Comments on: “A family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations” [J. Comput. Appl. Math. 224 (2009) 11–19]

In this note a simple counter example shows that the proof of Lemma 3.3 in [1, W. Cheng, Y. Xiao and Q. Hu, A family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations, J. Comput. Appl. Math. 224 (2009) 11–19] is not correct, which implies that Lemma 3.2 in [1] is not enough to ensure Lemma 3.3 in [1]. A new proof is given, which leads to a stronger result than Lemma 3.2 in [1]. And this result not only guarantees that Lemma 3.3 in [1] holds, but also improves the corresponding global convergence Theorem 3.1 in [1].