Matrix-free iterative processes with least-squares error damping for nonlinear systems of equations

New iterative processes for numerical solution of big nonlinear systems of equations are considered. The processes do not require factorization and storing of Jacobi matrix and employ a special technique of convergence acceleration which is called least-squares error damping and requires solution of auxiliary linear least-squares problems of low dimension. In linear case the resulting method is similar to the general minimal residual method (GMRES) with preconditioning. In nonlinear case, in contrast to popular Newton – Krylov method, the computational scheme do not involve operation of difference approximation of derivative operator. Numerical experiments include three nonlinear problems originating from two-dimensional elliptic partial differential equations and exhibit advantage of the proposed method compared to Newton – Krylov method.