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Recent technological developments have led to big data processing, whichresulted in significant computational difficulties when solving large-scalelinear systems or inverting matrices. As a result, fast approximate iterativematrix inversion methodologies via Graphical Processing Unit (GPU) accelerationhas been a subject of extensive research, to find solutions where classic anddirect inversion are too expensive to conduct. Some currently used methods areNeumann Series (NS), Newton iteration (NI), Chebyshev Iteration (CI), andSuccessive Over-Relaxation, to cite a few. In this work, we develop a newiterative algorithm based off the NS, which we named 'Nested Neumann' (NN).This new methodology generalizes higher orders of the NI (or CI), by takingadvantage of a computationally free iterative update of the preconditioningmatrix as a function of a given 'inception depth'. It has been mathematicallydemonstrated that the NN: (i) convergences given the preconditioning satisfiesthe spectral norm condition of the NS, (ii) has an order of rate of convergencehas 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 preferablytwo, depending on RAM constraints. Furthermore, we derive an explicit formulafor the NN, which is applicable to massive sparse matrices, given an increasein computational cost. Importantly, the NN finds an analytic equivalancystatement between the NS and the the NN (NI, CI, and higher orders), which isof importance for mMIMO systems. Finally, the NN method is applicable positivesemi-definite matrices for matrix inversion, and applicable to any linearsystem (sparse, non-sparse, complex, etc.).

Generalization of Higher Order Methods for Fast Iterative Matrix Inversion Suitable for GPU Acceleration