Optimal preconditioners for functions of matrices

The optimal preconditioner cU(A)cU(A) of a given matrix A was proposed in 1988 by T. Chan [6]. Since then, it has been proved to be efficient for solving a large class of structured systems. In this paper, we construct the optimal preconditioners for different functions of matrices. More precisely, let f   be a function of matrices from Cn×nCn×n to Cn×nCn×n. Given A∈Cn×nA∈Cn×n, there are two possible optimal preconditioners for f(A)f(A): cU(f(A))cU(f(A)) and f(cU(A))f(cU(A)). In the paper, we study properties of both cU(f(A))cU(f(A)) and f(cU(A))f(cU(A)) for different functions of matrices. Numerical experiments are given to illustrate the efficiency of the optimal preconditioners when they are used to solve f(A)x=bf(A)x=b.