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

Gerlach, Jürgen

SIAM Rev. 36, No. 2, 272-276 (1994).

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.

Reviewer: B.Döring (Düsseldorf)

MSC:

65H05

Numerical computation of solutions to single equations

Keywords:

accelerated convergence; one-point stationary iterative methods of higher order; Newton’s method; Halley’s method

Citations:

Zbl 0375.65025; Zbl 0486.65035