Solving Linear Equations by Classical Jacobi-SR Based Hybrid Evolutionary Algorithm with Uniform Adaptation Technique

TL;DR

This paper introduces a novel hybrid evolutionary algorithm using Jacobi-SR with uniform adaptation, which improves convergence speed and parallel implementation efficiency for solving large sparse linear systems.

Contribution

It proposes a new Jacobi-SR based hybrid evolutionary algorithm with uniform adaptation, enhancing convergence and parallelization over existing Gauss-Seidel based methods.

Findings

Outperforms classical Jacobi-SR and Gauss-Seidel-SR methods in convergence speed.

Theoretically proven convergence and adaptation theorems.

Efficient parallel implementation of the proposed algorithm.

Abstract

Solving a set of simultaneous linear equations is probably the most important topic in numerical methods. For solving linear equations, iterative methods are preferred over the direct methods especially when the coefficient matrix is sparse. The rate of convergence of iteration method is increased by using Successive Relaxation (SR) technique. But SR technique is very much sensitive to relaxation factor, {\omega}. Recently, hybridization of classical Gauss-Seidel based successive relaxation technique with evolutionary computation techniques have successfully been used to solve large set of linear equations in which relaxation factors are self-adapted. In this paper, a new hybrid algorithm is proposed in which uniform adaptive evolutionary computation techniques and classical Jacobi based SR technique are used instead of classical Gauss-Seidel based SR technique. The proposed Jacobi-SR based uniform adaptive hybrid algorithm, inherently, can be implemented in parallel processing environment efficiently. Whereas Gauss-Seidel-SR based hybrid algorithms cannot be implemented in parallel computing environment efficiently. The convergence theorem and adaptation theorem of the proposed algorithm are proved theoretically. And the performance of the proposed Jacobi-SR based uniform adaptive hybrid evolutionary algorithm is compared with Gauss-Seidel-SR based uniform adaptive hybrid evolutionary algorithm as well as with both classical Jacobi-SR method and Gauss-Seidel-SR method in the experimental domain. The proposed Jacobi-SR based hybrid algorithm outperforms the Gauss-Seidel-SR based hybrid algorithm as well as both classical Jacobi-SR method and Gauss-Seidel-SR method in terms of convergence speed and effectiveness.