Abstract:

 In this paper, a new modification of the standard Newton method for approximating the root of a univariate function is introduced. Two evaluations of function and two evaluations of its first derivative are required at a cost of one additional function and first derivative evaluations per iteration. The modified method converges faster with the order of convergence  2+\sqrt{6}  compared with 2 for the standard Newton method. Numerical examples demonstrate the new
algorithm has advantages in the iteration number, computation time and optimal value compared with the current algorithms. At last, the predictor-corrector improvement is generalized to multi-dimensional vector-valued functions, its convergence is proved using Taylor formula, and two two-dimensional examples are given to illustrate its convergence.

Key words: Newton method, order of convergence, nonlinear equation