Study of Solving Linear Equations by Hybrid Evolutionary Computation Techniques

Abstract

Solving a set of simultaneous linear equations is a fundamental problem that occurs in diverse applications. For solving large sets of linear equations, iterative methods are preferred over other methods specially when the coefficient matrix of the linear system is sparse. The rate of convergence of iterative (Jacobi & Gauss-Seidel) methods is increased by using successive relaxation (SR) technique. But SR technique is very sensitive to relaxation factor, . Recently, hybridization of evolutionary computation techniques with classical Gauss-Seidel-based SR method has successfully been used to solve large set of linear equations in which relaxation factors are self-adapted. Under this paradigm, this research work has developed a new class of hybrid evolutionary algorithms for solving system of linear equations. The first algorithm is the Jacobi-Based Uniform Adaptive (JBUA) hybrid algorithm, which has been developed within the framework of contemporary Gauss-Seidel-Based Uniform Adaptive (GSBUA) hybrid algorithm, and classical Jacobi method. The proposed JBUA hybrid algorithm can be implemented, inherently, in parallel processing environment efficiently whereas GSBUA hybrid algorithm cannot be implemented in parallel processing environment efficiently. The second algorithm is the Gauss-Seidel-Based Time-Variant Adaptive (GSBTVA) hybrid algorithm that has been developed within the framework of contemporary GSBUA hybrid algorithm and time-variant adaptive technique. In this algorithm two new time-variant adaptive operators have been introduced based on some observed biological evidences. The third algorithm is the Jacobi-Based Time-Variant Adaptive (JBTVA) hybrid algorithm that has been developed within the framework of GSBTVA and JBUA hybrid algorithms. This proposed JBTVA algorithm also can be implemented, inherently, in parallel processing environment efficiently. All the proposed hybrid algorithms have been tested on some test problems and compared with other hybrid evolutionary algorithms and classical iterative methods. Also the validity of the rapid convergence of the proposed algorithms are proved theoretically. The proposed hybrid algorithms outperform the contemporary GSBUA hybrid algorithm as well as classical iterative methods in terms of convergence speed and effectiveness.