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Three-step iterative methods for nonlinear equations. (English) Zbl 1113.65050

A new three step iterative method for solving nonlinear equations f(x)=0 is introduced based on the following scheme: Let x 0 be an initial guess sufficiently close to a simple root of the equation f(x)=0. The iterative step consists two predictor steps:

y n =x n -f(x n )/f(x n ),f ' (x n )0;z n =-(y n -x n ) 2 ·f '' (x n )/2·f ' (x n )

and one corrector step:

x n+1 =x n -f(x n )f ' (x n )-(y n +x n ) 2 ·f '' (x n )/2·f ' (x n )-(y n +z n -x n ) 2 ·f '' (x n )/2·f ' (x n ),

n=0,1,2,. The authors show that if the function f is sufficiently differentiable on an open interval which contains a single root, and if x 0 is sufficiently close to this root, then the proposed iterative algorithm has the fourth-order of convergence. Several numerical examples are given to illustrate the efficiency and performance of the new method.

65H05Single nonlinear equations (numerical methods)