# Conjugate Gradient for nonlinear equation system

Is it possible to apply adaptions of the conjugate gradient algorithm i.e. Fletcher-Reeves, Polak-Ribere or others to systems of nonlinear equations? How should the equation system be adjusted so one can use those algorithms?

I am asking this because in all of the literature, the pseudocode says to compute the conjugated directions by gradients, i.e.:

$$p_{k+1} = -\nabla f_{k+1} + \beta_{k+1} p_k$$ with $$\beta_{k+1} = \frac{\nabla f_{k+1}^T \nabla f_{k+1}}{\nabla f_{k}^T \nabla f_{k}}$$

But in an system of equations, $$\nabla f$$ results in the Jacobian. What do I miss?

• Nonlinear CG as you’ve described it with your formula is a method for function minimization. You can use it to minimize the sum of squares of the residuals if you want to. – Brian Borchers May 16 '20 at 20:21
• For $\pmb{F}(\pmb{x}) = \pmb{0}$, use $f(\pmb{x}) := \pmb{F}(\pmb{x})^{\top} \pmb{F}(\pmb{x})$. – Christoph May 17 '20 at 3:44