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An iterative method with cubic convergence for nonlinear equations. (English) Zbl 1113.65052

A new three step iterative method for solving a nonlinear equation f(x)=0 is introduced based on the following scheme: Let x 0 is an initial guess sufficiently close to simple root of the equation f(x)=0. The iterative step consists from 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 )/2f ' (x n ),

and one corrector step:

x n+1 =x n -f(x n )/f ' (x n )-(y n +z n -z n ) 2 ·f ' (x n )/2·f ' (x n ),n=0,1,2,·

The authors show that if the function f is sufficiently differentiable in the open interval, which contain a simple root of the equation f(x)=0 and if x 0 is sufficiently close to this root, then the proposed iterative algorithm has the order of convergence equal to three. Several numerical examples are given to illustrate the efficiency and performance of the new method.

65H05Single nonlinear equations (numerical methods)