APPLICATION OF LINEAR AND NONLINEAR ALGEBRAIC TECHNIQUES IN MODERN COMPUTATION

Abstract

The skills of representing, transforming, analysing and solving complex problems efficiently are very important for modern computation and rely heavily on mathematical tools. Linear and nonlinear algebraic tools are of key importance in computational science, engineering, artificial intelligence, data science, cryptography, computer graphics, robotics and scientific modelling. Vectors, matrices, tensors, linear transformations, eigenvalues, eigenvectors, matrix factorization and systems of linear equations are all addressed in the simple framework of linear algebra. They are broadly utilized in machine learning, image or signal processing, optimization, numerical simulation, and computer graphics. Nonlinear algebraic methods are also important in solving problems involving relationships that aren't proportional or aren't well modeled by simple linear relationships. In robotics, computer vision, cryptography, scientific computing, control systems and deep learning, we are seeing the use of nonlinear equations, polynomial systems, tensor algebra, nonlinear optimization, numerical iterative methods and algebraic geometry. This paper introduces the key ideas, techniques, algorithms and applications of linear algebra and nonlinear algebra relevant to current computation. It also identifies significant computational issues including high dimensionality, numerical instability, convergence problems, data uncertainty and computational complexity. The study demonstrates that algebraic methodology not only consists of mathematical theories but also of computational tools for modern technologies. The use of algebra as a solution to complex computational problems is likely to continue to grow in the future with the help of new developments in the field of randomized algorithms, symbolicnumeric computation, tensor methods, explainable artificial intelligence, high-performance computing and quantumassisted computation.