  In this paper we derive a Newton type method to solve the non-linear system
formed by combining the Tikhonov normal equations and Morozov's discrepancy
principle. We prove that by placing a bound on the step size of the Newton
iterations the method will always converge to the solution. By projecting the
problem onto a low dimensional Krylov subspace and using the method to solve
the projected non-linear system we show that we can reduce the computational
cost of the method.
