A note on Neuberger's double pass algorithm

TL;DR

This paper analyzes Neuberger's double pass algorithm for matrix-vector multiplication, showing its efficiency and independence from polynomial degree n, and identifying when it outperforms the single pass method.

Contribution

It demonstrates that the double pass algorithm's computational cost is independent of polynomial degree n and identifies the threshold where it becomes faster than the single pass.

Findings

Number of floating point operations is independent of degree n.

Double pass algorithm can approximate $H^{-1/2} Y$ with high precision.

Double pass outperforms single pass for n > 12-25.

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.