A SYSTEMATIC REVIEW ON PROGRESS OF TENSOR ANALYSIS, COMPUTATION AND APPLICATIONS IN MATHEMATICS

Tensor analysis has emerged as a foundational tool in mathematics, bridging abstract theory and computational practice. This systematic review explores the significant progress in tensor theory, computational strategies, and their diverse applications in mathematical disciplines. The paper surveys developments in tensor decompositions, multilinear algebra, and numerical methods that have enabled efficient representation and manipulation of high-dimensional data. Key tensor formats, like Tucker, Tensor Train (TT), and CANDECOMP/PARAFAC (CP) are analyzed for their theoretical underpinnings and computational effectiveness. The study highlights how these decompositions enhance problem-solving capabilities in areas such as numerical linear algebra, optimization, differential geometry, and functional analysis. Special emphasis is placed on the computational advancements that facilitate scalable tensor operations and the integration of tensor techniques in solving partial differential equations, machine learning problems, and multidimensional data modeling. Despite notable progress, the review identifies several research gaps, including the need for unified theoretical frameworks, robust algorithms for large-scale tensors, and deeper exploration of tensor applications in emerging mathematical fields. Future directions include development of adaptive tensor methods, theoretical convergence analyses, and cross-disciplinary applications involving stochastic modeling and quantum computing. By critically evaluating past and current research trends, this review provides a comprehensive understanding of the evolving landscape of tensor analysis and outlines strategic pathways for future investigation in mathematics.