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

2025/02/10 by A C PATERSON, Paterson, Ann, Jennifer Pestana +2
Computer Science · #65F05 (Primary) 15A09 (Secondary) #Blind Source Separation Techniques #FOS: Mathematics #G.1.3 #Matrix Theory and Algorithms #Neural Networks and Applications #Numerical Analysis (math.NA) #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2502.06377

openalex publication_date 2025/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Obtaining the inverse of a large symmetric positive definite matrix A∈ℝ^{p× p} 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 algorithm (IBMI), 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, non-overlapping approach converges for any positive definite matrix, while numerical results provide strong evidence that the multi-block, overlapping approach also converges for such matrices.

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