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Construction of iterative processes with high order of convergence. (English) Zbl 0908.65031

The authors consider a recursive procedure to construct iterative processes with increasing convergence order m. It is based on approximating the tangent to the curve y=f(x) at a point (x * ,0), giving rise to the function

g(x)=f(x)- k=2 m-1 f (k) (x * ) k!(x-x * ) k ·

A Newton sequence is then found for g(x). Rather than find g ' (x) from g(x), another function h(x) is introduced in such a form as to obtain order m of convergence for the iteration x n+1 =x n -g(x n )/h(x n ). Computational efficiency is analysed and the paper concludes with some numerical examples.

MSC:
65H05Single nonlinear equations (numerical methods)