On the rate of convergence of two-term recursions (Q1102474)

Let B be a compact interval in \({\mathbb{R}}\), \(M=B\times B\) and \(\phi\) : \(M\to B\) a map in \(C_ 3(M)\). Suppose that \(\xi\) is a fixed point of \(\phi\). We study the behaviour of the iterates \(x_{n+2}=\phi (x_{n+1},x_ n)\) \((x_ 0,x_ 1\in B)\). Of particular interest is the situation where \(\phi _ x(\xi,\xi)=\phi _ y(\xi,\xi)=0\). In case of the well-known ``Regula falsi'' we also have \(\phi _{xx}(\xi,\xi)=\phi _{yy}(\xi,\xi)=0\) and the order of convergence is \((1+\sqrt{5})\). We consider the case where \(\phi _{yy}(\xi,\xi)\neq 0\). It turns out that there is a constant \(\gamma\) \(\in (1,2)\) such that successive iterates gain factors \(\gamma\),2/\(\gamma\),\(\gamma\),2/\(\gamma\),... on the number of valid decimals. Depending on the initial values \(x_ 0,x_ 1\) the number \(\gamma\) may range over all of (1,2) such that it may occur that an additional iterative step has virtually no effect on the number of correct digits or nearly doubles them.