Journal of Engineering Mathematics & Statistics

Authors: Om Prakash Yadav, Arun Deshpande, Supriya Rajbhar, Aman Thakur

Abstract: Large systems of equations arise naturally in many branches of science and engineering, including fluid dynamics, structural analysis, climate modeling, power systems, machine learning, and biological simulations. As the size and complexity of these systems continue to grow, traditional sequential computational approaches become insufficient due to limitations in memory, time, and energy consumption. High Performance Computing (HPC) has emerged as a crucial enabler for solving such large-scale systems efficiently. This paper presents a comprehensive review of high-performance computing methods used for solving large systems of linear and nonlinear equations. The discussion includes parallel numerical algorithms, domain decomposition techniques, iterative solvers, sparse matrix methods, and the role of modern hardware architectures such as multicore processors, graphics processing units (GPUs), and distributed memory clusters. Performance metrics, scalability issues, and communication overheads are also examined. Through selected examples and comparative tables, the paper highlights the strengths and limitations of different HPC strategies. The review aims to provide researchers and practitioners with a structured understanding of current methodologies and practical considerations when applying HPC to large systems of equations.

Keywords: High performance computing, large systems of equations, parallel algorithms, iterative solvers, sparse matrices, scalability