## Iterations of Waring polynomials and convergence acceleration

*Chizhong Zhou*

`chizhongz@163.com`

Computer & Informati

**Yueyang Normal University**

Yueyang Normal University

PR China

### 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.

© ATCM, Inc. 2001.