=================================================================== RCS file: /home/cvs/OpenXM/src/asir-doc/parts/builtin/num.texi,v retrieving revision 1.1.1.1 retrieving revision 1.4 diff -u -p -r1.1.1.1 -r1.4 --- OpenXM/src/asir-doc/parts/builtin/num.texi 1999/12/08 05:47:44 1.1.1.1 +++ OpenXM/src/asir-doc/parts/builtin/num.texi 2001/03/12 05:01:19 1.4 @@ -1,5 +1,12 @@ +@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 @section $B?t$N1i;;(B +\E +\BEG +@node Numbers,,, Built-in Function +@section Numbers +\E @menu * idiv irem:: @@ -12,33 +19,40 @@ * mt_save mt_load:: * nm dn:: * conj real imag:: -* eval:: +* eval deval:: * pari:: * setprec:: * setmod:: * lrandom:: +* ntoint32 int32ton:: @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} @findex idiv @findex irem @table @t @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}) -:: $B@0?t=|;;$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 @table @var @item return -$B@0?t(B +\JP $B@0?t(B +\EG integer @item i1,i2 -$B@0?t(B +\JP $B@0?t(B +\EG integer @end table @itemize @bullet +\BJP @item @var{i1} $B$N(B @var{i2} $B$K$h$k@0?t=|;;$K$h$k>&(B, $B>jM>$r5a$a$k(B. @item @@ -50,6 +64,23 @@ @code{irem()} $B$NBe$o$j$KMQ$$$k$3$H$,$G$-$k(B. @item $BB?9`<0$N>l9g$O(B @code{sdiv}, @code{srem} $B$rMQ$$$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 @example @@ -64,31 +95,44 @@ @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{sdiv sdivm srem sremm sqr sqrm}, @fref{%}. @end table -@node fac,,, $B?t$N1i;;(B +\JP @node fac,,, $B?t$N1i;;(B +\EG @node fac,,, Numbers @subsection @code{fac} @findex fac @table @t @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 @table @var @item return -$B@0?t(B +\JP $B@0?t(B +\EG integer @item i -$B@0?t(B +\JP $B@0?t(B +\EG integer @end table @itemize @bullet +\BJP @item @var{i} $B$N3,>h$r7W;;$9$k(B. @item @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 @example @@ -96,26 +140,32 @@ 30414093201713378043612608166064768844377641568960512000000000000 @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} @findex igcd @findex igcdcntl @table @t @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}]) -:: $B@0?t(B GCD$B$N%"%k%4%j%:%`A*Br(B +\JP :: $B@0?t(B GCD$B$N%"%k%4%j%:%`A*Br(B +\EG :: Selects an algorithm for integer GCD. @end table @table @var @item return -$B@0?t(B +\JP $B@0?t(B +\EG integer @item i1,i2,i -$B@0?t(B +\JP $B@0?t(B +\EG integer @end table @itemize @bullet +\BJP @item @code{igcd} $B$O(B @var{i1} $B$H(B @var{i2} $B$N(B GCD $B$r5a$a$k(B. @item @@ -135,9 +185,38 @@ bmod GCD @item 3 accelerated integer GCD @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 @example @@ -161,61 +240,89 @@ accelerated integer GCD @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{gcd gcdz}. @end table -@node ilcm,,, $B?t$N1i;;(B +\JP @node ilcm,,, $B?t$N1i;;(B +\EG @node ilcm,,, Numbers @subsection @code{ilcm} @findex ilcm @table @t @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 @table @var @item return -$B@0?t(B +\JP $B@0?t(B +\EG integer @item i1,i2 -$B@0?t(B +\JP $B@0?t(B +\EG integer @end table @itemize @bullet +\BJP @item $B@0?t(B @var{i1}, @var{i2} $B$N:G>.8xG\?t$r5a$a$k(B. @item $B0lJ}$,(B 0 $B$N>l9g(B 0 $B$rJV$9(B. +\E +\BEG @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 @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{igcd igcdcntl}, @fref{mt_save mt_load}. @end table -@node inv,,, $B?t$N1i;;(B + +\JP @node inv,,, $B?t$N1i;;(B +\EG @node inv,,, Numbers @subsection @code{inv} @findex inv @table @t @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 @table @var @item return -$B@0?t(B +\JP $B@0?t(B +\EG integer @item i,m -$B@0?t(B +\JP $B@0?t(B +\EG integer @end table @itemize @bullet +\BJP @item @var{ia} @equiv{} 1 mod (@var{m}) $B$J$k@0?t(B @var{a} $B$r5a$a$k(B. @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 $B$=$N%A%'%C%/$O9T$o$J$$(B. +\E +\BEG +@item +This function computes an integer such that +@var{ia} @equiv{} 1 mod (@var{m}). +@item +The integer @var{i} and @var{m} must be mutually prime. +However, @code{inv()} does not check it. +\E @end itemize @example @@ -228,11 +335,13 @@ accelerated integer GCD @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{igcd igcdcntl}. @end table -@node prime lprime,,, $B?t$N1i;;(B +\JP @node prime lprime,,, $B?t$N1i;;(B +\EG @node prime lprime,,, Numbers @subsection @code{prime}, @code{lprime} @findex prime @findex lprime @@ -240,17 +349,21 @@ accelerated integer GCD @table @t @item prime(@var{index}) @item lprime(@var{index}) -:: $BAG?t$rJV$9(B +\JP :: $BAG?t$rJV$9(B +\EG :: Returns a prime number. @end table @table @var @item return -$B@0?t(B +\JP $B@0?t(B +\EG integer @item index -$B@0?t(B +\JP $B@0?t(B +\EG integer @end table @itemize @bullet +\BJP @item @code{prime()}, @code{lprime()} $B$$$:$l$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 @@ -259,8 +372,28 @@ 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 0 $B$rJV$9(B. @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. +\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 @example @@ -275,26 +408,30 @@ accelerated integer GCD @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{pari}. @end table -@node random,,, $B?t$N1i;;(B +\JP @node random,,, $B?t$N1i;;(B +\EG @node random,,, Numbers @subsection @code{random} @findex random @table @t @item radom([@var{seed}]) -:: $BMp?t$r@8@.$9$k(B. +\JP :: $BMp?t$r@8@.$9$k(B. @end table @table @var @item seed -@item return -$B<+A3?t(B +@itemx return +\JP $B<+A3?t(B +\EG non-negative integer @end table @itemize @bullet +\BJP @item $B:GBg(B 2^32-1 $B$NHsIi@0?t$NMp?t$r@8@.$9$k(B. @item @@ -310,63 +447,112 @@ default $B$N(B seed $B$O8GDj$N$?$a(B, $BH(B +\JP @item $B;2>H(B +\EG @item References @fref{lrandom}, @fref{mt_save mt_load}. @end table -@node lrandom,,, $B?t$N1i;;(B +\JP @node lrandom,,, $B?t$N1i;;(B +\EG @node lrandom,,, Numbers @subsection @code{lrandom} @findex lrandom @table @t @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 @table @var @item bit @item return -$B<+A3?t(B +\JP $B<+A3?t(B +\EG integer @end table @itemize @bullet +\BJP @item $B9b!9(B @var{bit} $B$NHsIi@0?t$NMp?t$r@8@.$9$k(B. @item @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 @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{random}, @fref{mt_save mt_load}. @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} @findex mt_save @findex mt_load @table @t @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}) -:: $B%U%!%$%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 @table @var @item return -0 $B$^$?$O(B 1 +\JP 0 $B$^$?$O(B 1 +\EG 0 or 1 @item fname -$BJ8;zNs(B +\JP $BJ8;zNs(B +\EG string @end table @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 $B$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 @example @@ -388,30 +574,37 @@ Copyright (C) FUJITSU LABORATORIES LIMITED. @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{random}, @fref{lrandom}. @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} @findex nm @findex dn @table @t @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}) -:: @var{rat} $B$NJ,Jl(B. +\JP :: @var{rat} $B$NJ,Jl(B. +\EG :: Denominator of @var{rat}. @end table @table @var @item return -$B@0?t$^$?$OB?9`<0(B +\JP $B@0?t$^$?$OB?9`<0(B +\EG integer or polynomial @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 @itemize @bullet +\BJP @item $BM?$($i$l$?M-M}?t$^$?M-M}<0$NJ,;R5Z$SJ,Jl$rJV$9(B. @item @@ -420,6 +613,24 @@ Copyright (C) FUJITSU LABORATORIES LIMITED. $BM-M}<0$N>l9g(B, $BC1$KJ,Jl(B, $BJ,;R$rH(B +\JP @item $B;2>H(B +\EG @item References @fref{red}. @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} @findex conj @table @t @item real(@var{comp}) -:: @var{comp} $B$N2A(B. +@item deval(@var{obj}) +\JP :: @var{obj} $B$NCM$NI>2A(B. +\EG :: Evaluate @var{obj} numerically. @end table @table @var @item return -$B?t$"$k$$$O<0(B +\JP $B?t$"$k$$$O<0(B +\EG number or expression @item obj -$B0lHL$N<0(B +\JP $B0lHL$N<0(B +\EG general expression @item prec -$B@0?t(B +\JP $B@0?t(B +\EG integer @end table @itemize @bullet +\BJP @item @var{obj} $B$K4^$^$l$kH!?t$NCM$r2DG=$J8B$jI>2A$9$k(B. @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 +@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$*$$$F$O(B, @var{prec} $B$r;XDj$7$?>l9g(B, $B7W;;$O(B, 10 $B?J(B @var{prec} $B7eDxEY$G9T$o$l$k(B. @var{prec} $B$N;XDj$,$J$$>l9g(B, $B8=:_@_Dj$5$l$F$$$k@:EY$G9T$o$l$k(B. (@xref{setprec}) @@ -510,7 +750,8 @@ y @code{exp}, @code{log}, @code{pow(a,b) (a^b)} @end table @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 @item @@i $B5u?tC10L(B @@ -519,6 +760,52 @@ y @item @@e $B<+A3BP?t$NDl(B @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 @example @@ -530,35 +817,47 @@ y 0.86602540378443864674620506632 [121] eval(sin(@@pi/3)-3^(1/2)/2,50); -2.78791084448179148471 E-58 +[122] eval(1/2); +1/2 +[123] deval(sin(1)^2+cos(1)^2); +1 @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{ctrl}, @fref{setprec}, @fref{pari}. @end table -@node pari,,, $B?t$N1i;;(B +\JP @node pari,,, $B?t$N1i;;(B +\EG @node pari,,, Numbers @subsection @code{pari} @findex pari @cindex PARI @table @t @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 @table @var @item return -@var{func} $BKh$K0[$J$k(B. +\JP @var{func} $BKh$K0[$J$k(B. +\EG Depends on @var{func}. @item func -@b{PARI} $B$NH!?tL>(B +\JP @b{PARI} $B$NH!?tL>(B +\EG Function name of @b{PARI}. @item arg -@var{func} $B$N0z?t(B +\JP @var{func} $B$N0z?t(B +\EG Arguments of @var{func}. @item prec -$B@0?t(B +\JP $B@0?t(B +\EG integer @end table @itemize @bullet +\BJP @item @b{PARI} $B$NH!?t$r8F$S=P$9(B. @@ -570,8 +869,8 @@ y $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$$$&(B @b{PARI}$B%i%$%V%i%j$N%$%s%?%U%'!<%9$K$h$j(B UNIX $B$N%"%W%j%1!<%7%g%s$H$7$F(B -$BMxMQ$9$k$3$H$b$G$-$k(B. $B8=:_$N%P!<%8%g%s$O(B @b{1.39} $B$G$$$/$D$+$N(B ftp -site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari}) +$BMxMQ$9$k$3$H$b$G$-$k(B. $B8=:_$N%P!<%8%g%s$O(B @b{2.0.17beta} $B$G$$$/$D$+$N(B ftp +site ($B$?$H$($P(B @code{ftp://megrez.ceremab.u-bordeaux.fr/pub/pari}) $B$+$i(B anonymous ftp $B$G$-$k(B. @item $B:G8e$N0z?t(B @var{prec} $B$G7W;;@:EY$r;XDj$G$-$k(B. @@ -580,6 +879,33 @@ site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari}) $B8=;~E@$GH$N$3$H(B. +\E +\BEG +@item +This command connects @b{Asir} to @b{PARI} system so that several +functions of @b{PARI} can be conveniently used from @b{Risa/Asir}. +@item +@b{PARI} @code{[Batut et al.]} is developed at Bordeaux University, and +distributed as a free software. Though it has a certain facility to computer +algebra, its major target is the operation of numbers (@b{bignum}, +@b{bigfloat}) related to the number theory. It facilitates various +function evaluations as well as arithmetic operations at a remarkable +speed. It can also be used from other external programs as a library. +It provides a language interface named @samp{gp} to its library, which +enables a user to use @b{PARI} as a calculator which runs on UNIX. +The current version is @b{2.0.17beta}. It can be obtained by several ftp +sites. (For example, @code{ftp://megrez.ceremab.u-bordeaux.fr/pub/pari}.) +@item +The last argument (optional) @var{int} specifies the precision in digits +for bigfloat operation. +If the precision is not explicitly specified, operation will be performed +with the precision set by @code{setprec()}. +@item +Currently available functions of @b{PARI} system are as follows. +Note these are only a part of functions in @b{PARI} system. +For details of individual functions, refer to the @b{PARI} manual. +(Some of them can be seen in the following example.) +\E @code{abs}, @code{adj}, @@ -714,44 +1040,60 @@ site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari}) @code{wf2}, @code{zeta} +\BJP @item @b{Asir} $B$GMQ$$$F$$$k$N$O(B @b{PARI} $B$N$[$s$N0lIt$N5!G=$G$"$k$,(B, $B:#8e(B $B$h$jB?$/$N5!G=$,MxMQ$G$-$k$h$&2~NI$9$kM=Dj$G$"$k(B. +\E +\BEG +@item +@b{Asir} currently uses only a very small subset of @b{PARI}. +We will improve @b{Asir} so that it can provide more functions of +@b{PARI}. +\E @end itemize @example -/* $B9TNs$N8GM-%Y%/%H%k$r5a$a$k(B. */ +\JP /* $B9TNs$N8GM-%Y%/%H%k$r5a$a$k(B. */ +\EG /* Eigen vectors of a numerical matrix */ [0] pari(eigen,newmat(2,2,[[1,1],[1,2]])); [ -1.61803398874989484819771921990 0.61803398874989484826 ] [ 1 1 ] -/* 1 $BJQ?tB?9`<0$N:,$r5a$a$k(B. */ +\JP /* 1 $BJQ?tB?9`<0$N:,$r5a$a$k(B. */ +\EG /* Roots of a polynomial */ [1] pari(roots,t^2-2); [ -1.41421356237309504876 1.41421356237309504876 ] @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{setprec}. @end table -@node setprec,,, $B?t$N1i;;(B +\JP @node setprec,,, $B?t$N1i;;(B +\EG @node setprec,,, Numbers @subsection @code{setprec} @findex setprec @cindex PARI @table @t @item setprec([@var{n}]) -:: @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B. +\JP :: @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B. +\EG :: Sets the precision for @b{bigfloat} operations to @var{n} digits. @end table @table @var @item return -$B@0?t(B +\JP $B@0?t(B +\EG integer @item n -$B@0?t(B +\JP $B@0?t(B +\EG integer @end table @itemize @bullet +\BJP @item $B0z?t$,$"$k>l9g(B, @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B. $B0z?t$N$"$k$J$7$K$+$+$o$i$:(B, $B0JA0$K@_Dj$5$l$F$$$?CM$rJV$9(B. @@ -763,6 +1105,24 @@ site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari}) @item $B@_Dj$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$G$"$k(B. +\E +\BEG +@item +When an argument is given, it +sets the precision for @b{bigfloat} operations to @var{n} digits. +The return value is always the previous precision in digits regardless of +the existence of an argument. + +@item +@b{Bigfloat} operations are done by @b{PARI}. (@xref{pari}) +@item +This is effective for computations in @b{bigfloat}. +Refer to @code{ctrl()} for turning on the `@b{bigfloat} flag.' +@item +There is no upper limit for precision digits. +It sets the precision to some digits around the specified precision. +Therefore, it is safe to specify a larger value. +\E @end itemize @example @@ -775,32 +1135,50 @@ site ($B$?$H$($P(B @code{math.ucla.edu:/pub/pari}) @end example @table @t -@item $B;2>H(B -@fref{ctrl}, @fref{eval}, @fref{pari}. +\JP @item $B;2>H(B +@fref{ctrl}, @fref{eval deval}, @fref{pari}. @end table -@node setmod,,, $B?t$N1i;;(B +\JP @node setmod,,, $B?t$N1i;;(B +\EG @node setmod,,, Numbers @subsection @code{setmod} @findex setmod @table @t @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 @table @var @item return -$B@0?t(B +\JP $B@0?t(B +\EG integer @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 @itemize @bullet +\BJP @item $BM-8BBN$r(B GF(@var{p}) $B$K@_Dj$9$k(B. $B@_DjCM$rJV$9(B. @item $BM-8BBN$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$$$kAG?t(B @var{p} $B$K$h$j(B GF(@var{p}) $B>e$G$N1i;;$,E,MQ$5$l$k(B. +@item +$B0L?t$NBg$-$JM-8BBN$K4X$7$F$O(B @pxref{$BM-8BBN$K4X$9$k1i;;(B}. +\E +\BEG +@item +Sets the ground field to GF(@var{p}) and returns the value @var{p}. +@item +A member of a finite field does not have any information +about the field and the arithmetic operations over GF(@var{p}) are applied +with @var{p} set at the time. +@item +As for large finite fields, @pxref{Finite fields}. +\E @end itemize @example @@ -816,7 +1194,56 @@ return to toplevel @end example @table @t -@item $B;2>H(B -@fref{dp_mod dp_rat}, @fref{$B?t$N7?(B}. +\JP @item $B;2>H(B +\EG @item References +\JP @fref{dp_mod dp_rat}, @fref{$B?t$N7?(B}. +\EG @fref{dp_mod dp_rat}, @fref{Types of numbers}. @end table +\JP @node ntoint32 int32ton,,, $B?t$N1i;;(B +\EG @node ntoint32 int32ton,,, Numbers +@subsection @code{ntoint32}, @code{int32ton} +@findex ntoint32 +@findex int32ton + +@table @t +@item ntoint32(@var{n}) +@itemx int32ton(@var{int32}) +\JP :: $BHsIi@0?t$HId9f$J$7(B 32bit $B@0?t$N4V$N7?JQ49(B. +\EG :: Type-conversion between a non-negative integer and an unsigned 32bit integer. +@end table + +@table @var +@item return +\JP $BId9f$J$7(B 32bit $B@0?t$^$?$OHsIi@0?t(B +\EG unsigned 32bit integer or non-negative integer +@item n +\JP 2^32 $BL$K~$NHsIi@0?t(B +\EG non-negative interger less than 2^32 +@item int32 +\JP $BId9f$J$7(B 32bit $B@0?t(B +\EG unsigned 32bit integer +@end table + +@itemize @bullet +\BJP +@item $BHsIi@0?t(B ($B<1JL;R(B 1) $B$NId9f$J$7(B 32bit $B@0?t(B ($B<1JL;R(B 10) $B$X$NJQ49(B, +$B$^$?$O$=$N5UJQ49$r9T$&(B. +@item 32bit $B@0?t$O(B @b{OpenXM} $B$N4pK\9=@.MWAG$G$"$j(B, $B@0?t$r$=$N7?$GAw?.(B +$B$9$kI,MW$,$"$k>l9g$KMQ$$$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