Obtaining the inverse of a large symmetric positive definite matrix $\mathcal{A}\in\mathbb{R}^{p\times p}$ 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.

翻译：求取大规模对称正定矩阵 $\mathcal{A}\in\mathbb{R}^{p\times p}$ 的逆矩阵是众多数学领域持续面临的挑战。直接方法的计算复杂度可能过高，导致无法实际计算其逆矩阵。本文提出一种新颖的迭代算法（IBMI），旨在近似大规模稠密对称正定矩阵的逆。该算法首先将矩阵划分为若干分块，随后利用分块矩阵求逆公式进行迭代计算，直至矩阵近似达到满意的精度水平。我们证明，对于任意正定矩阵，非重叠的双分块方法均能收敛；数值结果进一步表明，重叠的多分块方法对此类矩阵同样具有收敛性。