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.


65H05 Numerical computation of solutions to single equations
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