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version 1.1.1.1, 1999/12/08 05:47:44 version 1.4, 2001/03/12 05:01:19
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   @comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/num.texi,v 1.3 2000/01/13 08:29:57 noro Exp $
   \BJP
 @node $B?t$N1i;;(B,,, $BAH$_9~$_H!?t(B  @node $B?t$N1i;;(B,,, $BAH$_9~$_H!?t(B
 @section $B?t$N1i;;(B  @section $B?t$N1i;;(B
   \E
   \BEG
   @node Numbers,,, Built-in Function
   @section Numbers
   \E
   
 @menu  @menu
 * idiv irem::  * idiv irem::
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 * mt_save mt_load::  * mt_save mt_load::
 * nm dn::  * nm dn::
 * conj real imag::  * conj real imag::
 * eval::  * eval deval::
 * pari::  * pari::
 * setprec::  * setprec::
 * setmod::  * setmod::
 * lrandom::  * lrandom::
   * ntoint32 int32ton::
 @end menu  @end menu
   
 @node idiv irem,,, $B?t$N1i;;(B  \JP @node idiv irem,,, $B?t$N1i;;(B
   \EG @node idiv irem,,, Numbers
 @subsection @code{idiv}, @code{irem}  @subsection @code{idiv}, @code{irem}
 @findex idiv  @findex idiv
 @findex irem  @findex irem
   
 @table @t  @table @t
 @item idiv(@var{i1},@var{i2})  @item idiv(@var{i1},@var{i2})
 :: $B@0?t=|;;$K$h$k>&(B.  \JP :: $B@0?t=|;;$K$h$k>&(B.
   \EG :: Integer quotient of @var{i1} divided by @var{i2}.
 @item irem(@var{i1},@var{i2})  @item irem(@var{i1},@var{i2})
 :: $B@0?t=|;;$K$h$k>jM>(B.  \JP :: $B@0?t=|;;$K$h$k>jM>(B.
   \EG :: Integer remainder of @var{i1} divided by @var{i2}.
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @item i1,i2  @item i1,i2
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 @var{i1} $B$N(B @var{i2} $B$K$h$k@0?t=|;;$K$h$k>&(B, $B>jM>$r5a$a$k(B.  @var{i1} $B$N(B @var{i2} $B$K$h$k@0?t=|;;$K$h$k>&(B, $B>jM>$r5a$a$k(B.
 @item  @item
Line 50 
Line 64 
 @code{irem()} $B$NBe$o$j$KMQ$$$k$3$H$,$G$-$k(B.  @code{irem()} $B$NBe$o$j$KMQ$$$k$3$H$,$G$-$k(B.
 @item  @item
 $BB?9`<0$N>l9g$O(B @code{sdiv}, @code{srem} $B$rMQ$$$k(B.  $BB?9`<0$N>l9g$O(B @code{sdiv}, @code{srem} $B$rMQ$$$k(B.
   \E
   \BEG
   @item
   Integer quotient and remainder of @var{i1} divided by @var{i2}.
   @item
   @var{i2} must not be 0.
   @item
   If the dividend is negative, the results are obtained by changing the
   sign of the results for absolute values of the dividend.
   @item
   One can use
   @var{i1} @code{%} @var{i2}
   for replacement of @code{irem()} which only differs in the point that
   the result is always normalized to non-negative values.
   @item
   Use @code{sdiv()}, @code{srem()} for polynomial quotient.
   \E
 @end itemize  @end itemize
   
 @example  @example
Line 64 
Line 95 
 @end example  @end example
   
 @table @t  @table @t
 @item $B;2>H(B  \JP @item $B;2>H(B
   \EG @item References
 @fref{sdiv sdivm srem sremm sqr sqrm}, @fref{%}.  @fref{sdiv sdivm srem sremm sqr sqrm}, @fref{%}.
 @end table  @end table
   
 @node fac,,, $B?t$N1i;;(B  \JP @node fac,,, $B?t$N1i;;(B
   \EG @node fac,,, Numbers
 @subsection @code{fac}  @subsection @code{fac}
 @findex fac  @findex fac
   
 @table @t  @table @t
 @item fac(@var{i})  @item fac(@var{i})
 :: @var{i} $B$N3,>h(B.  \JP :: @var{i} $B$N3,>h(B.
   \EG :: The factorial of @var{i}.
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @item i  @item i
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 @var{i} $B$N3,>h$r7W;;$9$k(B.  @var{i} $B$N3,>h$r7W;;$9$k(B.
 @item  @item
 @var{i} $B$,Ii$N>l9g$O(B 0 $B$rJV$9(B.  @var{i} $B$,Ii$N>l9g$O(B 0 $B$rJV$9(B.
   \E
   \BEG
   @item
   The factorial of @var{i}.
   @item
   Returns 0 if the argument @var{i} is negative.
   \E
 @end itemize  @end itemize
   
 @example  @example
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Line 140 
 30414093201713378043612608166064768844377641568960512000000000000  30414093201713378043612608166064768844377641568960512000000000000
 @end example  @end example
   
 @node igcd igcdcntl,,, $B?t$N1i;;(B  \JP @node igcd igcdcntl,,, $B?t$N1i;;(B
   \EG @node igcd igcdcntl,,, Numbers
 @subsection @code{igcd},@code{igcdcntl}  @subsection @code{igcd},@code{igcdcntl}
 @findex igcd  @findex igcd
 @findex igcdcntl  @findex igcdcntl
   
 @table @t  @table @t
 @item igcd(@var{i1},@var{i2})  @item igcd(@var{i1},@var{i2})
 :: $B@0?t$N(B GCD ($B:GBg8xLs?t(B)  \JP :: $B@0?t$N(B GCD ($B:GBg8xLs?t(B)
   \EG :: The integer greatest common divisor of @var{i1} and @var{i2}.
 @item igcdcntl([@var{i}])  @item igcdcntl([@var{i}])
 :: $B@0?t(B GCD$B$N%"%k%4%j%:%`A*Br(B  \JP :: $B@0?t(B GCD$B$N%"%k%4%j%:%`A*Br(B
   \EG :: Selects an algorithm for integer GCD.
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @item i1,i2,i  @item i1,i2,i
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 @code{igcd} $B$O(B @var{i1} $B$H(B @var{i2} $B$N(B GCD $B$r5a$a$k(B.  @code{igcd} $B$O(B @var{i1} $B$H(B @var{i2} $B$N(B GCD $B$r5a$a$k(B.
 @item  @item
Line 135  bmod GCD
Line 185  bmod GCD
 @item 3  @item 3
 accelerated integer GCD  accelerated integer GCD
 @end table  @end table
 2, 3 $B$O(B K. Weber, ACM TOMS, Vol.21, No. 1 (1995), pp. 111-122 $B$K$h$k(B.  @code{2}, @code{3} $B$O(B @code{[Weber]} $B$K$h$k(B.
   
 $B$*$*$`$M(B 3 $B$,9bB.$@$,(B, $BNc30$b$"$k(B.  $B$*$*$`$M(B @code{3} $B$,9bB.$@$,(B, $BNc30$b$"$k(B.
   \E
   \BEG
   @item
   Function @code{igcd()} returns the integer greatest common divisor of
   the given two integers.
   @item
   An error will result if the argument is not an integer; the result is
   not valid even if one is returned.
   @item
   Use @code{gcd()}, @code{gcdz()} for polynomial GCD.
   
   @item
   Various method of integer GCD computation are implemented
   and they can be selected by @code{igcdcntl}.
   
   @table @code
   @item 0
   Euclid algorithm (default)
   @item 1
   binary GCD
   @item 2
   bmod GCD
   @item 3
   accelerated integer GCD
   @end table
   @code{2}, @code{3} are due to @code{[Weber]}.
   
   In most cases @code{3} is the fastest, but there are exceptions.
   \E
 @end itemize  @end itemize
   
 @example  @example
Line 161  accelerated integer GCD
Line 240  accelerated integer GCD
 @end example  @end example
   
 @table @t  @table @t
 @item $B;2>H(B  \JP @item $B;2>H(B
   \EG @item References
 @fref{gcd gcdz}.  @fref{gcd gcdz}.
 @end table  @end table
   
 @node ilcm,,, $B?t$N1i;;(B  \JP @node ilcm,,, $B?t$N1i;;(B
   \EG @node ilcm,,, Numbers
 @subsection @code{ilcm}  @subsection @code{ilcm}
 @findex ilcm  @findex ilcm
   
 @table @t  @table @t
 @item ilcm(@var{i1},@var{i2})  @item ilcm(@var{i1},@var{i2})
 :: $B:G>.8xG\?t$r5a$a$k(B.  \JP :: $B:G>.8xG\?t$r5a$a$k(B.
   \EG :: The integer least common multiple of @var{i1} and @var{i2}.
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @item i1,i2  @item i1,i2
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 $B@0?t(B @var{i1}, @var{i2} $B$N:G>.8xG\?t$r5a$a$k(B.  $B@0?t(B @var{i1}, @var{i2} $B$N:G>.8xG\?t$r5a$a$k(B.
 @item  @item
 $B0lJ}$,(B 0 $B$N>l9g(B 0 $B$rJV$9(B.  $B0lJ}$,(B 0 $B$N>l9g(B 0 $B$rJV$9(B.
   \E
   \BEG
 @item  @item
   This function computes the integer least common multiple of
   @var{i1}, @var{i2}.
   @item
   If one of argument is equal to 0, the return 0.
   \E
 @end itemize  @end itemize
   
 @table @t  @table @t
 @item $B;2>H(B  \JP @item $B;2>H(B
   \EG @item References
 @fref{igcd igcdcntl}, @fref{mt_save mt_load}.  @fref{igcd igcdcntl}, @fref{mt_save mt_load}.
 @end table  @end table
 @node inv,,, $B?t$N1i;;(B  
   \JP @node inv,,, $B?t$N1i;;(B
   \EG @node inv,,, Numbers
 @subsection @code{inv}  @subsection @code{inv}
 @findex inv  @findex inv
   
 @table @t  @table @t
 @item inv(@var{i},@var{m})  @item inv(@var{i},@var{m})
 :: @var{m} $B$rK!$H$9$k(B @var{i} $B$N5U?t(B  \JP :: @var{m} $B$rK!$H$9$k(B @var{i} $B$N5U?t(B
   \EG :: the inverse (reciprocal) of @var{i} modulo @var{m}.
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @item i,m  @item i,m
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 @var{ia} @equiv{} 1 mod (@var{m}) $B$J$k@0?t(B @var{a} $B$r5a$a$k(B.  @var{ia} @equiv{} 1 mod (@var{m}) $B$J$k@0?t(B @var{a} $B$r5a$a$k(B.
 @item  @item
 @var{i} $B$H(B @var{m} $B$O8_$$$KAG$G$J$1$l$P$J$i$J$$$,(B, @code{inv()} $B$O(B  @var{i} $B$H(B @var{m} $B$O8_$$$KAG$G$J$1$l$P$J$i$J$$$,(B, @code{inv()} $B$O(B
 $B$=$N%A%'%C%/$O9T$o$J$$(B.  $B$=$N%A%'%C%/$O9T$o$J$$(B.
   \E
   \BEG
   @item
   This function computes an integer such that
   @var{ia} @equiv{} 1 mod (@var{m}).
   @item
   The integer @var{i} and  @var{m} must be mutually prime.
   However, @code{inv()} does not check it.
   \E
 @end itemize  @end itemize
   
 @example  @example
Line 228  accelerated integer GCD
Line 335  accelerated integer GCD
 @end example  @end example
   
 @table @t  @table @t
 @item $B;2>H(B  \JP @item $B;2>H(B
   \EG @item References
 @fref{igcd igcdcntl}.  @fref{igcd igcdcntl}.
 @end table  @end table
   
 @node prime lprime,,, $B?t$N1i;;(B  \JP @node prime lprime,,, $B?t$N1i;;(B
   \EG @node prime lprime,,, Numbers
 @subsection @code{prime}, @code{lprime}  @subsection @code{prime}, @code{lprime}
 @findex prime  @findex prime
 @findex lprime  @findex lprime
Line 240  accelerated integer GCD
Line 349  accelerated integer GCD
 @table @t  @table @t
 @item prime(@var{index})  @item prime(@var{index})
 @item lprime(@var{index})  @item lprime(@var{index})
 :: $BAG?t$rJV$9(B  \JP :: $BAG?t$rJV$9(B
   \EG :: Returns a prime number.
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @item index  @item index
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 @code{prime()}, @code{lprime()} $B$$$:$l$b%7%9%F%`$,FbIt$K;}$D(B  @code{prime()}, @code{lprime()} $B$$$:$l$b%7%9%F%`$,FbIt$K;}$D(B
 $BAG?tI=$NMWAG$rJV$9(B. @code{index} $B$O(B 0 $B0J>e$N@0?t$G(B, $BAG?tI=(B  $BAG?tI=$NMWAG$rJV$9(B. @code{index} $B$O(B 0 $B0J>e$N@0?t$G(B, $BAG?tI=(B
Line 259  accelerated integer GCD
Line 372  accelerated integer GCD
 $BAG?t$+$iBg$-$$=g$K(B 999 $B8DJV$9(B. $B$=$l0J30$N%$%s%G%C%/%9$KBP$7$F$O(B  $BAG?t$+$iBg$-$$=g$K(B 999 $B8DJV$9(B. $B$=$l0J30$N%$%s%G%C%/%9$KBP$7$F$O(B
 0 $B$rJV$9(B.  0 $B$rJV$9(B.
 @item  @item
 $B$h$j0lHLE*$JAG?t@8@.H!?t$H$7$F$O(B, @code{pari(nextprime,@var{number})}  $B$h$j0lHLE*$JAG?t@8@.H!?t$H$7$F$O(B,
   @code{pari(nextprime,@var{number})}
 $B$,$"$k(B.  $B$,$"$k(B.
   \E
   \BEG
   @item
   The two functions, @code{prime()} and @code{lprime()}, returns
   an element stored in the system table of prime numbers.
   Here, @code{index} is a non-negative integer and be used as an index
   for the prime tables.
   The function @code{prime()} can return one of 1900 primes
   up to 16381 indexed so that the smaller one has smaller
   index.  The function @code{lprime()} can return one of 999 primes which
   are 8 digit sized and indexed so that the larger one has the smaller
   index.
   The two function always returns 0 for other indices.
   @item
   For more general function for prime generation, there is a @code{PARI}
   function
   
   @code{pari(nextprime,@var{number})}.
   \E
 @end itemize  @end itemize
   
 @example  @example
Line 275  accelerated integer GCD
Line 408  accelerated integer GCD
 @end example  @end example
   
 @table @t  @table @t
 @item $B;2>H(B  \JP @item $B;2>H(B
   \EG @item References
 @fref{pari}.  @fref{pari}.
 @end table  @end table
   
 @node random,,, $B?t$N1i;;(B  \JP @node random,,, $B?t$N1i;;(B
   \EG @node random,,, Numbers
 @subsection @code{random}  @subsection @code{random}
 @findex random  @findex random
   
 @table @t  @table @t
 @item radom([@var{seed}])  @item radom([@var{seed}])
 :: $BMp?t$r@8@.$9$k(B.  \JP :: $BMp?t$r@8@.$9$k(B.
 @end table  @end table
   
 @table @var  @table @var
 @item seed  @item seed
 @item return  @itemx return
 $B<+A3?t(B  \JP $B<+A3?t(B
   \EG non-negative integer
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 $B:GBg(B 2^32-1 $B$NHsIi@0?t$NMp?t$r@8@.$9$k(B.  $B:GBg(B 2^32-1 $B$NHsIi@0?t$NMp?t$r@8@.$9$k(B.
 @item  @item
Line 310  default $B$N(B seed $B$O8GDj$N$?$a(B, $B<o$r@_Dj$
Line 447  default $B$N(B seed $B$O8GDj$N$?$a(B, $B<o$r@_Dj$
 @code{mt_save} $B$K$h$j(B state $B$r%U%!%$%k$K(B save $B$G$-$k(B. $B$3$l$r(B @code{mt_load}  @code{mt_save} $B$K$h$j(B state $B$r%U%!%$%k$K(B save $B$G$-$k(B. $B$3$l$r(B @code{mt_load}
 $B$GFI$_9~$`$3$H$K$h$j(B, $B0[$k(B Asir $B%;%C%7%g%s4V$G0l$D$NMp?t$N7ONs$rC)$k$3$H$,(B  $B$GFI$_9~$`$3$H$K$h$j(B, $B0[$k(B Asir $B%;%C%7%g%s4V$G0l$D$NMp?t$N7ONs$rC)$k$3$H$,(B
 $B$G$-$k(B.  $B$G$-$k(B.
   \E
   \BEG
   @item
   Generates a random number which is a non-negative integer less than 2^32.
   @item
   If a non zero argument is specified, then after setting it as a random seed,
   a random number is generated.
   @item
   As the default seed is fixed, the sequence of the random numbers is
   always the same if a seed is not set.
   @item
   The algorithm is Mersenne Twister
   (http://www.math.keio.ac.jp/matsumoto/mt.html) by M. Matsumoto and
   T. Nishimura. The implementation is done also by themselves.
   @item
   The period of the random number sequence is 2^19937-1.
   @item
   One can save the state of the random number generator with @code{mt_save}.
   By loading the state file with @code{mt_load},
   one can trace a single random number sequence arcoss multiple sessions.
   \E
 @end itemize  @end itemize
   
 @table @t  @table @t
 @item $B;2>H(B  \JP @item $B;2>H(B
   \EG @item References
 @fref{lrandom}, @fref{mt_save mt_load}.  @fref{lrandom}, @fref{mt_save mt_load}.
 @end table  @end table
   
 @node lrandom,,, $B?t$N1i;;(B  \JP @node lrandom,,, $B?t$N1i;;(B
   \EG @node lrandom,,, Numbers
 @subsection @code{lrandom}  @subsection @code{lrandom}
 @findex lrandom  @findex lrandom
   
 @table @t  @table @t
 @item lradom(@var{bit})  @item lradom(@var{bit})
 :: $BB?G\D9Mp?t$r@8@.$9$k(B.  \JP :: $BB?G\D9Mp?t$r@8@.$9$k(B.
   \EG :: Generates a long random number.
 @end table  @end table
   
 @table @var  @table @var
 @item bit  @item bit
 @item return  @item return
 $B<+A3?t(B  \JP $B<+A3?t(B
   \EG integer
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 $B9b!9(B @var{bit} $B$NHsIi@0?t$NMp?t$r@8@.$9$k(B.  $B9b!9(B @var{bit} $B$NHsIi@0?t$NMp?t$r@8@.$9$k(B.
 @item  @item
 @code{random} $B$rJ#?t2s8F$S=P$7$F7k9g$7(B, $B;XDj$N(B bit $BD9$K%^%9%/$7$F$$$k(B.  @code{random} $B$rJ#?t2s8F$S=P$7$F7k9g$7(B, $B;XDj$N(B bit $BD9$K%^%9%/$7$F$$$k(B.
   \E
   \BEG
   @item
   Generates a non-negative integer of at most @var{bit} bits.
   @item
   The result is a concatination of outputs of @code{random}.
   \E
 @end itemize  @end itemize
   
 @table @t  @table @t
 @item $B;2>H(B  \JP @item $B;2>H(B
   \EG @item References
 @fref{random}, @fref{mt_save mt_load}.  @fref{random}, @fref{mt_save mt_load}.
 @end table  @end table
   
 @node mt_save mt_load,,, $B?t$N1i;;(B  \JP @node mt_save mt_load,,, $B?t$N1i;;(B
   \EG @node mt_save mt_load,,, Numbers
 @subsection @code{mt_save}, @code{mt_load}  @subsection @code{mt_save}, @code{mt_load}
 @findex mt_save  @findex mt_save
 @findex mt_load  @findex mt_load
   
 @table @t  @table @t
 @item mt_save(@var{fname})  @item mt_save(@var{fname})
 :: $BMp?t@8@.4o$N8=:_$N>uBV$r%U%!%$%k$K%;!<%V$9$k(B.  \JP :: $BMp?t@8@.4o$N8=:_$N>uBV$r%U%!%$%k$K%;!<%V$9$k(B.
   \EG :: Saves the state of the random number generator.
 @item mt_load(@var{fname})  @item mt_load(@var{fname})
 :: $B%U%!%$%k$K%;!<%V$5$l$?Mp?t@8@.4o$N>uBV$r%m!<%I$9$k(B.  \JP :: $B%U%!%$%k$K%;!<%V$5$l$?Mp?t@8@.4o$N>uBV$r%m!<%I$9$k(B.
   \EG :: Loads a saved state of the random number generator.
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 0 $B$^$?$O(B 1  \JP 0 $B$^$?$O(B 1
   \EG 0 or 1
 @item fname  @item fname
 $BJ8;zNs(B  \JP $BJ8;zNs(B
   \EG string
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
 @item $B$"$k>uBV$r%;!<%V$7(B, $B$=$N>uBV$r%m!<%I$9$k$3$H$G(B,  \BJP
   @item
   $B$"$k>uBV$r%;!<%V$7(B, $B$=$N>uBV$r%m!<%I$9$k$3$H$G(B,
 $B0l$D$N5?;wMp?t7ONs$r(B, $B?75,$N(B Asir $B%;%C%7%g%s$GB3$1$F$?$I$k$3$H$,(B  $B0l$D$N5?;wMp?t7ONs$r(B, $B?75,$N(B Asir $B%;%C%7%g%s$GB3$1$F$?$I$k$3$H$,(B
 $B$G$-$k(B.  $B$G$-$k(B.
   \E
   \BEG
   @item
   One can save the state of the random number generator with @code{mt_save}.
   By loading the state file with @code{mt_load},
   one can trace a single random number sequence arcoss multiple
   @b{Asir} sessions.
   \E
 @end itemize  @end itemize
   
 @example  @example
Line 388  Copyright (C) FUJITSU LABORATORIES LIMITED.
Line 574  Copyright (C) FUJITSU LABORATORIES LIMITED.
 @end example  @end example
   
 @table @t  @table @t
 @item $B;2>H(B  \JP @item $B;2>H(B
   \EG @item References
 @fref{random}, @fref{lrandom}.  @fref{random}, @fref{lrandom}.
 @end table  @end table
   
 @node nm dn,,, $B?t$N1i;;(B  \JP @node nm dn,,, $B?t$N1i;;(B
   \EG @node nm dn,,, Numbers
 @subsection @code{nm}, @code{dn}  @subsection @code{nm}, @code{dn}
 @findex nm  @findex nm
 @findex dn  @findex dn
   
 @table @t  @table @t
 @item nm(@var{rat})  @item nm(@var{rat})
 :: @var{rat} $B$NJ,;R(B.  \JP :: @var{rat} $B$NJ,;R(B.
   \EG :: Numerator of @var{rat}.
 @item dn(@var{rat})  @item dn(@var{rat})
 :: @var{rat} $B$NJ,Jl(B.  \JP :: @var{rat} $B$NJ,Jl(B.
   \EG :: Denominator of @var{rat}.
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 $B@0?t$^$?$OB?9`<0(B  \JP $B@0?t$^$?$OB?9`<0(B
   \EG integer or polynomial
 @item rat  @item rat
 $BM-M}?t$^$?$OM-M}<0(B  \JP $BM-M}?t$^$?$OM-M}<0(B
   \EG rational number or rational expression
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 $BM?$($i$l$?M-M}?t$^$?M-M}<0$NJ,;R5Z$SJ,Jl$rJV$9(B.  $BM?$($i$l$?M-M}?t$^$?M-M}<0$NJ,;R5Z$SJ,Jl$rJV$9(B.
 @item  @item
Line 420  Copyright (C) FUJITSU LABORATORIES LIMITED.
Line 613  Copyright (C) FUJITSU LABORATORIES LIMITED.
 $BM-M}<0$N>l9g(B, $BC1$KJ,Jl(B, $BJ,;R$r<h$j=P$9$@$1$G$"$k(B.  $BM-M}<0$N>l9g(B, $BC1$KJ,Jl(B, $BJ,;R$r<h$j=P$9$@$1$G$"$k(B.
 $BM-M}<0$KBP$7$F$O(B, $BLsJ,$O<+F0E*$K$O9T$o$l$J$$(B. @code{red()}  $BM-M}<0$KBP$7$F$O(B, $BLsJ,$O<+F0E*$K$O9T$o$l$J$$(B. @code{red()}
 $B$rL@<(E*$K8F$S=P$9I,MW$,$"$k(B.  $B$rL@<(E*$K8F$S=P$9I,MW$,$"$k(B.
   \E
   \BEG
   @item
   Numerator and denominator of a given rational expression.
   @item
   For a rational number, they return its numerator and denominator,
   respectively.  For a rational expression whose numerator and denominator
   may contain rational numbers, they do not separate those rational
   coefficients to numerators and denominators.
   @item
   For a rational number, the denominator is always kept positive, and
   the sign is contained in the numerator.
   @item
   @b{Risa/Asir} does not cancel the common divisors unless otherwise explicitly
   specified by the user.
   Therefore, @code{nm()} and @code{dn()} return the numerator and the
   denominator as it is, respectively.
   \E
 @end itemize  @end itemize
   
 @example  @example
Line 432  y
Line 643  y
 @end example  @end example
   
 @table @t  @table @t
 @item $B;2>H(B  \JP @item $B;2>H(B
   \EG @item References
 @fref{red}.  @fref{red}.
 @end table  @end table
   
 @node conj real imag,,, $B?t$N1i;;(B  \JP @node conj real imag,,, $B?t$N1i;;(B
   \EG @node conj real imag,,, Numbers
 @subsection @code{conj}, @code{real}, @code{imag}  @subsection @code{conj}, @code{real}, @code{imag}
 @findex conj  @findex conj
   
 @table @t  @table @t
 @item real(@var{comp})  @item real(@var{comp})
 :: @var{comp} $B$N<B?tItJ,(B.  \JP :: @var{comp} $B$N<B?tItJ,(B.
   \EG :: Real part of @var{comp}.
 @item imag(@var{comp})  @item imag(@var{comp})
 :: @var{comp} $B$N5u?tItJ,(B.  \JP :: @var{comp} $B$N5u?tItJ,(B.
   \EG :: Imaginary part of @var{comp}.
 @item conj(@var{comp})  @item conj(@var{comp})
 :: @var{comp} $B$N6&LrJ#AG?t(B.  \JP :: @var{comp} $B$N6&LrJ#AG?t(B.
   \EG :: Complex conjugate of @var{comp}.
 @end table  @end table
   
 @table @var  @table @var
 @item return comp  @item return comp
 $BJ#AG?t(B  \JP $BJ#AG?t(B
   \EG complex number
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 $BJ#AG?t$KBP$7(B, $B<BIt(B, $B5uIt(B, $B6&Lr$r5a$a$k(B.  $BJ#AG?t$KBP$7(B, $B<BIt(B, $B5uIt(B, $B6&Lr$r5a$a$k(B.
 @item  @item
 $B$3$l$i$O(B, $BB?9`<0$KBP$7$F$bF/$/(B.  $B$3$l$i$O(B, $BB?9`<0$KBP$7$F$bF/$/(B.
   \E
   \BEG
   @item
   Basic operations for complex numbers.
   @item
   These functions works also for polynomials with complex coefficients.
   \E
 @end itemize  @end itemize
   
 @example  @example
Line 468  y
Line 693  y
 [2,11,(2-11*@@i)]  [2,11,(2-11*@@i)]
 @end example  @end example
   
 @node eval,,, $B?t$N1i;;(B  \JP @node eval deval ,,, $B?t$N1i;;(B
 @subsection @code{eval}  \EG @node eval deval,,, Numbers
   @subsection @code{eval}, @code{deval}
 @findex eval  @findex eval
   @findex deval
 @cindex PARI  @cindex PARI
   
 @table @t  @table @t
 @item eval(@var{obj}[,@var{prec}])  @item eval(@var{obj}[,@var{prec}])
 :: @var{obj} $B$NCM$NI>2A(B.  @item deval(@var{obj})
   \JP :: @var{obj} $B$NCM$NI>2A(B.
   \EG :: Evaluate @var{obj} numerically.
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 $B?t$"$k$$$O<0(B  \JP $B?t$"$k$$$O<0(B
   \EG number or expression
 @item obj  @item obj
 $B0lHL$N<0(B  \JP $B0lHL$N<0(B
   \EG general expression
 @item prec  @item prec
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 @var{obj} $B$K4^$^$l$kH!?t$NCM$r2DG=$J8B$jI>2A$9$k(B.  @var{obj} $B$K4^$^$l$kH!?t$NCM$r2DG=$J8B$jI>2A$9$k(B.
 @item  @item
 $B7W;;$O(B @b{PARI} (@xref{pari}) $B$,9T$&(B.  @code{deval} $B$OG\@:EYIbF0>.?t$r7k2L$H$7$F(B
   @code{eval} $B$N>l9g(B, $BM-M}?t$O$=$N$^$^;D$k(B.
 @item  @item
   @code{eval} $B$K$*$$$F$O(B, $B7W;;$O(B @b{PARI} (@xref{pari}) $B$,9T$&(B.
   @code{deval} $B$K$*$$$F$O(B, $B7W;;$O(B C $B?t3X%i%$%V%i%j$N4X?t$rMQ$$$F9T$&(B.
   @item
   @code{deval} $B$OJ#AG?t$O07$($J$$(B.
   @item
   @code{eval} $B$K$*$$$F$O(B,
 @var{prec} $B$r;XDj$7$?>l9g(B, $B7W;;$O(B, 10 $B?J(B @var{prec} $B7eDxEY$G9T$o$l$k(B.  @var{prec} $B$r;XDj$7$?>l9g(B, $B7W;;$O(B, 10 $B?J(B @var{prec} $B7eDxEY$G9T$o$l$k(B.
 @var{prec} $B$N;XDj$,$J$$>l9g(B, $B8=:_@_Dj$5$l$F$$$k@:EY$G9T$o$l$k(B.  @var{prec} $B$N;XDj$,$J$$>l9g(B, $B8=:_@_Dj$5$l$F$$$k@:EY$G9T$o$l$k(B.
 (@xref{setprec})  (@xref{setprec})
Line 510  y
Line 750  y
 @code{exp}, @code{log}, @code{pow(a,b) (a^b)}  @code{exp}, @code{log}, @code{pow(a,b) (a^b)}
 @end table  @end table
 @item  @item
 $B0J2<$N5-9f$r?t$H$7$FI>2A$G$-$k(B.  $B0J2<$N5-9f$r?t$H$7$FI>2A$G$-$k(B. $B$?$@$7(B @code{@@i} $B$r07$($k$N$O(B
   @code{eval}, @code{deval} $B$N$_$G$"$k(B.
 @table @t  @table @t
 @item @@i  @item @@i
 $B5u?tC10L(B  $B5u?tC10L(B
Line 519  y
Line 760  y
 @item @@e  @item @@e
 $B<+A3BP?t$NDl(B  $B<+A3BP?t$NDl(B
 @end table  @end table
   \E
   \BEG
   @item
   Evaluates the value of the functions contained in @var{obj} as far as
   possible.
   @item
   @code{deval} returns
   double float. Rational numbers remain unchanged in results from @code{eval}.
   @item
   In @code{eval} the computation is done
   by @b{PARI} (@xref{pari}). In @code{deval} the computation is
   done by the C math library.
   @item
   @code{deval} cannot handle complex numbers.
   @item
   When @var{prec} is specified, computation will be performed with a
   precision of about @var{prec}-digits.
   If @var{prec} is not specified, computation is performed with the
   precision set currently. (@xref{setprec})
   @item
   Currently available numerical functions are listed below.
   Note they are only a small part of whole @b{PARI} functions.
   
   @table @t
   @code{sin}, @code{cos}, @code{tan},
   
   @code{asin}, @code{acos}, @code{atan},
   
   @code{sinh}, @code{cosh}, @code{tanh},
   @code{asinh}, @code{acosh}, @code{atanh},
   
   @code{exp}, @code{log}, @code{pow(a,b) (a^b)}
   @end table
   @item
   Symbols for special values are as the followings. Note that
   @code{@@i} cannot be handled by @code{deval}.
   @table @t
   @item @@i
   unit of imaginary number
   @item @@pi
   the number pi,
   the ratio of circumference to diameter
   @item @@e
   Napier's number (@t{exp}(1))
   @end table
   \E
 @end itemize  @end itemize
   
 @example  @example
Line 530  y
Line 817  y
 0.86602540378443864674620506632  0.86602540378443864674620506632
 [121] eval(sin(@@pi/3)-3^(1/2)/2,50);  [121] eval(sin(@@pi/3)-3^(1/2)/2,50);
 -2.78791084448179148471 E-58  -2.78791084448179148471 E-58
   [122] eval(1/2);
   1/2
   [123] deval(sin(1)^2+cos(1)^2);
   1
 @end example  @end example
   
 @table @t  @table @t
 @item $B;2>H(B  \JP @item $B;2>H(B
   \EG @item References
 @fref{ctrl}, @fref{setprec}, @fref{pari}.  @fref{ctrl}, @fref{setprec}, @fref{pari}.
 @end table  @end table
   
 @node pari,,, $B?t$N1i;;(B  \JP @node pari,,, $B?t$N1i;;(B
   \EG @node pari,,, Numbers
 @subsection @code{pari}  @subsection @code{pari}
 @findex pari  @findex pari
 @cindex PARI  @cindex PARI
   
 @table @t  @table @t
 @item pari(@var{func},@var{arg},@var{prec})  @item pari(@var{func},@var{arg},@var{prec})
 :: @b{PARI} $B$NH!?t(B @var{func} $B$r8F$S=P$9(B.  \JP :: @b{PARI} $B$NH!?t(B @var{func} $B$r8F$S=P$9(B.
   \EG :: Call @b{PARI} function @var{func}.
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 @var{func} $BKh$K0[$J$k(B.  \JP @var{func} $BKh$K0[$J$k(B.
   \EG Depends on @var{func}.
 @item func  @item func
 @b{PARI} $B$NH!?tL>(B  \JP @b{PARI} $B$NH!?tL>(B
   \EG Function name of @b{PARI}.
 @item arg  @item arg
 @var{func} $B$N0z?t(B  \JP @var{func} $B$N0z?t(B
   \EG Arguments of @var{func}.
 @item prec  @item prec
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 @b{PARI} $B$NH!?t$r8F$S=P$9(B.  @b{PARI} $B$NH!?t$r8F$S=P$9(B.
   
Line 570  y
Line 869  y
 $BH!?tCM$NI>2A$r9bB.$K9T$&$3$H$,$G$-$k(B. @b{PARI} $B$OB>$N%W%m%0%i%`$+$i(B  $BH!?tCM$NI>2A$r9bB.$K9T$&$3$H$,$G$-$k(B. @b{PARI} $B$OB>$N%W%m%0%i%`$+$i(B
 $B%5%V%k!<%A%s%i%$%V%i%j$H$7$FMQ$$$k$3$H$,$G$-(B, $B$^$?(B, @samp{gp} $B$H$$$&(B  $B%5%V%k!<%A%s%i%$%V%i%j$H$7$FMQ$$$k$3$H$,$G$-(B, $B$^$?(B, @samp{gp} $B$H$$$&(B
 @b{PARI}$B%i%$%V%i%j$N%$%s%?%U%'!<%9$K$h$j(B UNIX $B$N%"%W%j%1!<%7%g%s$H$7$F(B  @b{PARI}$B%i%$%V%i%j$N%$%s%?%U%'!<%9$K$h$j(B UNIX $B$N%"%W%j%1!<%7%g%s$H$7$F(B
 $BMxMQ$9$k$3$H$b$G$-$k(B. $B8=:_$N%P!<%8%g%s$O(B @b{1.39} $B$G$$$/$D$+$N(B ftp  $BMxMQ$9$k$3$H$b$G$-$k(B. $B8=:_$N%P!<%8%g%s$O(B @b{2.0.17beta} $B$G$$$/$D$+$N(B ftp
 site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})  site ($B$?$H$($P(B @code{ftp://megrez.ceremab.u-bordeaux.fr/pub/pari})
 $B$+$i(B anonymous ftp $B$G$-$k(B.  $B$+$i(B anonymous ftp $B$G$-$k(B.
 @item  @item
 $B:G8e$N0z?t(B @var{prec} $B$G7W;;@:EY$r;XDj$G$-$k(B.  $B:G8e$N0z?t(B @var{prec} $B$G7W;;@:EY$r;XDj$G$-$k(B.
Line 580  site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})
Line 879  site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})
 $B8=;~E@$G<B9T$G$-$k(B @b{PARI} $B$NH!?t$O<!$NDL$j$G$"$k(B. $B$$$:$l$b(B  $B8=;~E@$G<B9T$G$-$k(B @b{PARI} $B$NH!?t$O<!$NDL$j$G$"$k(B. $B$$$:$l$b(B
 1 $B0z?t$G(B @b{Asir} $B$,BP1~$G$-$k7?$N0z?t$r$H$kH!?t$G$"$k(B.  1 $B0z?t$G(B @b{Asir} $B$,BP1~$G$-$k7?$N0z?t$r$H$kH!?t$G$"$k(B.
 $B$J$*3F!9$N5!G=$K$D$$$F$O(B @b{PARI} $B$N%^%K%e%"%k$r;2>H$N$3$H(B.  $B$J$*3F!9$N5!G=$K$D$$$F$O(B @b{PARI} $B$N%^%K%e%"%k$r;2>H$N$3$H(B.
   \E
   \BEG
   @item
   This command connects @b{Asir} to @b{PARI} system so that several
   functions of @b{PARI} can be conveniently used from @b{Risa/Asir}.
   @item
   @b{PARI} @code{[Batut et al.]} is developed at Bordeaux University, and
   distributed as a free software.  Though it has a certain facility to computer
   algebra, its major target is the operation of numbers (@b{bignum},
   @b{bigfloat}) related to the number theory.  It facilitates various
   function evaluations as well as arithmetic operations at a remarkable
   speed.  It can also be used from other external programs as a library.
   It provides a language interface named @samp{gp} to its library, which
   enables a user to use @b{PARI} as a calculator which runs on UNIX.
   The current version is @b{2.0.17beta}.  It can be obtained by several ftp
   sites. (For example, @code{ftp://megrez.ceremab.u-bordeaux.fr/pub/pari}.)
   @item
   The last argument (optional) @var{int} specifies the precision in digits
   for bigfloat operation.
   If the precision is not explicitly specified, operation will be performed
   with the precision set by @code{setprec()}.
   @item
   Currently available functions of @b{PARI} system are as follows.
   Note these are only a part of functions in @b{PARI} system.
   For details of individual functions, refer to the @b{PARI} manual.
   (Some of them can be seen in the following example.)
   \E
   
 @code{abs},  @code{abs},
 @code{adj},  @code{adj},
Line 714  site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})
Line 1040  site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})
 @code{wf2},  @code{wf2},
 @code{zeta}  @code{zeta}
   
   \BJP
 @item  @item
 @b{Asir} $B$GMQ$$$F$$$k$N$O(B @b{PARI} $B$N$[$s$N0lIt$N5!G=$G$"$k$,(B, $B:#8e(B  @b{Asir} $B$GMQ$$$F$$$k$N$O(B @b{PARI} $B$N$[$s$N0lIt$N5!G=$G$"$k$,(B, $B:#8e(B
 $B$h$jB?$/$N5!G=$,MxMQ$G$-$k$h$&2~NI$9$kM=Dj$G$"$k(B.  $B$h$jB?$/$N5!G=$,MxMQ$G$-$k$h$&2~NI$9$kM=Dj$G$"$k(B.
   \E
   \BEG
   @item
   @b{Asir} currently uses only a very small subset of @b{PARI}.
   We will improve @b{Asir} so that it can provide more functions of
   @b{PARI}.
   \E
 @end itemize  @end itemize
   
 @example  @example
 /* $B9TNs$N8GM-%Y%/%H%k$r5a$a$k(B. */  \JP /* $B9TNs$N8GM-%Y%/%H%k$r5a$a$k(B. */
   \EG /* Eigen vectors of a numerical matrix */
 [0] pari(eigen,newmat(2,2,[[1,1],[1,2]]));  [0] pari(eigen,newmat(2,2,[[1,1],[1,2]]));
 [ -1.61803398874989484819771921990 0.61803398874989484826 ]  [ -1.61803398874989484819771921990 0.61803398874989484826 ]
 [ 1 1 ]  [ 1 1 ]
 /* 1 $BJQ?tB?9`<0$N:,$r5a$a$k(B. */  \JP /* 1 $BJQ?tB?9`<0$N:,$r5a$a$k(B. */
   \EG /* Roots of a polynomial */
 [1] pari(roots,t^2-2);  [1] pari(roots,t^2-2);
 [ -1.41421356237309504876 1.41421356237309504876 ]  [ -1.41421356237309504876 1.41421356237309504876 ]
 @end example  @end example
   
 @table @t  @table @t
 @item $B;2>H(B  \JP @item $B;2>H(B
   \EG @item References
 @fref{setprec}.  @fref{setprec}.
 @end table  @end table
   
 @node setprec,,, $B?t$N1i;;(B  \JP @node setprec,,, $B?t$N1i;;(B
   \EG @node setprec,,, Numbers
 @subsection @code{setprec}  @subsection @code{setprec}
 @findex setprec  @findex setprec
 @cindex PARI  @cindex PARI
   
 @table @t  @table @t
 @item setprec([@var{n}])  @item setprec([@var{n}])
 :: @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B.  \JP :: @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B.
   \EG :: Sets the precision for @b{bigfloat} operations to @var{n} digits.
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @item n  @item n
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 $B0z?t$,$"$k>l9g(B, @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B.  $B0z?t$,$"$k>l9g(B, @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B.
 $B0z?t$N$"$k$J$7$K$+$+$o$i$:(B, $B0JA0$K@_Dj$5$l$F$$$?CM$rJV$9(B.  $B0z?t$N$"$k$J$7$K$+$+$o$i$:(B, $B0JA0$K@_Dj$5$l$F$$$?CM$rJV$9(B.
Line 763  site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})
Line 1105  site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})
 @item  @item
 $B@_Dj$G$-$k7e?t$K>e8B$O$J$$$,(B, $B;XDj$7$?7e?t$K@_Dj$5$l$k$H$O(B  $B@_Dj$G$-$k7e?t$K>e8B$O$J$$$,(B, $B;XDj$7$?7e?t$K@_Dj$5$l$k$H$O(B
 $B8B$i$J$$(B. $BBg$-$a$NCM$r@_Dj$9$k$N$,0BA4$G$"$k(B.  $B8B$i$J$$(B. $BBg$-$a$NCM$r@_Dj$9$k$N$,0BA4$G$"$k(B.
   \E
   \BEG
   @item
   When an argument is given, it
   sets the precision for @b{bigfloat} operations to @var{n} digits.
   The return value is always the previous precision in digits regardless of
   the existence of an argument.
   
   @item
   @b{Bigfloat} operations are done by @b{PARI}. (@xref{pari})
   @item
   This is effective for computations in @b{bigfloat}.
   Refer to @code{ctrl()} for turning on the `@b{bigfloat} flag.'
   @item
   There is no upper limit for precision digits.
   It sets the precision to some digits around the specified precision.
   Therefore, it is safe to specify a larger value.
   \E
 @end itemize  @end itemize
   
 @example  @example
Line 775  site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})
Line 1135  site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari})
 @end example  @end example
   
 @table @t  @table @t
 @item $B;2>H(B  \JP @item $B;2>H(B
 @fref{ctrl}, @fref{eval}, @fref{pari}.  @fref{ctrl}, @fref{eval deval}, @fref{pari}.
 @end table  @end table
   
 @node setmod,,, $B?t$N1i;;(B  \JP @node setmod,,, $B?t$N1i;;(B
   \EG @node setmod,,, Numbers
 @subsection @code{setmod}  @subsection @code{setmod}
 @findex setmod  @findex setmod
   
 @table @t  @table @t
 @item setmod([@var{p}])  @item setmod([@var{p}])
 :: $BM-8BBN$r(B GF(@var{p}) $B$K@_Dj$9$k(B.  \JP :: $BM-8BBN$r(B GF(@var{p}) $B$K@_Dj$9$k(B.
   \EG :: Sets the ground field to GF(@var{p}).
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 $B@0?t(B  \JP $B@0?t(B
   \EG integer
 @item n  @item n
 2^27 $BL$K~$NAG?t(B  \JP 2^27 $BL$K~$NAG?t(B
   \EG prime less than 2^27
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
   \BJP
 @item  @item
 $BM-8BBN$r(B GF(@var{p}) $B$K@_Dj$9$k(B. $B@_DjCM$rJV$9(B.  $BM-8BBN$r(B GF(@var{p}) $B$K@_Dj$9$k(B. $B@_DjCM$rJV$9(B.
 @item  @item
 $BM-8BBN$N85$N7?$r;}$D?t$O(B, $B$=$l<+?H$O$I$NM-8BBN$KB0$9$k$+$N>pJs$r;}$?$:(B,  $BM-8BBN$N85$N7?$r;}$D?t$O(B, $B$=$l<+?H$O$I$NM-8BBN$KB0$9$k$+$N>pJs$r;}$?$:(B,
 $B8=:_@_Dj$5$l$F$$$kAG?t(B @var{p} $B$K$h$j(B GF(@var{p}) $B>e$G$N1i;;$,E,MQ$5$l$k(B.  $B8=:_@_Dj$5$l$F$$$kAG?t(B @var{p} $B$K$h$j(B GF(@var{p}) $B>e$G$N1i;;$,E,MQ$5$l$k(B.
   @item
   $B0L?t$NBg$-$JM-8BBN$K4X$7$F$O(B @pxref{$BM-8BBN$K4X$9$k1i;;(B}.
   \E
   \BEG
   @item
   Sets the ground field to GF(@var{p}) and returns the value @var{p}.
   @item
   A member of a finite field does not have any information
   about the field and the arithmetic operations over GF(@var{p}) are applied
   with @var{p} set at the time.
   @item
   As for large finite fields, @pxref{Finite fields}.
   \E
 @end itemize  @end itemize
   
 @example  @example
Line 816  return to toplevel
Line 1194  return to toplevel
 @end example  @end example
   
 @table @t  @table @t
 @item $B;2>H(B  \JP @item $B;2>H(B
 @fref{dp_mod dp_rat}, @fref{$B?t$N7?(B}.  \EG @item References
   \JP @fref{dp_mod dp_rat}, @fref{$B?t$N7?(B}.
   \EG @fref{dp_mod dp_rat}, @fref{Types of numbers}.
 @end table  @end table
   
   \JP @node ntoint32 int32ton,,, $B?t$N1i;;(B
   \EG @node ntoint32 int32ton,,, Numbers
   @subsection @code{ntoint32}, @code{int32ton}
   @findex ntoint32
   @findex int32ton
   
   @table @t
   @item ntoint32(@var{n})
   @itemx int32ton(@var{int32})
   \JP :: $BHsIi@0?t$HId9f$J$7(B 32bit $B@0?t$N4V$N7?JQ49(B.
   \EG :: Type-conversion between a non-negative integer and an unsigned 32bit integer.
   @end table
   
   @table @var
   @item return
   \JP $BId9f$J$7(B 32bit $B@0?t$^$?$OHsIi@0?t(B
   \EG unsigned 32bit integer or non-negative integer
   @item n
   \JP 2^32 $BL$K~$NHsIi@0?t(B
   \EG non-negative interger less than 2^32
   @item int32
   \JP $BId9f$J$7(B 32bit $B@0?t(B
   \EG unsigned 32bit integer
   @end table
   
   @itemize @bullet
   \BJP
   @item $BHsIi@0?t(B ($B<1JL;R(B 1) $B$NId9f$J$7(B 32bit $B@0?t(B ($B<1JL;R(B 10) $B$X$NJQ49(B,
   $B$^$?$O$=$N5UJQ49$r9T$&(B.
   @item 32bit $B@0?t$O(B @b{OpenXM} $B$N4pK\9=@.MWAG$G$"$j(B, $B@0?t$r$=$N7?$GAw?.(B
   $B$9$kI,MW$,$"$k>l9g$KMQ$$$k(B.
   \E
   \BEG
   @item These functions do conversions between non-negative
   integers (the type id 1) and unsigned 32bit integers (the type id 10).
   @item An unsigned 32bit integer is a fundamental construct of @b{OpenXM}
   and one often has to send an integer to a server as an unsigned 32bit
   integer. These functions are used in such a case.
   \E
   @end itemize
   
   @table @t
   \JP @item $B;2>H(B
   \EG @item References
   \JP @fref{$BJ,;67W;;(B}, @fref{$B?t$N7?(B}.
   \EG @fref{Distributed computation}, @fref{Types of numbers}.
   @end table

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