  Circulant preconditioners for functions of matrices have been recently of
interest. In particular, several authors proposed the use of the optimal
circulant preconditioners as well as the superoptimal circulant preconditioners
in this context and numerically illustrated that such preconditioners are
effective for certain functions of Toeplitz matrices. Motivated by their
results, we propose in this work the absolute value superoptimal circulant
preconditioners and provide several theorems that analytically show the
effectiveness of such circulant preconditioners for systems defined by
functions of Toeplitz matrices. Namely, we show that the eigenvalues of the
preconditioned matrices are clustered around $\pm 1$ and rapid convergence of
Krylov subspace methods can therefore be expected. Moreover, we show that our
results can be extended to functions of block Toeplitz matrices with Toeplitz
blocks provided that the optimal block circulant matrices with circulant blocks
are used as preconditioners. Numerical examples are given to support our
theoretical results.
