rate will be superlinear.
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Jin, X.Q. Hartley preconditioners for Toeplitz systems generated by positive continuous functions. BIT 34, 367–371 (1994). https://doi.org/10.1007/BF01935646
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DOI: https://doi.org/10.1007/BF01935646