On convergence criteria for Jarratt's method (Q1064014)

Let f(\(\chi)\) together with its first two derivatives be continuous in the domain \({\mathbb{D}}\) and additionally let \(\chi_ M\in {\mathbb{D}}\) be an extremum (or turning point) of this function. Also, let \(\chi_{n+1}=T(\chi_ n,\chi_{n-1},\chi_{n-2})\) be the value obtained by the method of \textit{P. Jarratt} [Computer J. 10, 82-84 (1967; Zbl 0154.169)] for computing the extremum (or turning point) of a function. Criteria are demonstrated which insure that, for any triple of initial assumptions \((\chi_ 1,\chi_ 0,\chi_{-1})\in {\mathbb{D}}\), Jarratt's method converges to the extremum of f(\(\chi)\), and that from and after some \(n=N_ 0\), the rate of convergence of this method increases steadily, finally becoming unbounded when the solution \(\chi_ M\) is attained.