Abstract
For any given n-by-n matrix A, a specific circulant preconditioner tF(A) introduced by Tyrtyshnikov [E. Tyrtyshnikov, Optimal and super-optimal circulant preconditioners, SIAM J. Matrix Anal. Appl. 13 (1992) 459–473] is defined to be the solution ofminC‖I-C-1A‖Fover all n-by-n nonsingular circulant matrices C. The preconditioner tF(A), called the superoptimal circulant preconditioner, has been proved to be a good preconditioner for a large class of structured systems. In this paper, we study this preconditioner in the general case by using the Moore-Penrose inverse. We give a formula for the superoptimal preconditioner and discuss the stability properties of this preconditioner. A spectral relation between the optimal and superoptimal preconditioned matrices in the general case is also given.
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