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Kaishav Gupta
Keywords:
Large-scale matrices, Numerical linear algebra, Sparse matrices, Krylov methods, Matrix decomposition, Parallel computing, Randomized algorithms, High-performance computing.
Abstract:
Large-scale matrix computations constitute the computational foundation of modern scientific computing, engineering analysis, artificial intelligence, machine learning, computational physics, big data analytics, and network science. The exponential growth of data generated from scientific experiments, industrial processes, and digital technologies has led to the emergence of matrices containing millions or even billions of elements. Traditional direct numerical techniques become computationally expensive for such large problems because they require significant memory resources and execution time. Consequently, modern numerical linear algebra has evolved to incorporate advanced iterative algorithms, sparse matrix methods, matrix decomposition techniques, randomized numerical algorithms, and parallel computing strategies that significantly improve computational efficiency while preserving numerical accuracy and stability. This review presents a comprehensive analytical perspective on modern numerical techniques used for solving large-scale matrix computations. It discusses the mathematical foundations, structural characteristics of large matrices, direct and iterative solution techniques, Krylov subspace methods, sparse matrix representations, randomized numerical linear algebra, high-performance computing, computational complexity, numerical stability, practical applications, current challenges, and future research trends. The article highlights how the integration of advanced algorithms with modern computing architectures has transformed computational science and enabled the solution of previously intractable problems across numerous scientific and engineering disciplines.
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International Journal of Recent Research and Review
ISSN: 2277-8322
Vol. XIX, Issue 2
June 2026
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PUBLISHED
June 2026
ISSUE
Vol. XIX, Issue 2
SECTION
Articles
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