How Many Steps Still Left to $x$*?

The high speed of $x_{k}\rightarrow x^\ast\in{\mathbb R}$ is usually measured using the $C$-, $Q$-, or $R$-orders: \[ \lim \frac {|x^\ast - x_{k+1}|}{|x^\ast x_k|^{p_0}}\in(0,+\infty), \qquad {\ln |x^\ast x_{k+1}|}{\ln x_k|} =q_0, \mbox{or} \quad \big\vert\ln x_k| \big \vert ^\frac1k =r_0. \] By connecting them to natural, term-by-term comparison errors two sequences, we find that $C$-orders---including (sub)linear---are in agreement. Weird relations may appear though for $Q$-orders: expect $|x^\ast x_k|=\mathcal{O}(|x^\ast y_k|^\alpha)$ $\forall\alpha>1$ imply “$\geq$" $Q$-orders $\{x_{k}\}$ vs. $\{y_{k}\}$; contrary shown by an example providing no infinite $Q$-orders. $R$-orders be even worse: with $R$-order have unbounded nonmonotone errors: \(\limsup \frac{|x^\ast x_k|}= +\infty\). Such aspects motivate study equivalent definitions, computational variants, and so on. These orders are also perspective from which analyze three basic iterative methods nonlinear equations $\mathbb R$. Newton method, widely known its quadratic convergence, fact attain any $C$-order $[1,+\infty]$ (including sublinear); briefly recall such convergence results, along connected (such as historical notes, asymptotic constants, floating point arithmetic issues, radius attraction balls), provide examples. This approach leads similar results successive approximations while secant method exhibits different behavior: it not $C$-orders, but only