Convergence rates for Kaczmarz-type algorithms

By making use of the normal and skew-Hermitian splitting (NSS) method as the inner solver for the modified Newton method, we establish a class of modified Newton-NSS method for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. Under proper conditions, the local convergence theorem is proved. Furthermore, the successive-overrelaxation (SOR) technique has been proved quite successfully in accelerating the convergence rate of the NSS or the Hermitian and skew-Hermitian splitting (HSS) iteration method, so we employ the SOR method in the NSS iteration, and we get a new method, which is called modified Newton SNSS method. Numerical results are given to examine its feasibility and effectiveness.