In this work, we studied convergence rates using quotient convergence factors and root convergence factors, as described by Ortega and Rheinboldt, for Hestenes’ Gram–Schmidt conjugate direction method without derivatives. We performed computations in order to make a comparison between this conjugate direction method, for minimizing a nonquadratic function f, and Newton’s method, for solving ∇f=0. Our primary purpose was to implement Hestenes’ CGS method with no derivatives and determine convergence rates.