Note on the Zolotarev optimal rational approximation for the overlap Dirac operator

Abstract

We discuss the salient features of the Zolotarev optimal rational approximation for the inverse square root function, in particular, for its applications in lattice QCD with the overlap Dirac quark. The theoretical error bound for the matrix-vector multiplication ${H}_{w}{(H}_{w}^{2}{)}^{\ensuremath{-}1/2}Y$ is derived. We check that the error bound is always satisfied amply, for any QCD gauge configurations we have tested. An empirical formula for the error bound is determined, together with its numerical values (by evaluating elliptic functions) listed in Table II as well as plotted in Fig. 3. Our results suggest that, with the Zolotarev approximation to ${(H}_{w}^{2}{)}^{\ensuremath{-}1/2},$ one can essentially preserve the exact chiral symmetry of the overlap Dirac operator to very high precision, for any gauge configuration on a finite lattice.