Developing Efficient Numerical Methods for Solving Large Linear Systems Using Spectroscopic Analysis Techniques

Abstract

This research is aimed at efficient numerical methods to solve large linear systems with spectroscopic analysis techniques. Jacobi method, Gauss–Seidel method, Conjugate Gradient method, GMRES and preconditioned iterative methods are compared by choosing various test systems, such as sparse, ill-conditioned, discredited systems, and symmetric positive definite systems. The criterion is based on the residual error, relative error, convergence rate, number of iterations, condition number, spectral radius, CPU time and memory usage. Results are presented, indicating that the preconditioned iterative methods can provide the most successful method, improving convergence, decreasing error, and increasing numerical stability. Based on the findings of the study, it is concluded that the spectroscopic analysis is proven to be a useful technique in understanding the behavior of the matrices involved, and also to increase the efficiency of the numerical solver.