Solving large linear systems of equations with the Jacobi method in Mathematica and MATLAB softwares: Serial vs. Parallel

In this paper , the iterative Jacobi method is applied to solve some linear equation systems with 64 to 512 equations in Mathematica 10.0 and MATLAB 2014a softwares. Two approaches are used for programming the method namely serial one and parallel one. The main goal of the paper is investigating advantages and disadvantages of the parallel Jacobi method over shared environments with parallel commands in comparison with the usual serial Jacobi method in the two softwares Results show that the parallel algorithm can converge faster to the exact solution than the serial one in Mathematica when size of matrix is increased and/or more accurate approximations are needed. Also using serial implementation in MATLAB returns the approximations for the unknowns much faster than Mathematica for same matrix sizes but it is vice versa for the parallel implementation at least for these sizes.