Stability of rounded off inverses under iteration

Abstract:

Let f be a monotone and strictly convex (or concave) function on a real interval and let g be the inverse function. Let $I(x) = x$. For $\phi$ a real valued function and N a positive integer let ${\phi _N}(x)$ denote the rounding of $\phi (x)$ to N significant figures. Let $h = {g_N} \circ {f_N}$ , the composition of ${f_N}$ and ${g_N}$. It is shown that \[ h \circ h \circ {I_N} = h \circ h \circ h \circ {I_N},\] and that equality can fail for fewer iterations.