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On odd/even and signum - libzahl - big integer library |
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--- |
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commit 8c2c44669b49e9f6bc95f08b2505b11b9b66082f |
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parent 821d9893f5749db996a3a384c50c75ed92bafe2e |
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Author: Mattias Andrée <[email protected]> |
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Date: Fri, 13 May 2016 16:56:12 +0200 |
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On odd/even and signum |
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Signed-off-by: Mattias Andrée <[email protected]> |
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Diffstat: |
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M doc/number-theory.tex | 90 +++++++++++++++++++++++++++++… |
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1 file changed, 87 insertions(+), 3 deletions(-) |
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--- |
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diff --git a/doc/number-theory.tex b/doc/number-theory.tex |
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@@ -1,7 +1,8 @@ |
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\chapter{Number theory} |
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\label{chap:Number theory} |
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-TODO |
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+In this chapter, you will learn about the |
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+number theoretic functions in libzahl. |
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\vspace{1cm} |
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\minitoc |
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@@ -11,14 +12,97 @@ TODO |
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\section{Odd or even} |
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\label{sec:Odd or even} |
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-TODO % zodd zeven zodd_nonzero zeven_nonzero |
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+There are four functions available for testing |
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+the oddness and evenness of an integer: |
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+ |
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+\begin{alltt} |
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+ int zodd(z_t a); |
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+ int zeven(z_t a); |
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+ int zodd_nonzero(z_t a); |
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+ int zeven_nonzero(z_t a); |
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+\end{alltt} |
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+ |
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+\noindent |
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+{\tt zodd} returns 1 if {\tt a} contains an |
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+odd value, or 0 if {\tt a} contains an even |
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+number. Conversely, {\tt zeven} returns 1 if |
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+{\tt a} contains an even value, or 0 if {\tt a} |
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+contains an odd number. {\tt zodd\_nonzero} and |
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+{\tt zeven\_nonzero} behave exactly like {\tt zodd} |
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+and {\tt zeven}, respectively, but assumes that |
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+{\tt a} contains a non-zero value, if not |
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+undefined behaviour is invoked, possibly in the |
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+form of a segmentation fault; they are thus |
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+sligtly faster than {\tt zodd} and {\tt zeven}. |
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+ |
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+It is discouraged to test the returned value |
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+against 1, we should always test against 0, |
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+treating all non-zero value as equivalent to 1. |
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+For clarity, we use also avoid testing that |
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+the returned value is zero, for example, rather |
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+than {\tt !zeven(a)} we write {\tt zodd(a)}. |
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\newpage |
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\section{Signum} |
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\label{sec:Signum} |
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-TODO % zsignum zzero |
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+There are two functions available for testing |
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+the sign of an integer, one of the can be used |
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+to retrieve the sign: |
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+ |
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+\begin{alltt} |
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+ int zsignum(z_t a); |
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+ int zzero(z_t a); |
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+\end{alltt} |
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+ |
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+\noindent |
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+{\tt zsignum} returns $-1$ if $a < 0$, |
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+$0$ if $a = 0$, and $+1$ if $a > 0$, that is, |
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+ |
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+\vspace{1em} |
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+\( \displaystyle{ |
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+ \mbox{sgn}~a = \left \lbrace \begin{array}{rl} |
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+ -1 & \textrm{if}~ a < 0 \\ |
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+ 0 & \textrm{if}~ a = 0 \\ |
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+ +1 & \textrm{if}~ a > 0 |
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+ \end{array} \right . |
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+}\) |
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+\vspace{1em} |
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+ |
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+\noindent |
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+It is discouraged to compare the returned value |
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+against $-1$ and $+1$; always compare against 0, |
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+for example: |
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+ |
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+\begin{alltt} |
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+ if (zsignum(a) > 0) "positive"; |
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+ if (zsignum(a) >= 0) "non-negative"; |
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+ if (zsignum(a) == 0) "zero"; |
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+ if (!zsignum(a)) "zero"; |
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+ if (zsignum(a) <= 0) "non-positive"; |
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+ if (zsignum(a) < 0) "negative"; |
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+ if (zsignum(a)) "non-zero"; |
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+\end{alltt} |
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+ |
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+\noindent |
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+However, when we are doing arithmetic with the |
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+signum, we may relay on the result never being |
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+any other value than $-1$, $0$, and $+0$. |
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+For example: |
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+ |
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+\begin{alltt} |
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+ zset(sgn, zsignum(a)); |
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+ zadd(b, sgn); |
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+\end{alltt} |
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+ |
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+{\tt zzero} returns 0 if $a = 0$ or 1 if |
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+$a \neq 0$. Like with {\tt zsignum}, avoid |
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+testing the returned value against 1, rather |
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+test that the returned value is not 0. When |
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+however we are doing arithmetic with the |
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+result, we may relay on the result never |
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+being any other value than 0 or 1. |
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\newpage |
