That is precisely how trust region methods work for nonlinear least squares problems and nonlinear systems. MINPACK, which both SciPy and Matlab use, defaults to this kind of TR scheme (without the matrix-free part). We do have TR comparisons in the paper.
Note that we don't however solve the normal form J'J system as that is generally bad (unless user opts in to doing so) and instead use least squares formulation which is more numeraically stable
Note that we don't however solve the normal form J'J system as that is generally bad (unless user opts in to doing so) and instead use least squares formulation which is more numeraically stable