The Fletcher-Reeves conjugate gradient algorithm

This algorithm calculates the mutually conjugate directions of search with respect to the Hessian matrix of $f$ directly from the function evaluation and the gradient evaluation, but without the direct evaluation of the Hessian of the function $f$.

Algorithm 5.34   Fletcher-Reeves conjugate gradient algorithm

The quantity $\dfrac{\Vert\nabla f(\mathbf{x}^i)\Vert^2}{\Vert\nabla f(\mathbf{x}^{i-1})\Vert^2}\mathbf{h}^{i-1}$ is added to the gradient at each iteration, and when $f$ is a quadratic form (positive definite), this results in a set of mutually conjugate vectors.