Iterations of Waring polynomials and convergence acceleration

Abstract

In this paper we introduce two polynomials: the $k$-Waring polynomial of the first kind \begin{equation*} p_k(x,y)=\sum_{i=0}^{[k/2]}(-1)^i\frac{k}{k-i}\binom{k-i}{i}x^{k-2i}y^i\;(k\in \Z^+), \end{equation*} and the $k$-Waring polynomial of the second kind \begin{equation*} q_k(x,y)=\sum_{i=0}^{[k/2]}\frac{k}{k-i}\binom{k-i}{i}\Delta^{\frac{k-1}{2}-i}x^{k-2i}y^i\;(k\in \Z^+,k \text{ is odd}), \end{equation*} where $\Delta =a^2+4b$. We also introduce two iterated sequences: the $k$-Waring sequence of the first kind $\{w_n(k;a,b)\}$ and the $k$-Waring sequence of the second kind $\{h_n(k;a,b)\}$ which are defined as \begin{equation*} w_{n+1}(k;a,b)=p_k(w_n(k;a,b),(-b)^n), \;w_0=a, \end{equation*} and \begin{equation*} h_{n+1}(k;a,b)=q_k(h_n(k;a,b),(-b)^n), \;h_0=1, \end{equation*} respectively. The explicit expressions of $w_n(k;a,b)$ and $h_n(k;a,b)$ are given. The applications of the two sequences to convergence acceleration including the Aitken acceleration are recommended. Fast algorithms for high accuracy computation of square-root and other quadratic irrational number are stated. As an example, we take $w_n(3;8,3)/(2\,h_n(3;8,3))$ as the approximate value of $\sqrt{19}$. When $n=3$ (i.e. Only $3$ iterations are needed) we get so accurate value that has 37 significant figures.