Neuberger’s double-pass algorithm

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)^-1/2Y≃R^(n-1,n)(H)⋅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≃12–25 for most platforms.