## Accelerated convergence in Newton’s method.(English)Zbl 0814.65046

G. H. Brown jun. [Am. Math. Monthly 84, 726-728 (1977; Zbl 0375.65025)] and G. Alefeld [ibid. 88, 530-536 (1981; Zbl 0486.65035)] applied Newton’s method to $$F(x):= f(x)/\sqrt{f'(x)}$$, $$f$$ being a real-valued function of a real variable, to solve $$f(x)= 0$$ approximately, and thus obtained Halley’s method for $$f$$. In the present paper this idea is used in a generalized form. It turns out that, among others, the following statement is true. Let $$m\in \mathbb{N}\backslash\{1, 2\}$$, $$f(a)= 0$$, $$f'(a)> 0$$, $$f''(a)=\cdots= f^{(m-1)}(a)= 0$$, $$f^{(m)}(a)\neq 0$$. Then Halley’s formula with 1/2 replaced by $$1/m$$ gives an iteration formula which converges of order at least $$m+1$$ to the (simple) zero $$a$$ of $$f$$ (provided the starting term in the sequence is chosen sufficiently close to $$a$$). Some examples with numerically computed errors are given.

### MSC:

 65H05 Numerical computation of solutions to single equations

### Citations:

Zbl 0375.65025; Zbl 0486.65035
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