On the Computation of Neumann Series

TL;DR

This paper introduces novel factorizations for Neumann series computation, reducing multiplication complexity through new bases and merging strategies, with applications in wireless communications and image rendering.

Contribution

It presents new factorization methods and bases that decrease the complexity of Neumann series calculations compared to traditional binary approaches.

Findings

Reduced multiplications from 2log2(N)-2 to ~1.72log2(N)-2 using a size-five basis.

Further reduction to ~1.70log2(N)-2 multiplications with basis merging.

Validated algorithms through simulations in wireless and image processing applications.

Abstract

This paper proposes new factorizations for computing the Neumann series. The factorizations are based on fast algorithms for small prime sizes series and the splitting of large sizes into several smaller ones. We propose a different basis for factorizations other than the well-known binary and ternary basis. We show that is possible to reduce the overall complexity for the usual binary decomposition from 2log2(N)-2 multiplications to around 1.72log2(N)-2 using a basis of size five. Merging different basis we can demonstrate that we can build fast algorithms for particular sizes. We also show the asymptotic case where one can reduce the number of multiplications to around 1.70log2(N)-2. Simulations are performed for applications in the context of wireless communications and image rendering, where is necessary perform large sized matrices inversion.