Solving systems of nonlinear equations using decomposition technique

A systematic way is presented for the construction of multi-step iterative method with frozen Jacobian. The inclusion of an auxiliary function is discussed. The presented analysis shows that how to incorporate auxiliary function in a way that we can keep the order of convergence and computational cost of Newton multi-step method. The auxiliary function provides us the way to overcome the singularity and ill-conditioning of the Jacobian. The order of convergence of proposed p-step iterative method is p + 1. Only one Jacobian inversion in the form of LU-factorization is required for a single iteration of the iterative method and in this way, it offers an efficient scheme. For the construction of our proposed iterative method, we used a decomposition technique that naturally provides different iterative schemes. We also computed the computational convergence order that confirms the claimed theoretical order of convergence. The developed iterative scheme is applied to large scale problems, and numerical results show that our iterative scheme is promising.