Monte Carlo Algorithm For Matrices in Solving Systems of Linear Equations, Determinants, Inverse, Eigen Values, and Eigenvectors. - Bamwine, Delik (Kabale University, 2024)
The This study sought to establish the general applicability of Monte Carlo algorithm for matrices in solving systems of linear equations, determinants, inverse, Eigen values, and Eigen vectors. Linear algebra operations play an important role in scientific computing and data analysis. With increasing data volume and complexity in the "Big Data" era, linear algebra operations are important tools to process massive datasets. On one hand, the advent of modern high-performance computing architectures with increasing computing power has greatly enhanced our capability to deal with a large volume of data. On the other hand, many classical, deterministic numerical linear algebra algorithms have difficulty scaling to handle large data sets. Monte Carlo methods, which are based on statistical sampling, exhibit many attractive properties in dealing with large volumes of datasets, including fast approximated results, memory efficiency, reduced data accesses, natural parallelism, and inherent fault tolerance. This research assessed Monte Carlo methods to accommodate a set of fundamental and ubiquitous large-scale linear algebra operations, including solving large-scale linear systems, constructing low-rank matrix approximation, and approximating the extreme eigenvalues/ eigenvectors, across modern distributed and parallel computing architectures. This research provides me with a solid base of knowledge in numerical linear algebra for parallel high-performance computing systems. Future research should focus on enhancing sampling efficiency in matrix-vector products along MCGMRES iterations and implementing the RSVD algorithm on big data analysis platforms.