Open AccessA New Improved Classical Iterative Algorithm for Solving System of Linear Equations
Author(s) -
Muhammad Shakeel Rind Baloch,
Zubair Ahmed Kalhoro,
Mir Sarfraz Khalil,
Prof. Abdul Wasim Shaikh
Publication year - 2022
Publication title -
proceedings of the pakistan academy of sciences. a, physical and computational sciences
Language(s) - English
Resource type - Journals
eISSN - 2518-4253
pISSN - 2518-4245
DOI - 10.53560/ppasa(58-4)638
Subject(s) - jacobi method , successive over relaxation , jacobi eigenvalue algorithm , relaxation (psychology) , iterative method , mathematics , gauss–seidel method , ode , matrix (chemical analysis) , diagonally dominant matrix , linear system , coefficient matrix , convergence (economics) , linear algebra , algorithm , local convergence , mathematical analysis , pure mathematics , eigenvalues and eigenvectors , geometry , materials science , economic growth , composite material , psychology , social psychology , quantum mechanics , physics , economics , invertible matrix
The fundamental problem of linear algebra is to solve the system of linear equations (SOLE’s). To solve SOLE’s, is one of the most crucial topics in iterative methods. The SOLE’s occurs throughout the natural sciences, social sciences, engineering, medicine and business. For the most part, iterative methods are used for solving sparse SOLE’s. In this research, an improved iterative scheme namely, ‘’a new improved classical iterative algorithm (NICA)’’ has been developed. The proposed iterative method is valid when the co-efficient matrix of SOLE’s is strictly diagonally dominant (SDD), irreducibly diagonally dominant (IDD), M-matrix, Symmetric positive definite with some conditions and H-matrix. Such types of SOLE’s does arise usually from ordinary differential equations (ODE’s) and partial differential equations (PDE’s). The proposed method reduces the number of iterations, decreases spectral radius and increases the rate of convergence. Some numerical examples are utilized to demonstrate the effectiveness of NICA over Jacobi (J), Gauss Siedel (GS), Successive Over Relaxation (SOR), Refinement of Jacobi (RJ), Second Refinement of Jacobi (SRJ), Generalized Jacobi (GJ) and Refinement of Generalized Jacobi (RGJ) methods.