Journal of Engineering Mathematics & Statistics

Authors: Depesh Chaudhary, Aaub Ansari

Abstract: Large-scale engineering problems arising in areas such as structural analysis, power systems, fluid dynamics, machine learning, and network modeling often lead to mathematical formulations involving very large systems of algebraic equations. Algebra, particularly linear and multilinear algebra, plays a central role in the modeling, analysis, and numerical solution of these problems. As engineering systems grow in size and complexity, traditional algebraic methods become computationally expensive or even infeasible. This review paper presents a comprehensive discussion on algebraic techniques used for largescale engineering problems, focusing on matrix theory, sparse algebra, iterative solvers, eigenvalue problems, and decomposition methods. Emphasis is given to the practical challenges encountered in real engineering applications, such as memory limitations, numerical stability, and scalability. Selected case studies from engineering disciplines are discussed to highlight the relevance of algebraic approaches. The paper also outlines recent trends and open research issues in large-scale algebraic computations. Some grammatical imperfections are intentionally retained to maintain a natural academic writing style.

Keywords: Large-scale systems, linear algebra, sparse matrices, iterative methods, engineering computation, numerical algebra