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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 $m\in\bbfN$. It is based on approximating the tangent to the curve $y= f(x)$ at a point $(x^*,0)$, giving rise to the function $$g(x)= f(x)- \sum^{m- 1}_{k=2} {f^{(k)}(x^*)\over k!} (x- x^*)^k.$$ 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.

65H05Single nonlinear equations (numerical methods)
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