On the sufficient descent condition of the Hager-Zhang conjugate gradient methods

Based on an eigenvalue study, the sufficient descent condition of an extended class of the Hager-Zhang nonlinear conjugate gradient methods is established. As an interesting result, it is shown that the search directions of the CG_Descent algorithm satisfy the sufficient descent condition d(k)(T) g(k) < -7/8 parallel to g(k)parallel to(2).