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	On greatest common divisor - libzahl - big integer library
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	commit 7214b27058765ea3892061e846c601499892c48d
	parent 8c2c44669b49e9f6bc95f08b2505b11b9b66082f
	Author: Mattias Andrée <[email protected]>
	Date: Fri, 13 May 2016 18:39:07 +0200

	On greatest common divisor

	Signed-off-by: Mattias Andrée <[email protected]>

	Diffstat:
	M doc/number-theory.tex \| 70 +++++++++++++++++++++++++++++…

	1 file changed, 69 insertions(+), 1 deletion(-)
	---
	diff --git a/doc/number-theory.tex b/doc/number-theory.tex
	@@ -109,7 +109,75 @@ being any other value than 0 or 1.
	\section{Greatest common divisor}
	\label{sec:Greatest common divisor}

	-TODO % zgcd
	+There is no single agreed upon definition
	+for the greatest common divisor of two
	+integer, that cover non-positive integers.
	+In libzahl we define it as
	+
	+\vspace{1em}
	+\( \displaystyle{
	+ \gcd(a, b) = \left \lbrace \begin{array}{rl}
	+ -k & \textrm{if}~ a < 0, b < 0 \\
	+ b & \textrm{if}~ a = 0 \\
	+ a & \textrm{if}~ b = 0 \\
	+ k & \textrm{otherwise}
	+ \end{array} \right .
	+}\),
	+\vspace{1em}
	+
	+\noindent
	+where $k$ is the largest integer that divides
	+both $\lvert a \rvert$ and $\lvert b \rvert$. This
	+definion ensures
	+
	+\vspace{1em}
	+\( \displaystyle{
	+ {a \over \gcd(a, b)} \left \lbrace \begin{array}{rl}
	+ > 0 & \textrm{if}~ a < 0, b < 0 \\
	+ < 0 & \textrm{if}~ a < 0, b > 0 \\
	+ = 1 & \textrm{if}~ b = 0, a \neq 0 \\
	+ = 0 & \textrm{if}~ a = 0, b \neq 0 \\
	+ \in \textbf{N} & \textrm{otherwise if}~ a \neq 0, b \neq 0
	+ \end{array} \right .
	+}\),
	+\vspace{1em}
	+
	+\noindent
	+and analogously for $b \over \gcd(a,\,b)$. Note however,
	+the convension $\gcd(0, 0) = 0$ is adhered. Therefore,
	+before dividing with $\gcd{a, b}$ you may want to check
	+whether $\gcd(a, b) = 0$. $\gcd(a, b)$ is calculated
	+with {\tt zgcd(a, b)}.
	+
	+{\tt zgcd} calculates the greatest common divisor using
	+the Binary GCD algorithm.
	+
	+\vspace{1em}
	+\hspace{-2.8ex}
	+\begin{minipage}{\linewidth}
	+\begin{algorithmic}
	+ \IF{$ab = 0$}
	+ \RETURN $a + b$
	+ \ELSIF{$a < 0$ \AND $b < 0$}
	+ \RETURN $-\gcd(\lvert a \rvert, \lvert b \rvert)$
	+ \ENDIF
	+ \STATE $s \gets \max s : 2^s \vert a, b$
	+ \STATE $u, v \gets \lvert a \rvert \div 2^s, \lvert b \rvert \div 2^s$
	+ \WHILE{$u \neq v$}
	+ \IF{$u > v$}
	+ \STATE $u \leftrightarrow v$
	+ \ENDIF
	+ \STATE $v \gets v - u$
	+ \STATE $v \gets v \div 2^x$, where $x = \max x : 2^x \vert v$
	+ \ENDWHILE
	+ \RETURN $u \cdot 2^s$
	+\end{algorithmic}
	+\end{minipage}
	+\vspace{1em}
	+
	+\noindent
	+$\max x : 2^x \vert z$ is returned by {\tt zlsb(z)}
	+\psecref{sec:Boundary}.


	\newpage