A $N$-Body Solver for Square Root Iteration

We develop the Sparse Approximate Matrix Multiply (SpAMM) n-body solver for first order Newton Schulz iteration of the matrix square root and inverse square root. The solver performs recursive two-sided metric queries on a modified Cauchy-Schwarz criterion (octree-occlusion-culling in the ijkcube), yielding a bounded relative error in the matrix-matrix product and reduced complexity for problems with a structured metric decay. This complexity reduction corresponds to the hierarchical resolution of product sub-volumes, which may be well localized. The main contributions of this paper are a new, bounded form of the SpAMM product and demonstration of a new, algebraic locality that develops under contractive identity iteration, involving the deflation of volumes onto plane diagonals of the resolvent, and a stronger bound on the SpAMM product. Also, we carry out a first order Frëchet analyses for single and dual channel instances of the square root iteration, and look at bifurcations due to ill-conditioning and a too-aggressive SpAMM approximation. Then, we show that extreme SpAMM approximation and contractive identity iteration can be achieved for ill-conditioned systems through regularization, and we demonstrate the potential for acceleration with a scoping, product representation of the inverse factor.