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Annotation of OpenXM/src/asir-doc/parts/builtin/num.texi, Revision 1.2

1.2     ! noro        1: @comment $OpenXM$
        !             2: \BJP
1.1       noro        3: @node $B?t$N1i;;(B,,, $BAH$_9~$_H!?t(B
                      4: @section $B?t$N1i;;(B
1.2     ! noro        5: \E
        !             6: \BEG
        !             7: @node Numbers,,, Built-in Function
        !             8: @section Numbers
        !             9: \E
1.1       noro       10:
                     11: @menu
                     12: * idiv irem::
                     13: * fac::
                     14: * igcd igcdcntl::
                     15: * ilcm::
                     16: * inv::
                     17: * prime lprime::
                     18: * random::
                     19: * mt_save mt_load::
                     20: * nm dn::
                     21: * conj real imag::
                     22: * eval::
                     23: * pari::
                     24: * setprec::
                     25: * setmod::
                     26: * lrandom::
                     27: @end menu
                     28:
1.2     ! noro       29: \JP @node idiv irem,,, $B?t$N1i;;(B
        !            30: \EG @node idiv irem,,, Numbers
1.1       noro       31: @subsection @code{idiv}, @code{irem}
                     32: @findex idiv
                     33: @findex irem
                     34:
                     35: @table @t
                     36: @item idiv(@var{i1},@var{i2})
1.2     ! noro       37: \JP :: $B@0?t=|;;$K$h$k>&(B.
        !            38: \EG :: Integer quotient of @var{i1} divided by @var{i2}.
1.1       noro       39: @item irem(@var{i1},@var{i2})
1.2     ! noro       40: \JP :: $B@0?t=|;;$K$h$k>jM>(B.
        !            41: \EG :: Integer remainder of @var{i1} divided by @var{i2}.
1.1       noro       42: @end table
                     43:
                     44: @table @var
                     45: @item return
1.2     ! noro       46: \JP $B@0?t(B
        !            47: \EG integer
1.1       noro       48: @item i1,i2
1.2     ! noro       49: \JP $B@0?t(B
        !            50: \EG integer
1.1       noro       51: @end table
                     52:
                     53: @itemize @bullet
1.2     ! noro       54: \BJP
1.1       noro       55: @item
                     56: @var{i1} $B$N(B @var{i2} $B$K$h$k@0?t=|;;$K$h$k>&(B, $B>jM>$r5a$a$k(B.
                     57: @item
                     58: @var{i2} $B$O(B 0 $B$G$"$C$F$O$J$i$J$$(B.
                     59: @item
                     60: $BHo=|?t$,Ii$N>l9g(B, $B@dBPCM$KBP$9$kCM$K%^%$%J%9$r$D$1$?CM$rJV$9(B.
                     61: @item
                     62: @var{i1} @code{%} @var{i2} $B$O(B, $B7k2L$,@5$K@55,2=$5$l$k$3$H$r=|$1$P(B
                     63: @code{irem()} $B$NBe$o$j$KMQ$$$k$3$H$,$G$-$k(B.
                     64: @item
                     65: $BB?9`<0$N>l9g$O(B @code{sdiv}, @code{srem} $B$rMQ$$$k(B.
1.2     ! noro       66: \E
        !            67: \BEG
        !            68: @item
        !            69: Integer quotient and remainder of @var{i1} divided by @var{i2}.
        !            70: @item
        !            71: @var{i2} must not be 0.
        !            72: @item
        !            73: If the dividend is negative, the results are obtained by changing the
        !            74: sign of the results for absolute values of the dividend.
        !            75: @item
        !            76: One can use
        !            77: @var{i1} @code{%} @var{i2}
        !            78: for replacement of @code{irem()} which only differs in the point that
        !            79: the result is always normalized to non-negative values.
        !            80: @item
        !            81: Use @code{sdiv()}, @code{srem()} for polynomial quotient.
        !            82: \E
1.1       noro       83: @end itemize
                     84:
                     85: @example
                     86: [0] idiv(100,7);
                     87: 14
                     88: [0] idiv(-100,7);
                     89: -14
                     90: [1] irem(100,7);
                     91: 2
                     92: [1] irem(-100,7);
                     93: -2
                     94: @end example
                     95:
                     96: @table @t
1.2     ! noro       97: \JP @item $B;2>H(B
        !            98: \EG @item References
1.1       noro       99: @fref{sdiv sdivm srem sremm sqr sqrm}, @fref{%}.
                    100: @end table
                    101:
1.2     ! noro      102: \JP @node fac,,, $B?t$N1i;;(B
        !           103: \EG @node fac,,, Numbers
1.1       noro      104: @subsection @code{fac}
                    105: @findex fac
                    106:
                    107: @table @t
                    108: @item fac(@var{i})
1.2     ! noro      109: \JP :: @var{i} $B$N3,>h(B.
        !           110: \EG :: The factorial of @var{i}.
1.1       noro      111: @end table
                    112:
                    113: @table @var
                    114: @item return
1.2     ! noro      115: \JP $B@0?t(B
        !           116: \EG integer
1.1       noro      117: @item i
1.2     ! noro      118: \JP $B@0?t(B
        !           119: \EG integer
1.1       noro      120: @end table
                    121:
                    122: @itemize @bullet
1.2     ! noro      123: \BJP
1.1       noro      124: @item
                    125: @var{i} $B$N3,>h$r7W;;$9$k(B.
                    126: @item
                    127: @var{i} $B$,Ii$N>l9g$O(B 0 $B$rJV$9(B.
1.2     ! noro      128: \E
        !           129: \BEG
        !           130: @item
        !           131: The factorial of @var{i}.
        !           132: @item
        !           133: Returns 0 if the argument @var{i} is negative.
        !           134: \E
1.1       noro      135: @end itemize
                    136:
                    137: @example
                    138: [0] fac(50);
                    139: 30414093201713378043612608166064768844377641568960512000000000000
                    140: @end example
                    141:
1.2     ! noro      142: \JP @node igcd igcdcntl,,, $B?t$N1i;;(B
        !           143: \EG @node igcd igcdcntl,,, Numbers
1.1       noro      144: @subsection @code{igcd},@code{igcdcntl}
                    145: @findex igcd
                    146: @findex igcdcntl
                    147:
                    148: @table @t
                    149: @item igcd(@var{i1},@var{i2})
1.2     ! noro      150: \JP :: $B@0?t$N(B GCD ($B:GBg8xLs?t(B)
        !           151: \EG :: The integer greatest common divisor of @var{i1} and @var{i2}.
1.1       noro      152: @item igcdcntl([@var{i}])
1.2     ! noro      153: \JP :: $B@0?t(B GCD$B$N%"%k%4%j%:%`A*Br(B
        !           154: \EG :: Selects an algorithm for integer GCD.
1.1       noro      155: @end table
                    156:
                    157: @table @var
                    158: @item return
1.2     ! noro      159: \JP $B@0?t(B
        !           160: \EG integer
1.1       noro      161: @item i1,i2,i
1.2     ! noro      162: \JP $B@0?t(B
        !           163: \EG integer
1.1       noro      164: @end table
                    165:
                    166: @itemize @bullet
1.2     ! noro      167: \BJP
1.1       noro      168: @item
                    169: @code{igcd} $B$O(B @var{i1} $B$H(B @var{i2} $B$N(B GCD $B$r5a$a$k(B.
                    170: @item
                    171: $B0z?t$,@0?t$G$J$$>l9g$O(B, $B%(%i!<$^$?$OL50UL#$J7k2L$rJV$9(B.
                    172: @item
                    173: $BB?9`<0$N>l9g$O(B, @code{gcd}, @code{gcdz} $B$rMQ$$$k(B.
                    174: @item
                    175: $B@0?t(B GCD $B$K$O$5$^$6$^$JJ}K!$,$"$j(B, @code{igcdcntl} $B$G@_Dj$G$-$k(B.
                    176:
                    177: @table @code
                    178: @item 0
                    179: Euclid $B8_=|K!(B (default)
                    180: @item 1
                    181: binary GCD
                    182: @item 2
                    183: bmod GCD
                    184: @item 3
                    185: accelerated integer GCD
                    186: @end table
1.2     ! noro      187: @code{2}, @code{3} $B$O(B @code{[Weber]} $B$K$h$k(B.
1.1       noro      188:
1.2     ! noro      189: $B$*$*$`$M(B @code{3} $B$,9bB.$@$,(B, $BNc30$b$"$k(B.
        !           190: \E
        !           191: \BEG
        !           192: @item
        !           193: Function @code{igcd()} returns the integer greatest common divisor of
        !           194: the given two integers.
        !           195: @item
        !           196: An error will result if the argument is not an integer; the result is
        !           197: not valid even if one is returned.
        !           198: @item
        !           199: Use @code{gcd()}, @code{gcdz()} for polynomial GCD.
        !           200:
        !           201: @item
        !           202: Various method of integer GCD computation are implemented
        !           203: and they can be selected by @code{igcdcntl}.
        !           204:
        !           205: @table @code
        !           206: @item 0
        !           207: Euclid algorithm (default)
        !           208: @item 1
        !           209: binary GCD
        !           210: @item 2
        !           211: bmod GCD
        !           212: @item 3
        !           213: accelerated integer GCD
        !           214: @end table
        !           215: @code{2}, @code{3} are due to @code{[Weber]}.
        !           216:
        !           217: In most cases @code{3} is the fastest, but there are exceptions.
        !           218: \E
1.1       noro      219: @end itemize
                    220:
                    221: @example
                    222: [0] A=lrandom(10^4)$
                    223: [1] B=lrandom(10^4)$
                    224: [2] C=lrandom(10^4)$
                    225: [3] D=A*C$
                    226: [4] E=A*B$
                    227: [5] cputime(1)$
                    228: [6] igcd(D,E)$
                    229: 0.6sec + gc : 1.93sec(2.531sec)
                    230: [7] igcdcntl(1)$
                    231: [8] igcd(D,E)$
                    232: 0.27sec(0.2635sec)
                    233: [9] igcdcntl(2)$
                    234: [10] igcd(D,E)$
                    235: 0.19sec(0.1928sec)
                    236: [11] igcdcntl(3)$
                    237: [12] igcd(D,E)$
                    238: 0.08sec(0.08023sec)
                    239: @end example
                    240:
                    241: @table @t
1.2     ! noro      242: \JP @item $B;2>H(B
        !           243: \EG @item References
1.1       noro      244: @fref{gcd gcdz}.
                    245: @end table
                    246:
1.2     ! noro      247: \JP @node ilcm,,, $B?t$N1i;;(B
        !           248: \EG @node ilcm,,, Numbers
1.1       noro      249: @subsection @code{ilcm}
                    250: @findex ilcm
                    251:
                    252: @table @t
                    253: @item ilcm(@var{i1},@var{i2})
1.2     ! noro      254: \JP :: $B:G>.8xG\?t$r5a$a$k(B.
        !           255: \EG :: The integer least common multiple of @var{i1} and @var{i2}.
1.1       noro      256: @end table
                    257:
                    258: @table @var
                    259: @item return
1.2     ! noro      260: \JP $B@0?t(B
        !           261: \EG integer
1.1       noro      262: @item i1,i2
1.2     ! noro      263: \JP $B@0?t(B
        !           264: \EG integer
1.1       noro      265: @end table
                    266:
                    267: @itemize @bullet
1.2     ! noro      268: \BJP
1.1       noro      269: @item
                    270: $B@0?t(B @var{i1}, @var{i2} $B$N:G>.8xG\?t$r5a$a$k(B.
                    271: @item
                    272: $B0lJ}$,(B 0 $B$N>l9g(B 0 $B$rJV$9(B.
1.2     ! noro      273: \E
        !           274: \BEG
1.1       noro      275: @item
1.2     ! noro      276: This function computes the integer least common multiple of
        !           277: @var{i1}, @var{i2}.
        !           278: @item
        !           279: If one of argument is equal to 0, the return 0.
        !           280: \E
1.1       noro      281: @end itemize
                    282:
                    283: @table @t
1.2     ! noro      284: \JP @item $B;2>H(B
        !           285: \EG @item References
1.1       noro      286: @fref{igcd igcdcntl}, @fref{mt_save mt_load}.
                    287: @end table
1.2     ! noro      288:
        !           289: \JP @node inv,,, $B?t$N1i;;(B
        !           290: \EG @node inv,,, Numbers
1.1       noro      291: @subsection @code{inv}
                    292: @findex inv
                    293:
                    294: @table @t
                    295: @item inv(@var{i},@var{m})
1.2     ! noro      296: \JP :: @var{m} $B$rK!$H$9$k(B @var{i} $B$N5U?t(B
        !           297: \EG :: the inverse (reciprocal) of @var{i} modulo @var{m}.
1.1       noro      298: @end table
                    299:
                    300: @table @var
                    301: @item return
1.2     ! noro      302: \JP $B@0?t(B
        !           303: \EG integer
1.1       noro      304: @item i,m
1.2     ! noro      305: \JP $B@0?t(B
        !           306: \EG integer
1.1       noro      307: @end table
                    308:
                    309: @itemize @bullet
1.2     ! noro      310: \BJP
1.1       noro      311: @item
                    312: @var{ia} @equiv{} 1 mod (@var{m}) $B$J$k@0?t(B @var{a} $B$r5a$a$k(B.
                    313: @item
                    314: @var{i} $B$H(B @var{m} $B$O8_$$$KAG$G$J$1$l$P$J$i$J$$$,(B, @code{inv()} $B$O(B
                    315: $B$=$N%A%'%C%/$O9T$o$J$$(B.
1.2     ! noro      316: \E
        !           317: \BEG
        !           318: @item
        !           319: This function computes an integer such that
        !           320: @var{ia} @equiv{} 1 mod (@var{m}).
        !           321: @item
        !           322: The integer @var{i} and  @var{m} must be mutually prime.
        !           323: However, @code{inv()} does not check it.
        !           324: \E
1.1       noro      325: @end itemize
                    326:
                    327: @example
                    328: [71] igcd(1234,4321);
                    329: 1
                    330: [72] inv(1234,4321);
                    331: 3239
                    332: [73] irem(3239*1234,4321);
                    333: 1
                    334: @end example
                    335:
                    336: @table @t
1.2     ! noro      337: \JP @item $B;2>H(B
        !           338: \EG @item References
1.1       noro      339: @fref{igcd igcdcntl}.
                    340: @end table
                    341:
1.2     ! noro      342: \JP @node prime lprime,,, $B?t$N1i;;(B
        !           343: \EG @node prime lprime,,, Numbers
1.1       noro      344: @subsection @code{prime}, @code{lprime}
                    345: @findex prime
                    346: @findex lprime
                    347:
                    348: @table @t
                    349: @item prime(@var{index})
                    350: @item lprime(@var{index})
1.2     ! noro      351: \JP :: $BAG?t$rJV$9(B
        !           352: \EG :: Returns a prime number.
1.1       noro      353: @end table
                    354:
                    355: @table @var
                    356: @item return
1.2     ! noro      357: \JP $B@0?t(B
        !           358: \EG integer
1.1       noro      359: @item index
1.2     ! noro      360: \JP $B@0?t(B
        !           361: \EG integer
1.1       noro      362: @end table
                    363:
                    364: @itemize @bullet
1.2     ! noro      365: \BJP
1.1       noro      366: @item
                    367: @code{prime()}, @code{lprime()} $B$$$:$l$b%7%9%F%`$,FbIt$K;}$D(B
                    368: $BAG?tI=$NMWAG$rJV$9(B. @code{index} $B$O(B 0 $B0J>e$N@0?t$G(B, $BAG?tI=(B
                    369: $B$N%$%s%G%C%/%9$KMQ$$$i$l$k(B. @code{prime()} $B$O(B 16381 $B$^$G(B
                    370: $B$NAG?t$r>.$5$$=g$K(B 1900 $B8D(B, @code{lprime()} $B$O(B, 10 $B?J(B 8 $B7e$G:GBg$N(B
                    371: $BAG?t$+$iBg$-$$=g$K(B 999 $B8DJV$9(B. $B$=$l0J30$N%$%s%G%C%/%9$KBP$7$F$O(B
                    372: 0 $B$rJV$9(B.
                    373: @item
1.2     ! noro      374: $B$h$j0lHLE*$JAG?t@8@.H!?t$H$7$F$O(B,
        !           375: @code{pari(nextprime,@var{number})}
1.1       noro      376: $B$,$"$k(B.
1.2     ! noro      377: \E
        !           378: \BEG
        !           379: @item
        !           380: The two functions, @code{prime()} and @code{lprime()}, returns
        !           381: an element stored in the system table of prime numbers.
        !           382: Here, @code{index} is a non-negative integer and be used as an index
        !           383: for the prime tables.
        !           384: The function @code{prime()} can return one of 1900 primes
        !           385: up to 16381 indexed so that the smaller one has smaller
        !           386: index.  The function @code{lprime()} can return one of 999 primes which
        !           387: are 8 digit sized and indexed so that the larger one has the smaller
        !           388: index.
        !           389: The two function always returns 0 for other indices.
        !           390: @item
        !           391: For more general function for prime generation, there is a @code{PARI}
        !           392: function
        !           393:
        !           394: @code{pari(nextprime,@var{number})}.
        !           395: \E
1.1       noro      396: @end itemize
                    397:
                    398: @example
                    399: [95] prime(0);
                    400: 2
                    401: [96] prime(1228);
                    402: 9973
                    403: [97] lprime(0);
                    404: 99999989
                    405: [98] lprime(999);
                    406: 0
                    407: @end example
                    408:
                    409: @table @t
1.2     ! noro      410: \JP @item $B;2>H(B
        !           411: \EG @item References
1.1       noro      412: @fref{pari}.
                    413: @end table
                    414:
1.2     ! noro      415: \JP @node random,,, $B?t$N1i;;(B
        !           416: \EG @node random,,, Numbers
1.1       noro      417: @subsection @code{random}
                    418: @findex random
                    419:
                    420: @table @t
                    421: @item radom([@var{seed}])
1.2     ! noro      422: \JP :: $BMp?t$r@8@.$9$k(B.
1.1       noro      423: @end table
                    424:
                    425: @table @var
                    426: @item seed
1.2     ! noro      427: @itemx return
        !           428: \JP $B<+A3?t(B
        !           429: \EG non-negative integer
1.1       noro      430: @end table
                    431:
                    432: @itemize @bullet
1.2     ! noro      433: \BJP
1.1       noro      434: @item
                    435: $B:GBg(B 2^32-1 $B$NHsIi@0?t$NMp?t$r@8@.$9$k(B.
                    436: @item
                    437: 0 $B$G$J$$0z?t$,$"$k;~(B, $B$=$NCM$r(B seed $B$H$7$F@_Dj$7$F$+$i(B, $BMp?t$r@8@.$9$k(B.
                    438: @item
                    439: default $B$N(B seed $B$O8GDj$N$?$a(B, $B<o$r@_Dj$7$J$1$l$P(B, $B@8@.$5$l$kMp?t$N(B
                    440: $B7ONs$O5/F0Kh$K0lDj$G$"$k(B.
                    441: @item
                    442: $B>>K\bC(B-$B@>B<Bs;N$K$h$k(B Mersenne Twister (http://www.math.keio.ac.jp/matsumoto/mt.html) $B%"%k%4%j%:%`$N(B, $BH`$i<+?H$K$h$k<BAu$rMQ$$$F$$$k(B.
                    443: @item
                    444: $B<~4|$O(B 2^19937-1 $B$HHs>o$KD9$$(B.
                    445: @item
                    446: @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}
                    447: $B$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
                    448: $B$G$-$k(B.
1.2     ! noro      449: \E
        !           450: \BEG
        !           451: @item
        !           452: Generates a random number which is a non-negative integer less than 2^32.
        !           453: @item
        !           454: If a non zero argument is specified, then after setting it as a random seed,
        !           455: a random number is generated.
        !           456: @item
        !           457: As the default seed is fixed, the sequence of the random numbers is
        !           458: always the same if a seed is not set.
        !           459: @item
        !           460: The algorithm is Mersenne Twister
        !           461: (http://www.math.keio.ac.jp/matsumoto/mt.html) by M. Matsumoto and
        !           462: T. Nishimura. The implementation is done also by themselves.
        !           463: @item
        !           464: The period of the random number sequence is 2^19937-1.
        !           465: @item
        !           466: One can save the state of the random number generator with @code{mt_save}.
        !           467: By loading the state file with @code{mt_load},
        !           468: one can trace a single random number sequence arcoss multiple sessions.
        !           469: \E
1.1       noro      470: @end itemize
                    471:
                    472: @table @t
1.2     ! noro      473: \JP @item $B;2>H(B
        !           474: \EG @item References
1.1       noro      475: @fref{lrandom}, @fref{mt_save mt_load}.
                    476: @end table
                    477:
1.2     ! noro      478: \JP @node lrandom,,, $B?t$N1i;;(B
        !           479: \EG @node lrandom,,, Numbers
1.1       noro      480: @subsection @code{lrandom}
                    481: @findex lrandom
                    482:
                    483: @table @t
                    484: @item lradom(@var{bit})
1.2     ! noro      485: \JP :: $BB?G\D9Mp?t$r@8@.$9$k(B.
        !           486: \EG :: Generates a long random number.
1.1       noro      487: @end table
                    488:
                    489: @table @var
                    490: @item bit
                    491: @item return
1.2     ! noro      492: \JP $B<+A3?t(B
        !           493: \EG integer
1.1       noro      494: @end table
                    495:
                    496: @itemize @bullet
1.2     ! noro      497: \BJP
1.1       noro      498: @item
                    499: $B9b!9(B @var{bit} $B$NHsIi@0?t$NMp?t$r@8@.$9$k(B.
                    500: @item
                    501: @code{random} $B$rJ#?t2s8F$S=P$7$F7k9g$7(B, $B;XDj$N(B bit $BD9$K%^%9%/$7$F$$$k(B.
1.2     ! noro      502: \E
        !           503: \BEG
        !           504: @item
        !           505: Generates a non-negative integer of at most @var{bit} bits.
        !           506: @item
        !           507: The result is a concatination of outputs of @code{random}.
        !           508: \E
1.1       noro      509: @end itemize
                    510:
                    511: @table @t
1.2     ! noro      512: \JP @item $B;2>H(B
        !           513: \EG @item References
1.1       noro      514: @fref{random}, @fref{mt_save mt_load}.
                    515: @end table
                    516:
1.2     ! noro      517: \JP @node mt_save mt_load,,, $B?t$N1i;;(B
        !           518: \EG @node mt_save mt_load,,, Numbers
1.1       noro      519: @subsection @code{mt_save}, @code{mt_load}
                    520: @findex mt_save
                    521: @findex mt_load
                    522:
                    523: @table @t
                    524: @item mt_save(@var{fname})
1.2     ! noro      525: \JP :: $BMp?t@8@.4o$N8=:_$N>uBV$r%U%!%$%k$K%;!<%V$9$k(B.
        !           526: \EG :: Saves the state of the random number generator.
1.1       noro      527: @item mt_load(@var{fname})
1.2     ! noro      528: \JP :: $B%U%!%$%k$K%;!<%V$5$l$?Mp?t@8@.4o$N>uBV$r%m!<%I$9$k(B.
        !           529: \EG :: Loads a saved state of the random number generator.
1.1       noro      530: @end table
                    531:
                    532: @table @var
                    533: @item return
1.2     ! noro      534: \JP 0 $B$^$?$O(B 1
        !           535: \EG 0 or 1
1.1       noro      536: @item fname
1.2     ! noro      537: \JP $BJ8;zNs(B
        !           538: \EG string
1.1       noro      539: @end table
                    540:
                    541: @itemize @bullet
1.2     ! noro      542: \BJP
        !           543: @item
        !           544: $B$"$k>uBV$r%;!<%V$7(B, $B$=$N>uBV$r%m!<%I$9$k$3$H$G(B,
1.1       noro      545: $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
                    546: $B$G$-$k(B.
1.2     ! noro      547: \E
        !           548: \BEG
        !           549: @item
        !           550: One can save the state of the random number generator with @code{mt_save}.
        !           551: By loading the state file with @code{mt_load},
        !           552: one can trace a single random number sequence arcoss multiple
        !           553: @b{Asir} sessions.
        !           554: \E
1.1       noro      555: @end itemize
                    556:
                    557: @example
                    558: [340] random();
                    559: 3510405877
                    560: [341] mt_save("/tmp/mt_state");
                    561: 1
                    562: [342] random();
                    563: 4290933890
                    564: [343] quit;
                    565: % asir
                    566: This is Asir, Version 991108.
                    567: Copyright (C) FUJITSU LABORATORIES LIMITED.
                    568: 3 March 1994. All rights reserved.
                    569: [340] mt_load("/tmp/mt_state");
                    570: 1
                    571: [341] random();
                    572: 4290933890
                    573: @end example
                    574:
                    575: @table @t
1.2     ! noro      576: \JP @item $B;2>H(B
        !           577: \EG @item References
1.1       noro      578: @fref{random}, @fref{lrandom}.
                    579: @end table
                    580:
1.2     ! noro      581: \JP @node nm dn,,, $B?t$N1i;;(B
        !           582: \EG @node nm dn,,, Numbers
1.1       noro      583: @subsection @code{nm}, @code{dn}
                    584: @findex nm
                    585: @findex dn
                    586:
                    587: @table @t
                    588: @item nm(@var{rat})
1.2     ! noro      589: \JP :: @var{rat} $B$NJ,;R(B.
        !           590: \EG :: Numerator of @var{rat}.
1.1       noro      591: @item dn(@var{rat})
1.2     ! noro      592: \JP :: @var{rat} $B$NJ,Jl(B.
        !           593: \EG :: Denominator of @var{rat}.
1.1       noro      594: @end table
                    595:
                    596: @table @var
                    597: @item return
1.2     ! noro      598: \JP $B@0?t$^$?$OB?9`<0(B
        !           599: \EG integer or polynomial
1.1       noro      600: @item rat
1.2     ! noro      601: \JP $BM-M}?t$^$?$OM-M}<0(B
        !           602: \EG rational number or rational expression
1.1       noro      603: @end table
                    604:
                    605: @itemize @bullet
1.2     ! noro      606: \BJP
1.1       noro      607: @item
                    608: $BM?$($i$l$?M-M}?t$^$?M-M}<0$NJ,;R5Z$SJ,Jl$rJV$9(B.
                    609: @item
                    610: $BM-M}?t$N>l9g(B, $BJ,Jl$O>o$K@5$G(B, $BId9f$OJ,;R$,;}$D(B.
                    611: @item
                    612: $BM-M}<0$N>l9g(B, $BC1$KJ,Jl(B, $BJ,;R$r<h$j=P$9$@$1$G$"$k(B.
                    613: $BM-M}<0$KBP$7$F$O(B, $BLsJ,$O<+F0E*$K$O9T$o$l$J$$(B. @code{red()}
                    614: $B$rL@<(E*$K8F$S=P$9I,MW$,$"$k(B.
1.2     ! noro      615: \E
        !           616: \BEG
        !           617: @item
        !           618: Numerator and denominator of a given rational expression.
        !           619: @item
        !           620: For a rational number, they return its numerator and denominator,
        !           621: respectively.  For a rational expression whose numerator and denominator
        !           622: may contain rational numbers, they do not separate those rational
        !           623: coefficients to numerators and denominators.
        !           624: @item
        !           625: For a rational number, the denominator is always kept positive, and
        !           626: the sign is contained in the numerator.
        !           627: @item
        !           628: @b{Risa/Asir} does not cancel the common divisors unless otherwise explicitly
        !           629: specified by the user.
        !           630: Therefore, @code{nm()} and @code{dn()} return the numerator and the
        !           631: denominator as it is, respectively.
        !           632: \E
1.1       noro      633: @end itemize
                    634:
                    635: @example
                    636: [2] [nm(-43/8),dn(-43/8)];
                    637: [-43,8]
                    638: [3] dn((x*z)/(x*y));
                    639: y*x
                    640: [3] dn(red((x*z)/(x*y)));
                    641: y
                    642: @end example
                    643:
                    644: @table @t
1.2     ! noro      645: \JP @item $B;2>H(B
        !           646: \EG @item References
1.1       noro      647: @fref{red}.
                    648: @end table
                    649:
1.2     ! noro      650: \JP @node conj real imag,,, $B?t$N1i;;(B
        !           651: \EG @node conj real imag,,, Numbers
1.1       noro      652: @subsection @code{conj}, @code{real}, @code{imag}
                    653: @findex conj
                    654:
                    655: @table @t
                    656: @item real(@var{comp})
1.2     ! noro      657: \JP :: @var{comp} $B$N<B?tItJ,(B.
        !           658: \EG :: Real part of @var{comp}.
1.1       noro      659: @item imag(@var{comp})
1.2     ! noro      660: \JP :: @var{comp} $B$N5u?tItJ,(B.
        !           661: \EG :: Imaginary part of @var{comp}.
1.1       noro      662: @item conj(@var{comp})
1.2     ! noro      663: \JP :: @var{comp} $B$N6&LrJ#AG?t(B.
        !           664: \EG :: Complex conjugate of @var{comp}.
1.1       noro      665: @end table
                    666:
                    667: @table @var
                    668: @item return comp
1.2     ! noro      669: \JP $BJ#AG?t(B
        !           670: \EG complex number
1.1       noro      671: @end table
                    672:
                    673: @itemize @bullet
1.2     ! noro      674: \BJP
1.1       noro      675: @item
                    676: $BJ#AG?t$KBP$7(B, $B<BIt(B, $B5uIt(B, $B6&Lr$r5a$a$k(B.
                    677: @item
                    678: $B$3$l$i$O(B, $BB?9`<0$KBP$7$F$bF/$/(B.
1.2     ! noro      679: \E
        !           680: \BEG
        !           681: @item
        !           682: Basic operations for complex numbers.
        !           683: @item
        !           684: These functions works also for polynomials with complex coefficients.
        !           685: \E
1.1       noro      686: @end itemize
                    687:
                    688: @example
                    689: [111] A=(2+@@i)^3;
                    690: (2+11*@@i)
                    691: [112] [real(A),imag(A),conj(A)];
                    692: [2,11,(2-11*@@i)]
                    693: @end example
                    694:
1.2     ! noro      695: \JP @node eval,,, $B?t$N1i;;(B
        !           696: \EG @node eval,,, Numbers
1.1       noro      697: @subsection @code{eval}
                    698: @findex eval
                    699: @cindex PARI
                    700:
                    701: @table @t
                    702: @item eval(@var{obj}[,@var{prec}])
1.2     ! noro      703: \JP :: @var{obj} $B$NCM$NI>2A(B.
        !           704: \EG :: Evaluate @var{obj} numerically.
1.1       noro      705: @end table
                    706:
                    707: @table @var
                    708: @item return
1.2     ! noro      709: \JP $B?t$"$k$$$O<0(B
        !           710: \EG number or expression
1.1       noro      711: @item obj
1.2     ! noro      712: \JP $B0lHL$N<0(B
        !           713: \EG general expression
1.1       noro      714: @item prec
1.2     ! noro      715: \JP $B@0?t(B
        !           716: \EG integer
1.1       noro      717: @end table
                    718:
                    719: @itemize @bullet
1.2     ! noro      720: \BJP
1.1       noro      721: @item
                    722: @var{obj} $B$K4^$^$l$kH!?t$NCM$r2DG=$J8B$jI>2A$9$k(B.
                    723: @item
                    724: $B7W;;$O(B @b{PARI} (@xref{pari}) $B$,9T$&(B.
                    725: @item
                    726: @var{prec} $B$r;XDj$7$?>l9g(B, $B7W;;$O(B, 10 $B?J(B @var{prec} $B7eDxEY$G9T$o$l$k(B.
                    727: @var{prec} $B$N;XDj$,$J$$>l9g(B, $B8=:_@_Dj$5$l$F$$$k@:EY$G9T$o$l$k(B.
                    728: (@xref{setprec})
                    729: @item
                    730: @table @t
                    731: @item $B07$($kH!?t$O(B, $B<!$NDL$j(B.
                    732: @code{sin}, @code{cos}, @code{tan},
                    733:
                    734: @code{asin}, @code{acos}, @code{atan},
                    735:
                    736: @code{sinh}, @code{cosh}, @code{tanh},
                    737:
                    738: @code{asinh}, @code{acosh}, @code{atanh},
                    739:
                    740: @code{exp}, @code{log}, @code{pow(a,b) (a^b)}
                    741: @end table
                    742: @item
                    743: $B0J2<$N5-9f$r?t$H$7$FI>2A$G$-$k(B.
                    744: @table @t
                    745: @item @@i
                    746: $B5u?tC10L(B
                    747: @item @@pi
                    748: $B1_<~N((B
                    749: @item @@e
                    750: $B<+A3BP?t$NDl(B
                    751: @end table
1.2     ! noro      752: \E
        !           753: \BEG
        !           754: @item
        !           755: Evaluates the value of the functions contained in @var{obj} as far as
        !           756: possible.
        !           757: @item
        !           758: The computation is done by @b{PARI} (@xref{pari}).
        !           759: @item
        !           760: When @var{prec} is specified, computation will be performed with a
        !           761: precision of about @var{prec}-digits.
        !           762: If @var{prec} is not specified, computation is performed with the
        !           763: precision set currently. (@xref{setprec})
        !           764: @item
        !           765: Currently available numerical functions are listed below.
        !           766: Note they are only a small part of whole @b{PARI} functions.
        !           767:
        !           768: @table @t
        !           769: @code{sin}, @code{cos}, @code{tan},
        !           770:
        !           771: @code{asin}, @code{acos}, @code{atan},
        !           772:
        !           773: @code{sinh}, @code{cosh}, @code{tanh},
        !           774: @code{asinh}, @code{acosh}, @code{atanh},
        !           775:
        !           776: @code{exp}, @code{log}, @code{pow(a,b) (a^b)}
        !           777: @end table
        !           778: @item
        !           779: Symbols for special values are as the followings.
        !           780: @table @t
        !           781: @item @@i
        !           782: unit of imaginary number
        !           783: @item @@pi
        !           784: the number pi,
        !           785: the ratio of circumference to diameter
        !           786: @item @@e
        !           787: Napier's number (@t{exp}(1))
        !           788: @end table
        !           789: \E
1.1       noro      790: @end itemize
                    791:
                    792: @example
                    793: [118] eval(exp(@@pi*@@i));
                    794: -1.0000000000000000000000000000
                    795: [119] eval(2^(1/2));
                    796: 1.414213562373095048763788073031
                    797: [120] eval(sin(@@pi/3));
                    798: 0.86602540378443864674620506632
                    799: [121] eval(sin(@@pi/3)-3^(1/2)/2,50);
                    800: -2.78791084448179148471 E-58
                    801: @end example
                    802:
                    803: @table @t
1.2     ! noro      804: \JP @item $B;2>H(B
        !           805: \EG @item References
1.1       noro      806: @fref{ctrl}, @fref{setprec}, @fref{pari}.
                    807: @end table
                    808:
1.2     ! noro      809: \JP @node pari,,, $B?t$N1i;;(B
        !           810: \EG @node pari,,, Numbers
1.1       noro      811: @subsection @code{pari}
                    812: @findex pari
                    813: @cindex PARI
                    814:
                    815: @table @t
                    816: @item pari(@var{func},@var{arg},@var{prec})
1.2     ! noro      817: \JP :: @b{PARI} $B$NH!?t(B @var{func} $B$r8F$S=P$9(B.
        !           818: \EG :: Call @b{PARI} function @var{func}.
1.1       noro      819: @end table
                    820:
                    821: @table @var
                    822: @item return
1.2     ! noro      823: \JP @var{func} $BKh$K0[$J$k(B.
        !           824: \EG Depends on @var{func}.
1.1       noro      825: @item func
1.2     ! noro      826: \JP @b{PARI} $B$NH!?tL>(B
        !           827: \EG Function name of @b{PARI}.
1.1       noro      828: @item arg
1.2     ! noro      829: \JP @var{func} $B$N0z?t(B
        !           830: \EG Arguments of @var{func}.
1.1       noro      831: @item prec
1.2     ! noro      832: \JP $B@0?t(B
        !           833: \EG integer
1.1       noro      834: @end table
                    835:
                    836: @itemize @bullet
1.2     ! noro      837: \BJP
1.1       noro      838: @item
                    839: @b{PARI} $B$NH!?t$r8F$S=P$9(B.
                    840:
                    841: @item
                    842: @b{PARI} @code{[Batut et al.]} $B$O(B Bordeaux $BBg3X$G3+H/$5$l%U(B
                    843: $B%j!<%=%U%H%&%'%"$H$7$F8x3+$5$l$F$$$k(B. @b{PARI} $B$O?t<0=hM}E*$J5!G=$rM-(B
                    844: $B$7$F$O$$$k$,(B, $B<g$J%?!<%2%C%H$O@0?tO@$K4XO"$7$??t(B (@b{bignum},
                    845: @b{bigfloat}) $B$N1i;;$G(B, $B;MB'1i;;$K8B$i$:(B@b{bigfloat} $B$K$h$k$5$^$6$^$J(B
                    846: $BH!?tCM$NI>2A$r9bB.$K9T$&$3$H$,$G$-$k(B. @b{PARI} $B$OB>$N%W%m%0%i%`$+$i(B
                    847: $B%5%V%k!<%A%s%i%$%V%i%j$H$7$FMQ$$$k$3$H$,$G$-(B, $B$^$?(B, @samp{gp} $B$H$$$&(B
                    848: @b{PARI}$B%i%$%V%i%j$N%$%s%?%U%'!<%9$K$h$j(B UNIX $B$N%"%W%j%1!<%7%g%s$H$7$F(B
1.2     ! noro      849: $BMxMQ$9$k$3$H$b$G$-$k(B. $B8=:_$N%P!<%8%g%s$O(B @b{2.0.17beta} $B$G$$$/$D$+$N(B ftp
        !           850: site ($B$?$H$($P(B @code{ftp://megrez.ceremab.u-bordeaux.fr/pub/pari})
1.1       noro      851: $B$+$i(B anonymous ftp $B$G$-$k(B.
                    852: @item
                    853: $B:G8e$N0z?t(B @var{prec} $B$G7W;;@:EY$r;XDj$G$-$k(B.
                    854: @var{prec} $B$r>JN,$7$?>l9g(B @code{setprec()} $B$G;XDj$7$?@:EY$H$J$k(B.
                    855: @item
                    856: $B8=;~E@$G<B9T$G$-$k(B @b{PARI} $B$NH!?t$O<!$NDL$j$G$"$k(B. $B$$$:$l$b(B
                    857: 1 $B0z?t$G(B @b{Asir} $B$,BP1~$G$-$k7?$N0z?t$r$H$kH!?t$G$"$k(B.
                    858: $B$J$*3F!9$N5!G=$K$D$$$F$O(B @b{PARI} $B$N%^%K%e%"%k$r;2>H$N$3$H(B.
1.2     ! noro      859: \E
        !           860: \BEG
        !           861: @item
        !           862: This command connects @b{Asir} to @b{PARI} system so that several
        !           863: functions of @b{PARI} can be conveniently used from @b{Risa/Asir}.
        !           864: @item
        !           865: @b{PARI} @code{[Batut et al.]} is developed at Bordeaux University, and
        !           866: distributed as a free software.  Though it has a certain facility to computer
        !           867: algebra, its major target is the operation of numbers (@b{bignum},
        !           868: @b{bigfloat}) related to the number theory.  It facilitates various
        !           869: function evaluations as well as arithmetic operations at a remarkable
        !           870: speed.  It can also be used from other external programs as a library.
        !           871: It provides a language interface named @samp{gp} to its library, which
        !           872: enables a user to use @b{PARI} as a calculator which runs on UNIX.
        !           873: The current version is @b{2.0.17beta}.  It can be obtained by several ftp
        !           874: sites. (For example, @code{ftp://megrez.ceremab.u-bordeaux.fr/pub/pari}.)
        !           875: @item
        !           876: The last argument (optional) @var{int} specifies the precision in digits
        !           877: for bigfloat operation.
        !           878: If the precision is not explicitly specified, operation will be performed
        !           879: with the precision set by @code{setprec()}.
        !           880: @item
        !           881: Currently available functions of @b{PARI} system are as follows.
        !           882: Note these are only a part of functions in @b{PARI} system.
        !           883: For details of individual functions, refer to the @b{PARI} manual.
        !           884: (Some of them can be seen in the following example.)
        !           885: \E
1.1       noro      886:
                    887: @code{abs},
                    888: @code{adj},
                    889: @code{arg},
                    890: @code{bigomega},
                    891: @code{binary},
                    892: @code{ceil},
                    893: @code{centerlift},
                    894: @code{cf},
                    895: @code{classno},
                    896: @code{classno2},
                    897: @code{conj},
                    898: @code{content},
                    899: @code{denom},
                    900: @code{det},
                    901: @code{det2},
                    902: @code{detr},
                    903: @code{dilog},
                    904: @code{disc},
                    905: @code{discf},
                    906: @code{divisors},
                    907: @code{eigen},
                    908: @code{eintg1},
                    909: @code{erfc},
                    910: @code{eta},
                    911: @code{floor},
                    912: @code{frac},
                    913: @code{galois},
                    914: @code{galoisconj},
                    915: @code{gamh},
                    916: @code{gamma},
                    917: @code{hclassno},
                    918: @code{hermite},
                    919: @code{hess},
                    920: @code{imag},
                    921: @code{image},
                    922: @code{image2},
                    923: @code{indexrank},
                    924: @code{indsort},
                    925: @code{initalg},
                    926: @code{isfund},
                    927: @code{isprime},
                    928: @code{ispsp},
                    929: @code{isqrt},
                    930: @code{issqfree},
                    931: @code{issquare},
                    932: @code{jacobi},
                    933: @code{jell},
                    934: @code{ker},
                    935: @code{keri},
                    936: @code{kerint},
                    937: @code{kerintg1},
                    938: @code{kerint2},
                    939: @code{kerr},
                    940: @code{length},
                    941: @code{lexsort},
                    942: @code{lift},
                    943: @code{lindep},
                    944: @code{lll},
                    945: @code{lllg1},
                    946: @code{lllgen},
                    947: @code{lllgram},
                    948: @code{lllgramg1},
                    949: @code{lllgramgen},
                    950: @code{lllgramint},
                    951: @code{lllgramkerim},
                    952: @iftex
                    953: @break
                    954: @end iftex
                    955: @code{lllgramkerimgen},
                    956: @code{lllint},
                    957: @code{lllkerim},
                    958: @code{lllkerimgen},
                    959: @code{lllrat},
                    960: @code{lngamma},
                    961: @code{logagm},
                    962: @code{mat},
                    963: @code{matinvr},
                    964: @code{matrixqz2},
                    965: @code{matrixqz3},
                    966: @code{matsize},
                    967: @code{modreverse},
                    968: @code{mu},
                    969: @code{nextprime},
                    970: @code{norm},
                    971: @code{norml2},
                    972: @code{numdiv},
                    973: @code{numer},
                    974: @code{omega},
                    975: @code{order},
                    976: @code{ordred},
                    977: @code{phi},
                    978: @code{pnqn},
                    979: @code{polred},
                    980: @code{polred2},
                    981: @code{primroot},
                    982: @code{psi},
                    983: @code{quadgen},
                    984: @code{quadpoly},
                    985: @code{real},
                    986: @code{recip},
                    987: @code{redcomp},
                    988: @code{redreal},
                    989: @code{regula},
                    990: @code{reorder},
                    991: @code{reverse},
                    992: @code{rhoreal},
                    993: @code{roots},
                    994: @code{rootslong},
                    995: @code{round},
                    996: @code{sigma},
                    997: @code{signat},
                    998: @code{simplify},
                    999: @code{smalldiscf},
                   1000: @code{smallfact},
                   1001: @code{smallpolred},
                   1002: @code{smallpolred2},
                   1003: @code{smith},
                   1004: @code{smith2},
                   1005: @code{sort},
                   1006: @code{sqr},
                   1007: @code{sqred},
                   1008: @code{sqrt},
                   1009: @code{supplement},
                   1010: @code{trace},
                   1011: @code{trans},
                   1012: @code{trunc},
                   1013: @code{type},
                   1014: @code{unit},
                   1015: @code{vec},
                   1016: @code{wf},
                   1017: @code{wf2},
                   1018: @code{zeta}
                   1019:
1.2     ! noro     1020: \BJP
1.1       noro     1021: @item
                   1022: @b{Asir} $B$GMQ$$$F$$$k$N$O(B @b{PARI} $B$N$[$s$N0lIt$N5!G=$G$"$k$,(B, $B:#8e(B
                   1023: $B$h$jB?$/$N5!G=$,MxMQ$G$-$k$h$&2~NI$9$kM=Dj$G$"$k(B.
1.2     ! noro     1024: \E
        !          1025: \BEG
        !          1026: @item
        !          1027: @b{Asir} currently uses only a very small subset of @b{PARI}.
        !          1028: We will improve @b{Asir} so that it can provide more functions of
        !          1029: @b{PARI}.
        !          1030: \E
1.1       noro     1031: @end itemize
                   1032:
                   1033: @example
1.2     ! noro     1034: \JP /* $B9TNs$N8GM-%Y%/%H%k$r5a$a$k(B. */
        !          1035: \EG /* Eigen vectors of a numerical matrix */
1.1       noro     1036: [0] pari(eigen,newmat(2,2,[[1,1],[1,2]]));
                   1037: [ -1.61803398874989484819771921990 0.61803398874989484826 ]
                   1038: [ 1 1 ]
1.2     ! noro     1039: \JP /* 1 $BJQ?tB?9`<0$N:,$r5a$a$k(B. */
        !          1040: \EG /* Roots of a polynomial */
1.1       noro     1041: [1] pari(roots,t^2-2);
                   1042: [ -1.41421356237309504876 1.41421356237309504876 ]
                   1043: @end example
                   1044:
                   1045: @table @t
1.2     ! noro     1046: \JP @item $B;2>H(B
        !          1047: \EG @item References
1.1       noro     1048: @fref{setprec}.
                   1049: @end table
                   1050:
1.2     ! noro     1051: \JP @node setprec,,, $B?t$N1i;;(B
        !          1052: \EG @node setprec,,, Numbers
1.1       noro     1053: @subsection @code{setprec}
                   1054: @findex setprec
                   1055: @cindex PARI
                   1056:
                   1057: @table @t
                   1058: @item setprec([@var{n}])
1.2     ! noro     1059: \JP :: @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B.
        !          1060: \EG :: Sets the precision for @b{bigfloat} operations to @var{n} digits.
1.1       noro     1061: @end table
                   1062:
                   1063: @table @var
                   1064: @item return
1.2     ! noro     1065: \JP $B@0?t(B
        !          1066: \EG integer
1.1       noro     1067: @item n
1.2     ! noro     1068: \JP $B@0?t(B
        !          1069: \EG integer
1.1       noro     1070: @end table
                   1071:
                   1072: @itemize @bullet
1.2     ! noro     1073: \BJP
1.1       noro     1074: @item
                   1075: $B0z?t$,$"$k>l9g(B, @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B.
                   1076: $B0z?t$N$"$k$J$7$K$+$+$o$i$:(B, $B0JA0$K@_Dj$5$l$F$$$?CM$rJV$9(B.
                   1077: @item
                   1078: @b{bigfloat} $B$N7W;;$O(B @b{PARI} (@xref{pari}) $B$K$h$C$F9T$o$l$k(B.
                   1079: @item
                   1080: @b{bigfloat} $B$G$N7W;;$KBP$7M-8z$G$"$k(B.
                   1081: @b{bigfloat} $B$N(B flag $B$r(B on $B$K$9$kJ}K!$O(B, @code{ctrl} $B$r;2>H(B.
                   1082: @item
                   1083: $B@_Dj$G$-$k7e?t$K>e8B$O$J$$$,(B, $B;XDj$7$?7e?t$K@_Dj$5$l$k$H$O(B
                   1084: $B8B$i$J$$(B. $BBg$-$a$NCM$r@_Dj$9$k$N$,0BA4$G$"$k(B.
1.2     ! noro     1085: \E
        !          1086: \BEG
        !          1087: @item
        !          1088: When an argument is given, it
        !          1089: sets the precision for @b{bigfloat} operations to @var{n} digits.
        !          1090: The return value is always the previous precision in digits regardless of
        !          1091: the existence of an argument.
        !          1092:
        !          1093: @item
        !          1094: @b{Bigfloat} operations are done by @b{PARI}. (@xref{pari})
        !          1095: @item
        !          1096: This is effective for computations in @b{bigfloat}.
        !          1097: Refer to @code{ctrl()} for turning on the `@b{bigfloat} flag.'
        !          1098: @item
        !          1099: There is no upper limit for precision digits.
        !          1100: It sets the precision to some digits around the specified precision.
        !          1101: Therefore, it is safe to specify a larger value.
        !          1102: \E
1.1       noro     1103: @end itemize
                   1104:
                   1105: @example
                   1106: [1] setprec();
                   1107: 9
                   1108: [2] setprec(100);
                   1109: 9
                   1110: [3] setprec(100);
                   1111: 96
                   1112: @end example
                   1113:
                   1114: @table @t
1.2     ! noro     1115: \JP @item $B;2>H(B
1.1       noro     1116: @fref{ctrl}, @fref{eval}, @fref{pari}.
                   1117: @end table
                   1118:
1.2     ! noro     1119: \JP @node setmod,,, $B?t$N1i;;(B
        !          1120: \EG @node setmod,,, Numbers
1.1       noro     1121: @subsection @code{setmod}
                   1122: @findex setmod
                   1123:
                   1124: @table @t
                   1125: @item setmod([@var{p}])
1.2     ! noro     1126: \JP :: $BM-8BBN$r(B GF(@var{p}) $B$K@_Dj$9$k(B.
        !          1127: \EG :: Sets the ground field to GF(@var{p}).
1.1       noro     1128: @end table
                   1129:
                   1130: @table @var
                   1131: @item return
1.2     ! noro     1132: \JP $B@0?t(B
        !          1133: \EG integer
1.1       noro     1134: @item n
1.2     ! noro     1135: \JP 2^27 $BL$K~$NAG?t(B
        !          1136: \EG prime less than 2^27
1.1       noro     1137: @end table
                   1138:
                   1139: @itemize @bullet
1.2     ! noro     1140: \BJP
1.1       noro     1141: @item
                   1142: $BM-8BBN$r(B GF(@var{p}) $B$K@_Dj$9$k(B. $B@_DjCM$rJV$9(B.
                   1143: @item
                   1144: $BM-8BBN$N85$N7?$r;}$D?t$O(B, $B$=$l<+?H$O$I$NM-8BBN$KB0$9$k$+$N>pJs$r;}$?$:(B,
                   1145: $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.
1.2     ! noro     1146: @item
        !          1147: $B0L?t$NBg$-$JM-8BBN$K4X$7$F$O(B @pxref{$BM-8BBN$K4X$9$k1i;;(B}.
        !          1148: \E
        !          1149: \BEG
        !          1150: @item
        !          1151: Sets the ground field to GF(@var{p}) and returns the value @var{p}.
        !          1152: @item
        !          1153: A member of a finite field does not have any information
        !          1154: about the field and the arithmetic operations over GF(@var{p}) are applied
        !          1155: with @var{p} set at the time.
        !          1156: @item
        !          1157: As for large finite fields, @pxref{Finite fields}.
        !          1158: \E
1.1       noro     1159: @end itemize
                   1160:
                   1161: @example
                   1162: [0] A=dp_mod(dp_ptod(2*x,[x]),3,[]);
                   1163: (2)*<<1>>
                   1164: [1] A+A;
                   1165: addmi : invalid modulus
                   1166: return to toplevel
                   1167: [1] setmod(3);
                   1168: 3
                   1169: [2] A+A;
                   1170: (1)*<<1>>
                   1171: @end example
                   1172:
                   1173: @table @t
1.2     ! noro     1174: \JP @item $B;2>H(B
        !          1175: \EG @item References
        !          1176: \JP @fref{dp_mod dp_rat}, @fref{$B?t$N7?(B}.
        !          1177: \EG @fref{dp_mod dp_rat}, @fref{Types of numbers}.
1.1       noro     1178: @end table
                   1179: