  Recent technological developments have led to big data processing, which
resulted in significant computational difficulties when solving large-scale
linear systems or inverting matrices. As a result, fast approximate iterative
matrix inversion methodologies via Graphical Processing Unit (GPU) acceleration
has been a subject of extensive research, to find solutions where classic and
direct inversion are too expensive to conduct. Some currently used methods are
Neumann Series (NS), Newton iteration (NI), Chebyshev Iteration (CI), and
Successive Over-Relaxation, to cite a few. In this work, we develop a new
iterative algorithm based off the NS, which we named 'Nested Neumann' (NN).
This new methodology generalizes higher orders of the NI (or CI), by taking
advantage of a computationally free iterative update of the preconditioning
matrix as a function of a given 'inception depth'. It has been mathematically
demonstrated that the NN: (i) convergences given the preconditioning satisfies
the spectral norm condition of the NS, (ii) has an order of rate of convergence
has been shown to be equivalent to the order (inception depth plus one), and
(iii) has an optimal inception depth is an inception depth of one or preferably
two, depending on RAM constraints. Furthermore, we derive an explicit formula
for the NN, which is applicable to massive sparse matrices, given an increase
in computational cost. Importantly, the NN finds an analytic equivalancy
statement between the NS and the the NN (NI, CI, and higher orders), which is
of importance for mMIMO systems. Finally, the NN method is applicable positive
semi-definite matrices for matrix inversion, and applicable to any linear
system (sparse, non-sparse, complex, etc.).
