A note on Neuberger's double pass algorithm

Abstract

We analyze Neuberger's double pass algorithm for the matrix-vector multiplication R(H).Y (where R(H) is (n-1,n)-th degree rational polynomial of positive definite operator H), and show that the number of floating point operations is independent of the degree n, provided that the number of sites is much larger than the number of iterations in the conjugate gradient. This implies that the matrix-vector product $(H{)}^{-1/2}Y\simeq R^{(n-1,n)}(H)\cdot Y$ can be approximated to very high precision with sufficiently large n, without noticeably extra costs. Further, we show that there exists a threshold $n_T$ such that the double pass is faster than the single pass for $n>n_T$, where $n_T\simeq 12-25$ for most platforms.

18 pages, v3: CPU time formulas are obtained, to appear in Physical Review E

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• Optimal lattice domain-wall fermions
• A note on Zolotarev optimal rational approximation for the overlap Dirac operator
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Cited by in corpus (7)

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• Study of $Thet{a}^{+}(udud\bar{s})$ in lattice QCD with exact chiral symmetry
• Efficient implementation of the overlap operator on multi-GPUs
• Electric polarizability of hadrons with overlap fermions on multi-GPUs
• Double-pass variants for multi-shift BiCGstab(ell)