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We present a method for the numerical calculation of derivatives of functionsof general complex matrices. The method can be used in combination with anyalgorithm that evaluates or approximates the desired matrix function, inparticular with implicit Krylov-Ritz-type approximations. An important use casefor the method is the evaluation of the overlap Dirac operator in latticeQuantum Chromodynamics (QCD) at finite chemical potential, which requires theapplication of the sign function of a non-Hermitian matrix to some sourcevector. While the sign function of non-Hermitian matrices in practice cannot beefficiently approximated with source-independent polynomials or rationalfunctions, sufficiently good approximating polynomials can still be constructedfor each particular source vector. Our method allows for an efficientcalculation of the derivatives of such implicit approximations with respect tothe gauge field or other external parameters, which is necessary for thecalculation of conserved lattice currents or the fermionic force in HybridMonte-Carlo or Langevin simulations. We also give an explicit deflationprescription for the case when one knows several eigenvalues and eigenvectorsof the matrix being the argument of the differentiated function. We test themethod for the two-sided Lanczos approximation of the finite-density overlapDirac operator on realistic $SU(3)$ gauge field configurations on lattices withsizes as large as $14\times14^3$ and $6\times18^3$.

computer physics communications

A numerical method to compute derivatives of functions of large complex  matrices and its application to the overlap Dirac operator at finite chemical  potential