Superoptimal Preconditioners for Functions of Matrices

For any given matrix A ∈ Cn×n, a preconditioner tU (A) called the superoptimal preconditioner was proposed in 1992 by Tyrtyshnikov [20]. It has been shown that tU (A) is an efficient preconditioner for solving various structured systems, for instance, Toeplitzlike systems. In this paper, we construct the superoptimal preconditioners for different functions of matrices. Let f be a function of matrices from Cn×n to Cn×n. For any A ∈ Cn×n, one may construct two superoptimal preconditioners for f(A): tU (f(A)) and f(tU (A)). We establish basic properties of tU (f(A)) and f(tU (A)) for different functions of matrices. Some numerical tests demonstrate that the proposed preconditioners are very efficient for solving the system f(A)x = b. AMS classification: 65F10; 65F15; 65L05; 65N22.