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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)/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{1,2}, f(a)=0, f ' (a)>0, f '' (a)==f (m-1) (a)=0, f (m) (a)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.

65H05Single nonlinear equations (numerical methods)