### Speaker

Mr
Matthias Puhr
(University of Regensburg)

### Description

We present a method for the numerical calculation of derivatives of
functions of general complex matrices which also works for implicit
matrix function approximations such as Krylov-Ritz type algorithms. An
important use case for the method is the overlap Dirac operator at
finite quark chemical potential. The evaluation of the overlap Dirac
operator at finite chemical potential calls for the computation of the
product of the sign function of a non-Hermitian matrix with some source
vector. For non-Hermitian matrices the sign function can no longer be
efficiently approximated with polynomials or rational functions.
Instead one invokes implicit approximation algorithms, like Krylov-Ritz
methods, that depend on the source vector. Our method allows for an
efficient calculation of the derivatives of such implicit
approximations, which is necessary for the computation of conserved
lattice currents or the fermionic force in Hybrid Monte-Carlo or
Langevin simulations. We also give an explicit deflation prescription for the case when one
knows several eigenvalues and eigenvectors of the matrix being the argument
of the differentiated function. To show that the method is efficient and well
suited for practical calculations we provide test results for the
two-sided Lanczos approximation of the finite-density overlap Dirac
operator on $SU(3)$ gauge field configurations on lattices with sizes up
to $ 14\times14^3 $

### Primary author

Mr
Matthias Puhr
(University of Regensburg)

### Co-author

Dr
Pavel Buividovich
(University of Regensburg)