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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 $F\left(x\right):=f\left(x\right)/\sqrt{{f}^{\text{'}}\left(x\right)}$, $f$ being a real-valued function of a real variable, to solve $f\left(x\right)=0$ 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 $m\in ℕ\setminus \left\{1,2\right\}$, $f\left(a\right)=0$, ${f}^{\text{'}}\left(a\right)>0$, ${f}^{\text{'}\text{'}}\left(a\right)=\cdots ={f}^{\left(m-1\right)}\left(a\right)=0$, ${f}^{\left(m\right)}\left(a\right)\ne 0$. 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 Single nonlinear equations (numerical methods)