Improbability of nonconvergence in a cubic root-finding method (Q1107954)

This paper is concerned with the structure of the nonconvergent set NC(Ef) of initial points for which the Euler method for finding the roots of real functions \[ x_{k+1}=Ef(x_ k),\quad where\quad Ef(x)=x- [f(x)/f'(x)][1+f(x)\cdot f''(x)/2(f'(x))^ 2] \] fails to converge. The theorem is proved, that NC(Ef) is a Cantor set of Lebesgue measure zero. As referred in the paper, the conditions of the theorem (they are satisfied, for example, if f is a polynomial with all roots real) are adapted from similar conditions used by \textit{D. Saari} and \textit{J. Urenko} [Am. Math. Mon. 91, 3-17 (1984; Zbl 0532.58016)] and \textit{J. Urenko} [J. Math. Anal. Appl. 117, 42-47 (1986; Zbl 0634.58008)], where a similar result was shown for Newton's method. The theorem is established by reducing questions about the Euler method to a problem on Newton's method.