A Neumann series is a mathematical series that sums k-times repeated applications of an operator. This has the generator form

is the k is the identity operator. This is a special case of the generalization of a geometric series

The series is named after the mathematician Carl Neumann, who used it in 1877 in the context of potential theory. The Neumann series is used in functional analysis. It is closely connected to the resolvent formalism for studying the spectrum of bounded operators and, applied from the left to a function, it forms the Liouville-Neumann series that formally solves Fredholm integral equations.

Properties

is a bounded linear operator on the normed vector space. If the Neumann series converges in the operator norm is invertible and its inverse is the series:

1. ,

is the identity operator. To see why, consider the partial sums

1. .

Then we have

This result on operators is analogous to geometric series.

is a Banach space converges. However, there are also results which give weaker conditions under which the series converges.

Example

be given by:

goes to infinity, the spectral norm as can be checked numerically (e.g., by computing the singular values), confirming that the Neumann series converges.

Approximate matrix inversion

A truncated Neumann series can be used for approximate matrix inversion, consider that

may give a practical approximation to the inverse matrix:

The set of invertible operators is open

is convergent. Therefore, we have

Taking the norms, we get

can be bounded by

Applications

The Neumann series has been used for linear data detection in massive multiuser multiple-input multiple-output (MIMO) wireless systems. Using a truncated Neumann series avoids computation of an explicit matrix inverse, which reduces the complexity of linear data detection from cubic to square.

Another application is the theory of propagation graphs which takes advantage of Neumann series to derive closed form expressions for transfer functions.

References