  The overlap Dirac operator in lattice QCD requires the computation of the
sign function of a matrix. While this matrix is usually Hermitian, it becomes
non-Hermitian in the presence of a quark chemical potential. We show how the
action of the sign function of a non-Hermitian matrix on an arbitrary vector
can be computed efficiently on large lattices by an iterative method. A Krylov
subspace approximation based on the Arnoldi algorithm is described for the
evaluation of a generic matrix function. The efficiency of the method is
spoiled when the matrix has eigenvalues close to a function discontinuity. This
is cured by adding a small number of critical eigenvectors to the Krylov
subspace, for which we propose two different deflation schemes. The ensuing
modified Arnoldi method is then applied to the sign function, which has a
discontinuity along the imaginary axis. The numerical results clearly show the
improved efficiency of the method. Our modification is particularly effective
when the action of the sign function of the same matrix has to be computed many
times on different vectors, e.g., if the overlap Dirac operator is inverted
using an iterative method.
