Method for Solving Large Linear Algebraic Equation Systems Based on Kaczmarz– K ‐Means Algorithm

• Published in: Journal of applied mathematics Vol. 2025; no. 1
• Main Author: Li, Hao
• Format: Journal Article
• Language: English
• Published: New York John Wiley & Sons, Inc 01.01.2025; Wiley
• Subjects: Algebra; Algorithms; Cluster analysis; Clustering; Computer graphics; Efficiency; Iterative methods; Linear algebra; Linear equations; Methods; Numerical analysis; Ordinary differential equations; Partial differential equations; Probability; Sparsity
• ISSN: 1110-757X; 1687-0042
• DOI: 10.1155/jama/6462454

In response to the problems in solving large‐scale linear algebraic equations, this study adopts two block discrimination criteria, the Euclidean clustering and the cosine clustering, to decompose them into row vectors and construct the K ‐means clustering algorithm. Based on the uniformly distributed block Kaczmarz algorithm, the weight coefficients are integrated into the probability criterion to construct a new and efficient weight probability. Thus, a block Kaczmarz algorithm based on weight probability is constructed. Experimental results showed that the method was three to five times faster than the greedy Kaczmarz method. In any convergence situation, the uniformly distributed Kaczmarz method had smaller computational complexity and iteration times compared with the weighted probability block Kaczmarz method, with a minimum acceleration ratio of 2.21 and a maximum acceleration ratio of 8.44. This study can establish and analyze mathematical models of various complex systems, reduce memory consumption and computation time when solving large linear equation systems, and provide solutions and scientific guidance for solving large linear equation systems in fields such as civil engineering and electronic engineering.