A NOTE ON THE OPTIMAL PARAMETER OF BABAIE-KAFAKI'S THREE-TERM CONJUGATE GRADIENT METHOD

Minimizing the condition number is often used in conjugate gradient methods to improve computational efficiency. In this paper, based on an eigenvalue study and a singular value study, respectively, we discuss the condition number of the conjugate gradient method proposed by Babaie-Kafaki. The obtained results improve the method, since the condition number of the corresponding iteration matrix attains its minimum value. Moreover, we propose a modified Hestenes-Stiefel type three-term conjugate gradient method with adaptive strategy, in which the nice properties of the sufficient descent condition and adaptive conjugacy condition can be retained, accelerating the convergence or reducing the condition number of iteration matrix. Under mild conditions, we show that the proposed method converges globally for general objective functions. Numerical experiments indicate that the method is practically promising.