In this paper, we introduce novel fast matrix inversion algorithms that leverage triangular decomposition and recurrent formalism, incorporating Strassen's fast matrix multiplication. Our research places particular emphasis on triangular matrices, where we propose a novel computational approach based on combinatorial techniques for finding the inverse of a general non-singular triangular matrix. Unlike iterative methods, our combinatorial approach for (block) triangular-type matrices enables direct computation of the matrix inverse through a nonlinear combination of carefully selected combinatorial entries from the initial matrix. This unique characteristic makes our proposed method fully parallelizable, offering significant potential for efficient implementation on parallel computing architectures. Our approach demonstrates intriguing features that allow the derivation of recurrent relations for constructing the matrix inverse. By combining the (block) combinatorial approach, with a recursive triangular split method for inverting triangular matrices, we develop potentially competitive algorithms that strike a balance between efficiency and accuracy. We provide rigorous mathematical proofs of the newly presented method. Additionally, we conduct extensive numerical tests to showcase its applicability and efficiency. The comprehensive evaluation and experimental results presented in this paper confirm the practical utility of our proposed algorithms, demonstrating their superiority over classical approaches in terms of computational efficiency.

翻译：本文提出了新颖的快速矩阵求逆算法，该算法利用三角分解与递归形式，融合了Strassen快速矩阵乘法。研究特别聚焦于三角矩阵，提出了一种基于组合技术的全新计算方法，用于求解一般非奇异三角矩阵的逆。与迭代方法不同，我们的（分块）三角型矩阵组合方法，通过初始矩阵中精心选取的组合项的非线性组合，直接计算矩阵的逆。这一独特特性使所提方法完全可并行化，为并行计算架构的高效实现提供了巨大潜力。我们的方法展现出引人注目的特性，能够推导出构造矩阵逆的递归关系。通过将（分块）组合方法与求逆三角矩阵的递归三角分裂法相结合，我们开发出在效率与精度之间取得平衡的潜在竞争性算法。我们对新方法给出了严格的数学证明，并开展了广泛的数值测试以展示其适用性与效率。本文呈现的综合评估与实验结果，证实了所提算法的实际效用，表明其在计算效率上优于经典方法。