For any given matrix A ∈ℂnxn, a preconditioner tU(A) called the
superoptimal preconditioner was proposed in 1992 by Tyrtyshnikov. It has been
shown that tU(A) is an efficient preconditioner for
solving various structured systems, for instance, Toeplitz-like systems. In this
paper, we construct the superoptimal preconditioners for different functions of
matrices. Let f be a function of matrices from ℂnxn to ℂnxn. For any A ∈ ℂ nxn, 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.