Two-step relaxation Newton algorithm for solving nonlinear algebraic equations

We introduce a new algorithm, namely two-step relaxation Newton, for solving algebraic nonlinear equations f(x)=0. This new algorithm is derived by combining two different relaxation Newton algorithms introduced by Wu et al. (Appl. Math. Comput. 201:553–560, 2008), and therefore with special choice of the so called splitting function it can be implemented simultaneously, stably with much less memory storage and CPU time compared with the Newton–Raphson method. Global convergence of this algorithm is established and numerical experiments show that this new algorithm is feasible and effective, and outperforms the original relaxation Newton algorithm and the Newton–Raphson method in the sense of iteration number and CPU time.