Based on a singular value study, lower and upper bounds for the condition number of the matrix which generates search directions of the Hager-Zhang conjugate gradient method are obtained. Then, based on the insight gained by our analysis, a modified version of the Hager-Zhang method is proposed, using an adaptive switch form the Hager-Zhang method to the Hestenes-Stiefel method when the mentioned condition number is large. A brief global convergence analysis is made for the uniformly convex objective functions. Numerical comparisons between the implementations of the proposed method and the Hager-Zhang method are made on a set of unconstrained optimization test problems of the CUTEr collection, using the performance profile introduced by Dolan and Mor´e. Comparative testing results are reported.

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