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The overlap Dirac operator in lattice QCD requires the computation of thesign function of a matrix. While this matrix is usually Hermitian, it becomesnon-Hermitian in the presence of a quark chemical potential. We show how theaction of the sign function of a non-Hermitian matrix on an arbitrary vectorcan be computed efficiently on large lattices by an iterative method. A Krylovsubspace approximation based on the Arnoldi algorithm is described for theevaluation of a generic matrix function. The efficiency of the method isspoiled when the matrix has eigenvalues close to a function discontinuity. Thisis cured by adding a small number of critical eigenvectors to the Krylovsubspace, for which we propose two different deflation schemes. The ensuingmodified Arnoldi method is then applied to the sign function, which has adiscontinuity along the imaginary axis. The numerical results clearly show theimproved efficiency of the method. Our modification is particularly effectivewhen the action of the sign function of the same matrix has to be computed manytimes on different vectors, e.g., if the overlap Dirac operator is invertedusing an iterative method.

An iterative method to compute the sign function of a non-Hermitian  matrix and its application to the overlap Dirac operator at nonzero chemical  potential