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\magnification=\magstep3\hsize=19truecm\vsize=19truecm\nopagenumbers\parindent=0mm\font\eins=cmb10 scaled \magstep 3\font\zwei=cmr12\font\mini=cmr7\def\frac#1#2{{{#1} \over {#2}}}\hbox{}\vfill
\centerline{\eins Binary Splitting}\bigskip\bigskipRecursive algorithm:\medskip\centerline{$\displaystyle S_{[n_1,n_2)} = {\sum\limits_{n=n_1}^{n_2-1} \frac{a(n)}{b(n)} \, \frac{p(n_1) \cdots p(n)}{q(n_1) \cdots q(n)}}$}\medskipCompute $P = {p(n_1) \cdots p(n_2-1)}$, $Q = {q(n_1) \cdots q(n_2-1)}$,\vskip 0cm$B = {b(n_1) \cdots b(n_2-1)}$ and $T$ with\medskip\centerline{$\displaystyle S_{[n_1,n_2)} = \frac{T}{B \cdot Q}$}\bigskip\quad $n_2 - n_1 < 4$ \quad $\rightarrow$ directly\medskip\quad $n_2 - n_1 \geq 4$ \quad $\rightarrow$ split\medskip\centerline{$P = P_L \cdot P_R$}\centerline{$Q = Q_L \cdot Q_R$}\centerline{$B = B_L \cdot B_R$}\centerline{$T = B_R \cdot Q_R \cdot T_L + B_L \cdot P_L \cdot T_R$}
\vfill\hbox{}\eject
\end
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