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Zbl 0908.65031
Ezquerro, J.A.; Hernández, M.A.; Salanova, M.A.
Construction of iterative processes with high order of convergence.
(English)
[J] Int. J. Comput. Math. 69, No.1-2, 191-201 (1998). ISSN 0020-7160; ISSN 1029-0265/e

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.
[A.Swift (Palmerston North)]
MSC 2000:
*65H05 Single nonlinear equations (numerical methods)

Keywords: nonlinear equations; Newton's method; high-order convergence; numerical examples

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