Algorithms for nonlinear least-squares problems the gauss-newton method

x*, giving rapid local convergence. When H(x*) = 0, the convergence is actually quadratic.
When n and m are both large and the Jacobian J(x) is sparse, the cost of computing steps exactly by factoring either J_k or J_k^TJ_k at each iteration may become quite expensive relative to the cost of function and gradient evaluations. In this case, we can design inexact variants of the Gauss–Newton algorithm that are analogous to the inexact Newton algorithms discussed in Chapter 7. We simply replace the Hessian ∇²f(x_k) in these methods by its approximation J_k^TJ_k . The positive semidefiniteness of this approximation simplifies the resulting algorithms in several places.