In this paper, we propose two hybrid gradient based methods and genetic algorithm for solving systems of linear equations with fast convergence. The first proposed hybrid method is obtained by using the steepest descent method and the second one by the Cauchy-Barzilai-Borwein method. These algorithms are based on minimizing the residual of solution which has genetic characteristics. They are compared with the normal genetic algorithm and standard gradient based methods in order to show the accuracy and the convergence speed of them. Since the conjugate gradient method is recommended for solving large sparse and symmetric positive definite matrices, we also compare the numerical results of our proposed algorithms with this method. The numerical results demonstrate the robustness and efficiency of the proposed algorithms. Moreover, we observe that our hybridization of the CBB method and genetic algorithm gives more accurate results with faster convergence than other mentioned methods in all given cases

The steepest descent (SD) method is well-known as the simplest method in optimization. In this paper, we propose a new SD search direction for solving system of linear equations Ax = b. We also prove that the proposed SD method with exact line search satisfies descent condition and possesses global convergence properties. This proposed method is motivated by previous work on the SD method by Zubai'ah-Mustafa-Rivaie-Ismail (ZMRI) [2]. Numerical comparisons with a classical SD algorithm and ZMRI algorithm show that this algorithm is very effective depending on the number of iterations (NOI) and CPU time.