PhD's web site

Maple package dcfun

Publications

MathExp Team

With $\vartheta =2\pi /5$, define two matrices $$A_0=\left(\begin{array}{cc}\cos \vartheta & 0\\ 0& \cos \vartheta \end{array}\right),A_1=\left(\begin{array}{cc}0& -\sin \vartheta \\ \sin \vartheta & 0\end{array}\right),$$ and an initial vector $$v_0={\left(\begin{array}{cc}1& 0\end{array}\right)}^T.$$ Next consider the sequence $(v_n)$ defined by $v_0$ and the iteration $$v_{2n}=A_0{v}_n,v_{2n+1}=A_1{v}_n\text{for }n\ge 0\text{.}$$ The picture shows the values of the vector sequence $s_N$ for $2^{11}\le N<2^{15}$ where $s_N$ is defined as the summatory function $$s_N=\sum _{n=0}^N{v}_n,N\ge 0.$$

It can be shown (See here.) that the sequence $s_N$ has the following expansion $$s_N\underset{N\to +\mathrm{\infty }}{=}\left(\begin{array}{cc}\cos (\vartheta {\log }_{2}N)& -\sin (\vartheta {\log }_{2}N)\\ \sin (\vartheta {\log }_{2}N)& \cos (\vartheta {\log }_{2}N)\end{array}\right)\mathrm{\Phi }({\log }_{2}N)+O(N^{-0.07}),$$ where $\mathrm{\Phi }(t)$ is a $1$-periodic vector function. This explains why, when $N$ is doubled, the drawing is rotated by a fifth of a turn.