# Neumann series

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## Neumann series

[′nȯi‚män ‚sir·ēz]
(mathematics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Neumann series expansion was proposed in [13] to replace the matrix inversion in MMSE detection, the performance and computational complexity of which scaled with the number of selected terms of Neumann series.
According to the Neumann series expansion [13], the inversion of matrix W can be simplified as
By decomposing W such that W = D + E, where D and E are the main diagonal and hollow component of W respectively, we can compute the first t terms of the Neumann series expansion as follows:
Thus, we aim to reduce the complexity of the ZF precoding algorithm by replacing direct inversion by simple mathematical approximations such as Neumann series.
According to [14], [G.sup.-1] in (7) is approximated using Neumann series as
The complexity of hybrid precoder based on ZF algorithm incurred by direct inversion of large size matrix is greatly reduced by Neumann series approximation.
Consequently, the Neumann series [[summation].sup.[infinity].sub.j=0] [V.sup.j] f is convergent.
Suzuki, "On the convergence of Neumann series in Banach space," Mathematische Annalen, vol.
Truncated Neumann series in [11,12] was proposed to obtain near-optimal performance.
In this paper, we propose a new joint Newton iteration and Neumann series method, where Newton iteration method is utilized to provide an efficient searching direction for the Neumann series.
We will represent the eigenvector as a Neumann series in an operator that is the sum of well localized pieces and then we estimate the terms in the series.
The result is, again, that the eigenvector is majorized by the exponential of the first term of the Neumann series. However in this case we obtain a different type of estimate for the first term.

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