The author studies a conjugate gradient algorithm for unconstrained optimization. The article begins with an introduction to the general nonlinear unconstrained optimization problem and a short review of the existing literature. The second section outlines the hybrid conjugate gradient algorithm as a convex combination of two existing algorithms. This is followed by the third and fourth sections which present the details of the algorithm itself and study its convergence properties, where several theorems are presented with proof. The paper concludes with an extensive section containing results of the numerical experimentation and a list of relevant references.