An Iterative Block Matrix Inversion (IBMI) Algorithm for Symmetric Positive Definite Matrices with Applications to Covariance Matrices

Obtaining the inverse of a large symmetric positive definite matrix [Formula: see text] is a continual challenge across many mathematical disciplines. The computational complexity associated with direct methods can be prohibitively expensive, making it infeasible to compute the inverse. In this paper, we present a novel iterative block matrix inversion (IBMI) algorithm, which is designed to approximate the inverse of a large, dense, symmetric positive definite matrix. The matrix is first partitioned into blocks, and an iterative process using block matrix inversion is repeated until the matrix approximation reaches a satisfactory level of accuracy. We demonstrate that the two-block, nonoverlapping approach converges for any positive definite matrix, while numerical results provide strong evidence that the multiblock, overlapping approach also converges for such matrices.