A Gauss–Newton-like conjugate gradient method for large-scale nonlinear equations and image restoration problems

In this paper, we present a Gauss–Newton-like conjugate gradient method for solving large-scale nonlinear equations. This new method can essentially be regarded as a spectral three-term conjugate gradient method, where the spectral parameter is designed based on an approximate Gauss–Newton direction and the secant equation. Global convergence is established under appropriate conditions. Numerical experiments demonstrate that the presented method is more effective than other existing methods in solving large-scale nonlinear equations. Moreover, this new method exhibits significant advantages in image restoration problems.