On the Computation of Neumann Series
This work provides an incremental improvement in the computational efficiency of Neumann series, which is relevant for applications requiring large matrix inversion.
The paper proposes new factorizations for computing Neumann series that reduce the number of multiplications from 2log2(N)-2 to around 1.72log2(N)-2 using a basis of size five, and asymptotically to 1.70log2(N)-2, with applications in wireless communications and image rendering.
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.