Annotation of OpenXM/src/asir-doc/parts/builtin/num.texi, Revision 1.11
1.11 ! noro 1: @comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/num.texi,v 1.10 2003/12/20 20:02:28 ohara Exp $
1.2 noro 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::
1.10 ohara 16: * isqrt::
1.1 noro 17: * inv::
18: * prime lprime::
19: * random::
20: * mt_save mt_load::
21: * nm dn::
22: * conj real imag::
1.4 noro 23: * eval deval::
1.1 noro 24: * pari::
25: * setprec::
26: * setmod::
27: * lrandom::
1.3 noro 28: * ntoint32 int32ton::
1.1 noro 29: @end menu
30:
1.2 noro 31: \JP @node idiv irem,,, $B?t$N1i;;(B
32: \EG @node idiv irem,,, Numbers
1.1 noro 33: @subsection @code{idiv}, @code{irem}
34: @findex idiv
35: @findex irem
36:
37: @table @t
38: @item idiv(@var{i1},@var{i2})
1.2 noro 39: \JP :: $B@0?t=|;;$K$h$k>&(B.
40: \EG :: Integer quotient of @var{i1} divided by @var{i2}.
1.1 noro 41: @item irem(@var{i1},@var{i2})
1.2 noro 42: \JP :: $B@0?t=|;;$K$h$k>jM>(B.
43: \EG :: Integer remainder of @var{i1} divided by @var{i2}.
1.1 noro 44: @end table
45:
46: @table @var
47: @item return
1.2 noro 48: \JP $B@0?t(B
49: \EG integer
1.8 noro 50: @item i1 i2
1.2 noro 51: \JP $B@0?t(B
52: \EG integer
1.1 noro 53: @end table
54:
55: @itemize @bullet
1.2 noro 56: \BJP
1.1 noro 57: @item
58: @var{i1} $B$N(B @var{i2} $B$K$h$k@0?t=|;;$K$h$k>&(B, $B>jM>$r5a$a$k(B.
59: @item
60: @var{i2} $B$O(B 0 $B$G$"$C$F$O$J$i$J$$(B.
61: @item
62: $BHo=|?t$,Ii$N>l9g(B, $B@dBPCM$KBP$9$kCM$K%^%$%J%9$r$D$1$?CM$rJV$9(B.
63: @item
64: @var{i1} @code{%} @var{i2} $B$O(B, $B7k2L$,@5$K@55,2=$5$l$k$3$H$r=|$1$P(B
65: @code{irem()} $B$NBe$o$j$KMQ$$$k$3$H$,$G$-$k(B.
66: @item
67: $BB?9`<0$N>l9g$O(B @code{sdiv}, @code{srem} $B$rMQ$$$k(B.
1.2 noro 68: \E
69: \BEG
70: @item
71: Integer quotient and remainder of @var{i1} divided by @var{i2}.
72: @item
73: @var{i2} must not be 0.
74: @item
75: If the dividend is negative, the results are obtained by changing the
76: sign of the results for absolute values of the dividend.
77: @item
78: One can use
79: @var{i1} @code{%} @var{i2}
80: for replacement of @code{irem()} which only differs in the point that
81: the result is always normalized to non-negative values.
82: @item
83: Use @code{sdiv()}, @code{srem()} for polynomial quotient.
84: \E
1.1 noro 85: @end itemize
86:
87: @example
88: [0] idiv(100,7);
89: 14
90: [0] idiv(-100,7);
91: -14
92: [1] irem(100,7);
93: 2
94: [1] irem(-100,7);
95: -2
96: @end example
97:
98: @table @t
1.2 noro 99: \JP @item $B;2>H(B
100: \EG @item References
1.1 noro 101: @fref{sdiv sdivm srem sremm sqr sqrm}, @fref{%}.
102: @end table
103:
1.2 noro 104: \JP @node fac,,, $B?t$N1i;;(B
105: \EG @node fac,,, Numbers
1.1 noro 106: @subsection @code{fac}
107: @findex fac
108:
109: @table @t
110: @item fac(@var{i})
1.2 noro 111: \JP :: @var{i} $B$N3,>h(B.
112: \EG :: The factorial of @var{i}.
1.1 noro 113: @end table
114:
115: @table @var
116: @item return
1.2 noro 117: \JP $B@0?t(B
118: \EG integer
1.1 noro 119: @item i
1.2 noro 120: \JP $B@0?t(B
121: \EG integer
1.1 noro 122: @end table
123:
124: @itemize @bullet
1.2 noro 125: \BJP
1.1 noro 126: @item
127: @var{i} $B$N3,>h$r7W;;$9$k(B.
128: @item
129: @var{i} $B$,Ii$N>l9g$O(B 0 $B$rJV$9(B.
1.2 noro 130: \E
131: \BEG
132: @item
133: The factorial of @var{i}.
134: @item
135: Returns 0 if the argument @var{i} is negative.
136: \E
1.1 noro 137: @end itemize
138:
139: @example
140: [0] fac(50);
141: 30414093201713378043612608166064768844377641568960512000000000000
142: @end example
143:
1.2 noro 144: \JP @node igcd igcdcntl,,, $B?t$N1i;;(B
145: \EG @node igcd igcdcntl,,, Numbers
1.1 noro 146: @subsection @code{igcd},@code{igcdcntl}
147: @findex igcd
148: @findex igcdcntl
149:
150: @table @t
151: @item igcd(@var{i1},@var{i2})
1.2 noro 152: \JP :: $B@0?t$N(B GCD ($B:GBg8xLs?t(B)
153: \EG :: The integer greatest common divisor of @var{i1} and @var{i2}.
1.1 noro 154: @item igcdcntl([@var{i}])
1.2 noro 155: \JP :: $B@0?t(B GCD$B$N%"%k%4%j%:%`A*Br(B
156: \EG :: Selects an algorithm for integer GCD.
1.1 noro 157: @end table
158:
159: @table @var
160: @item return
1.2 noro 161: \JP $B@0?t(B
162: \EG integer
1.8 noro 163: @item i1 i2 i
1.2 noro 164: \JP $B@0?t(B
165: \EG integer
1.1 noro 166: @end table
167:
168: @itemize @bullet
1.2 noro 169: \BJP
1.1 noro 170: @item
171: @code{igcd} $B$O(B @var{i1} $B$H(B @var{i2} $B$N(B GCD $B$r5a$a$k(B.
172: @item
173: $B0z?t$,@0?t$G$J$$>l9g$O(B, $B%(%i!<$^$?$OL50UL#$J7k2L$rJV$9(B.
174: @item
175: $BB?9`<0$N>l9g$O(B, @code{gcd}, @code{gcdz} $B$rMQ$$$k(B.
176: @item
177: $B@0?t(B GCD $B$K$O$5$^$6$^$JJ}K!$,$"$j(B, @code{igcdcntl} $B$G@_Dj$G$-$k(B.
178:
179: @table @code
180: @item 0
181: Euclid $B8_=|K!(B (default)
182: @item 1
183: binary GCD
184: @item 2
185: bmod GCD
186: @item 3
187: accelerated integer GCD
188: @end table
1.2 noro 189: @code{2}, @code{3} $B$O(B @code{[Weber]} $B$K$h$k(B.
1.1 noro 190:
1.2 noro 191: $B$*$*$`$M(B @code{3} $B$,9bB.$@$,(B, $BNc30$b$"$k(B.
192: \E
193: \BEG
194: @item
195: Function @code{igcd()} returns the integer greatest common divisor of
196: the given two integers.
197: @item
198: An error will result if the argument is not an integer; the result is
199: not valid even if one is returned.
200: @item
201: Use @code{gcd()}, @code{gcdz()} for polynomial GCD.
202:
203: @item
204: Various method of integer GCD computation are implemented
205: and they can be selected by @code{igcdcntl}.
206:
207: @table @code
208: @item 0
209: Euclid algorithm (default)
210: @item 1
211: binary GCD
212: @item 2
213: bmod GCD
214: @item 3
215: accelerated integer GCD
216: @end table
217: @code{2}, @code{3} are due to @code{[Weber]}.
218:
219: In most cases @code{3} is the fastest, but there are exceptions.
220: \E
1.1 noro 221: @end itemize
222:
223: @example
224: [0] A=lrandom(10^4)$
225: [1] B=lrandom(10^4)$
226: [2] C=lrandom(10^4)$
227: [3] D=A*C$
228: [4] E=A*B$
229: [5] cputime(1)$
230: [6] igcd(D,E)$
231: 0.6sec + gc : 1.93sec(2.531sec)
232: [7] igcdcntl(1)$
233: [8] igcd(D,E)$
234: 0.27sec(0.2635sec)
235: [9] igcdcntl(2)$
236: [10] igcd(D,E)$
237: 0.19sec(0.1928sec)
238: [11] igcdcntl(3)$
239: [12] igcd(D,E)$
240: 0.08sec(0.08023sec)
241: @end example
242:
243: @table @t
1.2 noro 244: \JP @item $B;2>H(B
245: \EG @item References
1.1 noro 246: @fref{gcd gcdz}.
247: @end table
248:
1.2 noro 249: \JP @node ilcm,,, $B?t$N1i;;(B
250: \EG @node ilcm,,, Numbers
1.1 noro 251: @subsection @code{ilcm}
252: @findex ilcm
253:
254: @table @t
255: @item ilcm(@var{i1},@var{i2})
1.2 noro 256: \JP :: $B:G>.8xG\?t$r5a$a$k(B.
257: \EG :: The integer least common multiple of @var{i1} and @var{i2}.
1.1 noro 258: @end table
259:
260: @table @var
261: @item return
1.2 noro 262: \JP $B@0?t(B
263: \EG integer
1.8 noro 264: @item i1 i2
1.2 noro 265: \JP $B@0?t(B
266: \EG integer
1.1 noro 267: @end table
268:
269: @itemize @bullet
1.2 noro 270: \BJP
1.1 noro 271: @item
272: $B@0?t(B @var{i1}, @var{i2} $B$N:G>.8xG\?t$r5a$a$k(B.
273: @item
274: $B0lJ}$,(B 0 $B$N>l9g(B 0 $B$rJV$9(B.
1.2 noro 275: \E
276: \BEG
1.1 noro 277: @item
1.2 noro 278: This function computes the integer least common multiple of
279: @var{i1}, @var{i2}.
280: @item
281: If one of argument is equal to 0, the return 0.
282: \E
1.1 noro 283: @end itemize
284:
285: @table @t
1.2 noro 286: \JP @item $B;2>H(B
287: \EG @item References
1.1 noro 288: @fref{igcd igcdcntl}, @fref{mt_save mt_load}.
1.9 ohara 289: @end table
290:
291: \JP @node isqrt,,, $B?t$N1i;;(B
292: \EG @node isqrt,,, Numbers
293: @subsection @code{isqrt}
294: @findex isqrt
295:
296: @table @t
297: @item isqrt(@var{n})
298: \JP :: $BJ?J}:,$r1[$($J$$:GBg$N@0?t$r5a$a$k(B.
299: \EG :: The integer square root of @var{n}.
300: @end table
301:
302: @table @var
303: @item return
304: \JP $BHsIi@0?t(B
305: \EG non-negative integer
306: @item n
307: \JP $BHsIi@0?t(B
308: \EG non-negative integer
1.1 noro 309: @end table
1.2 noro 310:
311: \JP @node inv,,, $B?t$N1i;;(B
312: \EG @node inv,,, Numbers
1.1 noro 313: @subsection @code{inv}
314: @findex inv
315:
316: @table @t
317: @item inv(@var{i},@var{m})
1.2 noro 318: \JP :: @var{m} $B$rK!$H$9$k(B @var{i} $B$N5U?t(B
319: \EG :: the inverse (reciprocal) of @var{i} modulo @var{m}.
1.1 noro 320: @end table
321:
322: @table @var
323: @item return
1.2 noro 324: \JP $B@0?t(B
325: \EG integer
1.8 noro 326: @item i m
1.2 noro 327: \JP $B@0?t(B
328: \EG integer
1.1 noro 329: @end table
330:
331: @itemize @bullet
1.2 noro 332: \BJP
1.1 noro 333: @item
334: @var{ia} @equiv{} 1 mod (@var{m}) $B$J$k@0?t(B @var{a} $B$r5a$a$k(B.
335: @item
336: @var{i} $B$H(B @var{m} $B$O8_$$$KAG$G$J$1$l$P$J$i$J$$$,(B, @code{inv()} $B$O(B
337: $B$=$N%A%'%C%/$O9T$o$J$$(B.
1.2 noro 338: \E
339: \BEG
340: @item
341: This function computes an integer such that
342: @var{ia} @equiv{} 1 mod (@var{m}).
343: @item
344: The integer @var{i} and @var{m} must be mutually prime.
345: However, @code{inv()} does not check it.
346: \E
1.1 noro 347: @end itemize
348:
349: @example
350: [71] igcd(1234,4321);
351: 1
352: [72] inv(1234,4321);
353: 3239
354: [73] irem(3239*1234,4321);
355: 1
356: @end example
357:
358: @table @t
1.2 noro 359: \JP @item $B;2>H(B
360: \EG @item References
1.1 noro 361: @fref{igcd igcdcntl}.
362: @end table
363:
1.2 noro 364: \JP @node prime lprime,,, $B?t$N1i;;(B
365: \EG @node prime lprime,,, Numbers
1.1 noro 366: @subsection @code{prime}, @code{lprime}
367: @findex prime
368: @findex lprime
369:
370: @table @t
371: @item prime(@var{index})
372: @item lprime(@var{index})
1.2 noro 373: \JP :: $BAG?t$rJV$9(B
374: \EG :: Returns a prime number.
1.1 noro 375: @end table
376:
377: @table @var
378: @item return
1.2 noro 379: \JP $B@0?t(B
380: \EG integer
1.1 noro 381: @item index
1.2 noro 382: \JP $B@0?t(B
383: \EG integer
1.1 noro 384: @end table
385:
386: @itemize @bullet
1.2 noro 387: \BJP
1.1 noro 388: @item
389: @code{prime()}, @code{lprime()} $B$$$:$l$b%7%9%F%`$,FbIt$K;}$D(B
390: $BAG?tI=$NMWAG$rJV$9(B. @code{index} $B$O(B 0 $B0J>e$N@0?t$G(B, $BAG?tI=(B
391: $B$N%$%s%G%C%/%9$KMQ$$$i$l$k(B. @code{prime()} $B$O(B 16381 $B$^$G(B
392: $B$NAG?t$r>.$5$$=g$K(B 1900 $B8D(B, @code{lprime()} $B$O(B, 10 $B?J(B 8 $B7e$G:GBg$N(B
393: $BAG?t$+$iBg$-$$=g$K(B 999 $B8DJV$9(B. $B$=$l0J30$N%$%s%G%C%/%9$KBP$7$F$O(B
394: 0 $B$rJV$9(B.
395: @item
1.2 noro 396: $B$h$j0lHLE*$JAG?t@8@.H!?t$H$7$F$O(B,
397: @code{pari(nextprime,@var{number})}
1.1 noro 398: $B$,$"$k(B.
1.2 noro 399: \E
400: \BEG
401: @item
402: The two functions, @code{prime()} and @code{lprime()}, returns
403: an element stored in the system table of prime numbers.
404: Here, @code{index} is a non-negative integer and be used as an index
405: for the prime tables.
406: The function @code{prime()} can return one of 1900 primes
407: up to 16381 indexed so that the smaller one has smaller
408: index. The function @code{lprime()} can return one of 999 primes which
409: are 8 digit sized and indexed so that the larger one has the smaller
410: index.
411: The two function always returns 0 for other indices.
412: @item
413: For more general function for prime generation, there is a @code{PARI}
414: function
415:
416: @code{pari(nextprime,@var{number})}.
417: \E
1.1 noro 418: @end itemize
419:
420: @example
421: [95] prime(0);
422: 2
423: [96] prime(1228);
424: 9973
425: [97] lprime(0);
426: 99999989
427: [98] lprime(999);
428: 0
429: @end example
430:
431: @table @t
1.2 noro 432: \JP @item $B;2>H(B
433: \EG @item References
1.1 noro 434: @fref{pari}.
435: @end table
436:
1.2 noro 437: \JP @node random,,, $B?t$N1i;;(B
438: \EG @node random,,, Numbers
1.1 noro 439: @subsection @code{random}
440: @findex random
441:
442: @table @t
1.5 takayama 443: @item random([@var{seed}])
1.2 noro 444: \JP :: $BMp?t$r@8@.$9$k(B.
1.1 noro 445: @end table
446:
447: @table @var
448: @item seed
1.2 noro 449: @itemx return
450: \JP $B<+A3?t(B
451: \EG non-negative integer
1.1 noro 452: @end table
453:
454: @itemize @bullet
1.2 noro 455: \BJP
1.1 noro 456: @item
457: $B:GBg(B 2^32-1 $B$NHsIi@0?t$NMp?t$r@8@.$9$k(B.
458: @item
459: 0 $B$G$J$$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.
460: @item
461: 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
462: $B7ONs$O5/F0Kh$K0lDj$G$"$k(B.
463: @item
464: $B>>K\bC(B-$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.
465: @item
466: $B<~4|$O(B 2^19937-1 $B$HHs>o$KD9$$(B.
467: @item
468: @code{mt_save} $B$K$h$j(B state $B$r%U%!%$%k$K(B save $B$G$-$k(B. $B$3$l$r(B @code{mt_load}
469: $B$GFI$_9~$`$3$H$K$h$j(B, $B0[$k(B Asir $B%;%C%7%g%s4V$G0l$D$NMp?t$N7ONs$rC)$k$3$H$,(B
470: $B$G$-$k(B.
1.2 noro 471: \E
472: \BEG
473: @item
474: Generates a random number which is a non-negative integer less than 2^32.
475: @item
476: If a non zero argument is specified, then after setting it as a random seed,
477: a random number is generated.
478: @item
479: As the default seed is fixed, the sequence of the random numbers is
480: always the same if a seed is not set.
481: @item
482: The algorithm is Mersenne Twister
483: (http://www.math.keio.ac.jp/matsumoto/mt.html) by M. Matsumoto and
484: T. Nishimura. The implementation is done also by themselves.
485: @item
486: The period of the random number sequence is 2^19937-1.
487: @item
488: One can save the state of the random number generator with @code{mt_save}.
489: By loading the state file with @code{mt_load},
490: one can trace a single random number sequence arcoss multiple sessions.
491: \E
1.1 noro 492: @end itemize
493:
494: @table @t
1.2 noro 495: \JP @item $B;2>H(B
496: \EG @item References
1.1 noro 497: @fref{lrandom}, @fref{mt_save mt_load}.
498: @end table
499:
1.2 noro 500: \JP @node lrandom,,, $B?t$N1i;;(B
501: \EG @node lrandom,,, Numbers
1.1 noro 502: @subsection @code{lrandom}
503: @findex lrandom
504:
505: @table @t
1.5 takayama 506: @item lrandom(@var{bit})
1.2 noro 507: \JP :: $BB?G\D9Mp?t$r@8@.$9$k(B.
508: \EG :: Generates a long random number.
1.1 noro 509: @end table
510:
511: @table @var
512: @item bit
513: @item return
1.2 noro 514: \JP $B<+A3?t(B
515: \EG integer
1.1 noro 516: @end table
517:
518: @itemize @bullet
1.2 noro 519: \BJP
1.1 noro 520: @item
521: $B9b!9(B @var{bit} $B$NHsIi@0?t$NMp?t$r@8@.$9$k(B.
522: @item
523: @code{random} $B$rJ#?t2s8F$S=P$7$F7k9g$7(B, $B;XDj$N(B bit $BD9$K%^%9%/$7$F$$$k(B.
1.2 noro 524: \E
525: \BEG
526: @item
527: Generates a non-negative integer of at most @var{bit} bits.
528: @item
529: The result is a concatination of outputs of @code{random}.
530: \E
1.1 noro 531: @end itemize
532:
533: @table @t
1.2 noro 534: \JP @item $B;2>H(B
535: \EG @item References
1.1 noro 536: @fref{random}, @fref{mt_save mt_load}.
537: @end table
538:
1.2 noro 539: \JP @node mt_save mt_load,,, $B?t$N1i;;(B
540: \EG @node mt_save mt_load,,, Numbers
1.1 noro 541: @subsection @code{mt_save}, @code{mt_load}
542: @findex mt_save
543: @findex mt_load
544:
545: @table @t
546: @item mt_save(@var{fname})
1.2 noro 547: \JP :: $BMp?t@8@.4o$N8=:_$N>uBV$r%U%!%$%k$K%;!<%V$9$k(B.
548: \EG :: Saves the state of the random number generator.
1.1 noro 549: @item mt_load(@var{fname})
1.2 noro 550: \JP :: $B%U%!%$%k$K%;!<%V$5$l$?Mp?t@8@.4o$N>uBV$r%m!<%I$9$k(B.
551: \EG :: Loads a saved state of the random number generator.
1.1 noro 552: @end table
553:
554: @table @var
555: @item return
1.2 noro 556: \JP 0 $B$^$?$O(B 1
557: \EG 0 or 1
1.1 noro 558: @item fname
1.2 noro 559: \JP $BJ8;zNs(B
560: \EG string
1.1 noro 561: @end table
562:
563: @itemize @bullet
1.2 noro 564: \BJP
565: @item
566: $B$"$k>uBV$r%;!<%V$7(B, $B$=$N>uBV$r%m!<%I$9$k$3$H$G(B,
1.1 noro 567: $B0l$D$N5?;wMp?t7ONs$r(B, $B?75,$N(B Asir $B%;%C%7%g%s$GB3$1$F$?$I$k$3$H$,(B
568: $B$G$-$k(B.
1.2 noro 569: \E
570: \BEG
571: @item
572: One can save the state of the random number generator with @code{mt_save}.
573: By loading the state file with @code{mt_load},
574: one can trace a single random number sequence arcoss multiple
575: @b{Asir} sessions.
576: \E
1.1 noro 577: @end itemize
578:
579: @example
580: [340] random();
581: 3510405877
582: [341] mt_save("/tmp/mt_state");
583: 1
584: [342] random();
585: 4290933890
586: [343] quit;
587: % asir
588: This is Asir, Version 991108.
589: Copyright (C) FUJITSU LABORATORIES LIMITED.
590: 3 March 1994. All rights reserved.
591: [340] mt_load("/tmp/mt_state");
592: 1
593: [341] random();
594: 4290933890
595: @end example
596:
597: @table @t
1.2 noro 598: \JP @item $B;2>H(B
599: \EG @item References
1.1 noro 600: @fref{random}, @fref{lrandom}.
601: @end table
602:
1.2 noro 603: \JP @node nm dn,,, $B?t$N1i;;(B
604: \EG @node nm dn,,, Numbers
1.1 noro 605: @subsection @code{nm}, @code{dn}
606: @findex nm
607: @findex dn
608:
609: @table @t
610: @item nm(@var{rat})
1.2 noro 611: \JP :: @var{rat} $B$NJ,;R(B.
612: \EG :: Numerator of @var{rat}.
1.1 noro 613: @item dn(@var{rat})
1.2 noro 614: \JP :: @var{rat} $B$NJ,Jl(B.
615: \EG :: Denominator of @var{rat}.
1.1 noro 616: @end table
617:
618: @table @var
619: @item return
1.2 noro 620: \JP $B@0?t$^$?$OB?9`<0(B
621: \EG integer or polynomial
1.1 noro 622: @item rat
1.2 noro 623: \JP $BM-M}?t$^$?$OM-M}<0(B
624: \EG rational number or rational expression
1.1 noro 625: @end table
626:
627: @itemize @bullet
1.2 noro 628: \BJP
1.1 noro 629: @item
630: $BM?$($i$l$?M-M}?t$^$?M-M}<0$NJ,;R5Z$SJ,Jl$rJV$9(B.
631: @item
632: $BM-M}?t$N>l9g(B, $BJ,Jl$O>o$K@5$G(B, $BId9f$OJ,;R$,;}$D(B.
633: @item
634: $BM-M}<0$N>l9g(B, $BC1$KJ,Jl(B, $BJ,;R$r<h$j=P$9$@$1$G$"$k(B.
635: $BM-M}<0$KBP$7$F$O(B, $BLsJ,$O<+F0E*$K$O9T$o$l$J$$(B. @code{red()}
636: $B$rL@<(E*$K8F$S=P$9I,MW$,$"$k(B.
1.2 noro 637: \E
638: \BEG
639: @item
640: Numerator and denominator of a given rational expression.
641: @item
642: For a rational number, they return its numerator and denominator,
643: respectively. For a rational expression whose numerator and denominator
644: may contain rational numbers, they do not separate those rational
645: coefficients to numerators and denominators.
646: @item
647: For a rational number, the denominator is always kept positive, and
648: the sign is contained in the numerator.
649: @item
650: @b{Risa/Asir} does not cancel the common divisors unless otherwise explicitly
651: specified by the user.
652: Therefore, @code{nm()} and @code{dn()} return the numerator and the
653: denominator as it is, respectively.
654: \E
1.1 noro 655: @end itemize
656:
657: @example
658: [2] [nm(-43/8),dn(-43/8)];
659: [-43,8]
660: [3] dn((x*z)/(x*y));
661: y*x
662: [3] dn(red((x*z)/(x*y)));
663: y
664: @end example
665:
666: @table @t
1.2 noro 667: \JP @item $B;2>H(B
668: \EG @item References
1.1 noro 669: @fref{red}.
670: @end table
671:
1.2 noro 672: \JP @node conj real imag,,, $B?t$N1i;;(B
673: \EG @node conj real imag,,, Numbers
1.1 noro 674: @subsection @code{conj}, @code{real}, @code{imag}
675: @findex conj
676:
677: @table @t
678: @item real(@var{comp})
1.2 noro 679: \JP :: @var{comp} $B$N<B?tItJ,(B.
680: \EG :: Real part of @var{comp}.
1.1 noro 681: @item imag(@var{comp})
1.2 noro 682: \JP :: @var{comp} $B$N5u?tItJ,(B.
683: \EG :: Imaginary part of @var{comp}.
1.1 noro 684: @item conj(@var{comp})
1.2 noro 685: \JP :: @var{comp} $B$N6&LrJ#AG?t(B.
686: \EG :: Complex conjugate of @var{comp}.
1.1 noro 687: @end table
688:
689: @table @var
690: @item return comp
1.2 noro 691: \JP $BJ#AG?t(B
692: \EG complex number
1.1 noro 693: @end table
694:
695: @itemize @bullet
1.2 noro 696: \BJP
1.1 noro 697: @item
698: $BJ#AG?t$KBP$7(B, $B<BIt(B, $B5uIt(B, $B6&Lr$r5a$a$k(B.
699: @item
700: $B$3$l$i$O(B, $BB?9`<0$KBP$7$F$bF/$/(B.
1.2 noro 701: \E
702: \BEG
703: @item
704: Basic operations for complex numbers.
705: @item
706: These functions works also for polynomials with complex coefficients.
707: \E
1.1 noro 708: @end itemize
709:
710: @example
711: [111] A=(2+@@i)^3;
712: (2+11*@@i)
713: [112] [real(A),imag(A),conj(A)];
714: [2,11,(2-11*@@i)]
715: @end example
716:
1.4 noro 717: \JP @node eval deval ,,, $B?t$N1i;;(B
718: \EG @node eval deval,,, Numbers
719: @subsection @code{eval}, @code{deval}
1.1 noro 720: @findex eval
1.4 noro 721: @findex deval
1.1 noro 722: @cindex PARI
723:
724: @table @t
725: @item eval(@var{obj}[,@var{prec}])
1.4 noro 726: @item deval(@var{obj})
1.2 noro 727: \JP :: @var{obj} $B$NCM$NI>2A(B.
728: \EG :: Evaluate @var{obj} numerically.
1.1 noro 729: @end table
730:
731: @table @var
732: @item return
1.2 noro 733: \JP $B?t$"$k$$$O<0(B
734: \EG number or expression
1.1 noro 735: @item obj
1.2 noro 736: \JP $B0lHL$N<0(B
737: \EG general expression
1.1 noro 738: @item prec
1.2 noro 739: \JP $B@0?t(B
740: \EG integer
1.1 noro 741: @end table
742:
743: @itemize @bullet
1.2 noro 744: \BJP
1.1 noro 745: @item
746: @var{obj} $B$K4^$^$l$kH!?t$NCM$r2DG=$J8B$jI>2A$9$k(B.
747: @item
1.4 noro 748: @code{deval} $B$OG\@:EYIbF0>.?t$r7k2L$H$7$F(B
749: @code{eval} $B$N>l9g(B, $BM-M}?t$O$=$N$^$^;D$k(B.
1.1 noro 750: @item
1.11 ! noro 751: @code{eval} $B$K$*$$$F$O(B, $B7W;;$O(B @b{MPFR} $B%i%$%V%i%j$,9T$&(B.
1.4 noro 752: @code{deval} $B$K$*$$$F$O(B, $B7W;;$O(B C $B?t3X%i%$%V%i%j$N4X?t$rMQ$$$F9T$&(B.
753: @item
754: @code{deval} $B$OJ#AG?t$O07$($J$$(B.
755: @item
756: @code{eval} $B$K$*$$$F$O(B,
1.1 noro 757: @var{prec} $B$r;XDj$7$?>l9g(B, $B7W;;$O(B, 10 $B?J(B @var{prec} $B7eDxEY$G9T$o$l$k(B.
758: @var{prec} $B$N;XDj$,$J$$>l9g(B, $B8=:_@_Dj$5$l$F$$$k@:EY$G9T$o$l$k(B.
1.11 ! noro 759: (@xref{setprec}, @xref{setbprec}.)
1.1 noro 760: @item
761: @table @t
762: @item $B07$($kH!?t$O(B, $B<!$NDL$j(B.
763: @code{sin}, @code{cos}, @code{tan},
764:
765: @code{asin}, @code{acos}, @code{atan},
766:
767: @code{sinh}, @code{cosh}, @code{tanh},
768:
769: @code{asinh}, @code{acosh}, @code{atanh},
770:
771: @code{exp}, @code{log}, @code{pow(a,b) (a^b)}
772: @end table
773: @item
1.4 noro 774: $B0J2<$N5-9f$r?t$H$7$FI>2A$G$-$k(B. $B$?$@$7(B @code{@@i} $B$r07$($k$N$O(B
775: @code{eval}, @code{deval} $B$N$_$G$"$k(B.
1.1 noro 776: @table @t
777: @item @@i
778: $B5u?tC10L(B
779: @item @@pi
780: $B1_<~N((B
781: @item @@e
782: $B<+A3BP?t$NDl(B
783: @end table
1.2 noro 784: \E
785: \BEG
786: @item
787: Evaluates the value of the functions contained in @var{obj} as far as
788: possible.
789: @item
1.4 noro 790: @code{deval} returns
791: double float. Rational numbers remain unchanged in results from @code{eval}.
792: @item
793: In @code{eval} the computation is done
1.11 ! noro 794: by @b{MPFR} library. In @code{deval} the computation is
1.4 noro 795: done by the C math library.
796: @item
797: @code{deval} cannot handle complex numbers.
1.2 noro 798: @item
799: When @var{prec} is specified, computation will be performed with a
800: precision of about @var{prec}-digits.
801: If @var{prec} is not specified, computation is performed with the
1.7 noro 802: precision set currently. (@xref{setprec}.)
1.2 noro 803: @item
804: Currently available numerical functions are listed below.
805:
806: @table @t
807: @code{sin}, @code{cos}, @code{tan},
808:
809: @code{asin}, @code{acos}, @code{atan},
810:
811: @code{sinh}, @code{cosh}, @code{tanh},
812: @code{asinh}, @code{acosh}, @code{atanh},
813:
814: @code{exp}, @code{log}, @code{pow(a,b) (a^b)}
815: @end table
816: @item
1.4 noro 817: Symbols for special values are as the followings. Note that
818: @code{@@i} cannot be handled by @code{deval}.
1.2 noro 819: @table @t
820: @item @@i
821: unit of imaginary number
822: @item @@pi
823: the number pi,
824: the ratio of circumference to diameter
825: @item @@e
826: Napier's number (@t{exp}(1))
827: @end table
828: \E
1.1 noro 829: @end itemize
830:
831: @example
832: [118] eval(exp(@@pi*@@i));
833: -1.0000000000000000000000000000
834: [119] eval(2^(1/2));
835: 1.414213562373095048763788073031
836: [120] eval(sin(@@pi/3));
837: 0.86602540378443864674620506632
838: [121] eval(sin(@@pi/3)-3^(1/2)/2,50);
839: -2.78791084448179148471 E-58
1.4 noro 840: [122] eval(1/2);
841: 1/2
842: [123] deval(sin(1)^2+cos(1)^2);
843: 1
1.1 noro 844: @end example
845:
846: @table @t
1.2 noro 847: \JP @item $B;2>H(B
848: \EG @item References
1.11 ! noro 849: @fref{ctrl}, @fref{setprec}, @fref{setbprec}.
1.1 noro 850: @end table
851:
1.2 noro 852: \JP @node pari,,, $B?t$N1i;;(B
853: \EG @node pari,,, Numbers
1.1 noro 854: @subsection @code{pari}
855: @findex pari
856: @cindex PARI
857:
858: @table @t
859: @item pari(@var{func},@var{arg},@var{prec})
1.2 noro 860: \JP :: @b{PARI} $B$NH!?t(B @var{func} $B$r8F$S=P$9(B.
861: \EG :: Call @b{PARI} function @var{func}.
1.1 noro 862: @end table
863:
864: @table @var
865: @item return
1.2 noro 866: \JP @var{func} $BKh$K0[$J$k(B.
867: \EG Depends on @var{func}.
1.1 noro 868: @item func
1.2 noro 869: \JP @b{PARI} $B$NH!?tL>(B
870: \EG Function name of @b{PARI}.
1.1 noro 871: @item arg
1.2 noro 872: \JP @var{func} $B$N0z?t(B
873: \EG Arguments of @var{func}.
1.1 noro 874: @item prec
1.2 noro 875: \JP $B@0?t(B
876: \EG integer
1.1 noro 877: @end table
878:
879: @itemize @bullet
1.2 noro 880: \BJP
1.1 noro 881: @item
882: @b{PARI} $B$NH!?t$r8F$S=P$9(B.
883:
884: @item
885: @b{PARI} @code{[Batut et al.]} $B$O(B Bordeaux $BBg3X$G3+H/$5$l%U(B
886: $B%j!<%=%U%H%&%'%"$H$7$F8x3+$5$l$F$$$k(B. @b{PARI} $B$O?t<0=hM}E*$J5!G=$rM-(B
887: $B$7$F$O$$$k$,(B, $B<g$J%?!<%2%C%H$O@0?tO@$K4XO"$7$??t(B (@b{bignum},
888: @b{bigfloat}) $B$N1i;;$G(B, $B;MB'1i;;$K8B$i$:(B@b{bigfloat} $B$K$h$k$5$^$6$^$J(B
889: $BH!?tCM$NI>2A$r9bB.$K9T$&$3$H$,$G$-$k(B. @b{PARI} $B$OB>$N%W%m%0%i%`$+$i(B
890: $B%5%V%k!<%A%s%i%$%V%i%j$H$7$FMQ$$$k$3$H$,$G$-(B, $B$^$?(B, @samp{gp} $B$H$$$&(B
891: @b{PARI}$B%i%$%V%i%j$N%$%s%?%U%'!<%9$K$h$j(B UNIX $B$N%"%W%j%1!<%7%g%s$H$7$F(B
1.11 ! noro 892: $BMxMQ$9$k$3$H$b$G$-$k(B.
1.1 noro 893: @item
894: $B:G8e$N0z?t(B @var{prec} $B$G7W;;@:EY$r;XDj$G$-$k(B.
895: @var{prec} $B$r>JN,$7$?>l9g(B @code{setprec()} $B$G;XDj$7$?@:EY$H$J$k(B.
896: @item
897: $B8=;~E@$G<B9T$G$-$k(B @b{PARI} $B$NH!?t$O<!$NDL$j$G$"$k(B. $B$$$:$l$b(B
898: 1 $B0z?t$G(B @b{Asir} $B$,BP1~$G$-$k7?$N0z?t$r$H$kH!?t$G$"$k(B.
899: $B$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 900: \E
901: \BEG
902: @item
903: This command connects @b{Asir} to @b{PARI} system so that several
904: functions of @b{PARI} can be conveniently used from @b{Risa/Asir}.
905: @item
906: @b{PARI} @code{[Batut et al.]} is developed at Bordeaux University, and
907: distributed as a free software. Though it has a certain facility to computer
908: algebra, its major target is the operation of numbers (@b{bignum},
909: @b{bigfloat}) related to the number theory. It facilitates various
910: function evaluations as well as arithmetic operations at a remarkable
911: speed. It can also be used from other external programs as a library.
912: It provides a language interface named @samp{gp} to its library, which
913: enables a user to use @b{PARI} as a calculator which runs on UNIX.
914: @item
915: The last argument (optional) @var{int} specifies the precision in digits
916: for bigfloat operation.
917: If the precision is not explicitly specified, operation will be performed
918: with the precision set by @code{setprec()}.
919: @item
920: Currently available functions of @b{PARI} system are as follows.
921: Note these are only a part of functions in @b{PARI} system.
922: For details of individual functions, refer to the @b{PARI} manual.
923: (Some of them can be seen in the following example.)
924: \E
1.1 noro 925:
926: @code{abs},
927: @code{adj},
928: @code{arg},
929: @code{bigomega},
930: @code{binary},
931: @code{ceil},
932: @code{centerlift},
933: @code{cf},
934: @code{classno},
935: @code{classno2},
936: @code{conj},
937: @code{content},
938: @code{denom},
939: @code{det},
940: @code{det2},
941: @code{detr},
942: @code{dilog},
943: @code{disc},
944: @code{discf},
945: @code{divisors},
946: @code{eigen},
947: @code{eintg1},
948: @code{erfc},
949: @code{eta},
950: @code{floor},
951: @code{frac},
952: @code{galois},
953: @code{galoisconj},
954: @code{gamh},
955: @code{gamma},
956: @code{hclassno},
957: @code{hermite},
958: @code{hess},
959: @code{imag},
960: @code{image},
961: @code{image2},
962: @code{indexrank},
963: @code{indsort},
964: @code{initalg},
965: @code{isfund},
966: @code{isprime},
967: @code{ispsp},
968: @code{isqrt},
969: @code{issqfree},
970: @code{issquare},
971: @code{jacobi},
972: @code{jell},
973: @code{ker},
974: @code{keri},
975: @code{kerint},
976: @code{kerintg1},
977: @code{kerint2},
978: @code{kerr},
979: @code{length},
980: @code{lexsort},
981: @code{lift},
982: @code{lindep},
983: @code{lll},
984: @code{lllg1},
985: @code{lllgen},
986: @code{lllgram},
987: @code{lllgramg1},
988: @code{lllgramgen},
989: @code{lllgramint},
990: @code{lllgramkerim},
991: @iftex
992: @break
993: @end iftex
994: @code{lllgramkerimgen},
995: @code{lllint},
996: @code{lllkerim},
997: @code{lllkerimgen},
998: @code{lllrat},
999: @code{lngamma},
1000: @code{logagm},
1001: @code{mat},
1002: @code{matrixqz2},
1003: @code{matrixqz3},
1004: @code{matsize},
1005: @code{modreverse},
1006: @code{mu},
1007: @code{nextprime},
1008: @code{norm},
1009: @code{norml2},
1010: @code{numdiv},
1011: @code{numer},
1012: @code{omega},
1013: @code{order},
1014: @code{ordred},
1015: @code{phi},
1016: @code{pnqn},
1017: @code{polred},
1018: @code{polred2},
1019: @code{primroot},
1020: @code{psi},
1021: @code{quadgen},
1022: @code{quadpoly},
1023: @code{real},
1024: @code{recip},
1025: @code{redcomp},
1026: @code{redreal},
1027: @code{regula},
1028: @code{reorder},
1029: @code{reverse},
1030: @code{rhoreal},
1031: @code{roots},
1032: @code{rootslong},
1033: @code{round},
1034: @code{sigma},
1035: @code{signat},
1036: @code{simplify},
1037: @code{smalldiscf},
1038: @code{smallfact},
1039: @code{smallpolred},
1040: @code{smallpolred2},
1041: @code{smith},
1042: @code{smith2},
1043: @code{sort},
1044: @code{sqr},
1045: @code{sqred},
1046: @code{sqrt},
1047: @code{supplement},
1048: @code{trace},
1049: @code{trans},
1050: @code{trunc},
1051: @code{type},
1052: @code{unit},
1053: @code{vec},
1054: @code{wf},
1055: @code{wf2},
1056: @code{zeta}
1057:
1.2 noro 1058: \BJP
1.1 noro 1059: @item
1.11 ! noro 1060: @b{Asir} $B$GMQ$$$F$$$k$N$O(B @b{PARI} $B$N$[$s$N0lIt$N5!G=$G$"$k(B.
1.2 noro 1061: \E
1062: \BEG
1063: @item
1064: @b{Asir} currently uses only a very small subset of @b{PARI}.
1065: \E
1.1 noro 1066: @end itemize
1067:
1068: @example
1.2 noro 1069: \JP /* $B9TNs$N8GM-%Y%/%H%k$r5a$a$k(B. */
1070: \EG /* Eigen vectors of a numerical matrix */
1.1 noro 1071: [0] pari(eigen,newmat(2,2,[[1,1],[1,2]]));
1072: [ -1.61803398874989484819771921990 0.61803398874989484826 ]
1073: [ 1 1 ]
1.2 noro 1074: \JP /* 1 $BJQ?tB?9`<0$N:,$r5a$a$k(B. */
1075: \EG /* Roots of a polynomial */
1.1 noro 1076: [1] pari(roots,t^2-2);
1077: [ -1.41421356237309504876 1.41421356237309504876 ]
1078: @end example
1079:
1080: @table @t
1.2 noro 1081: \JP @item $B;2>H(B
1082: \EG @item References
1.1 noro 1083: @fref{setprec}.
1084: @end table
1085:
1.11 ! noro 1086: \JP @node setbprec setprec,,, $B?t$N1i;;(B
! 1087: \EG @node setbprec setprec,,, Numbers
! 1088: @subsection @code{setbprec}, @code{setprec}
! 1089: @findex setbprec
1.1 noro 1090: @findex setprec
1091:
1092: @table @t
1.11 ! noro 1093: @item setbprec([@var{n}])
! 1094: @itemx setprec([@var{n}])
! 1095: \JP :: @b{setbprec}, @b{setprec} $B$O(B @b{bigfloat} $B$N@:EY$r$=$l$>$l(B 2 $B?J(B, 10$B?J(B @var{n} $B7e$K@_Dj$9$k(B.
! 1096: \EG :: @b{setbprec}, @b{setprec} set the precision for @b{bigfloat} operations to @var{n} bits, @var{n} digits respectively.
1.1 noro 1097: @end table
1098:
1099: @table @var
1100: @item return
1.2 noro 1101: \JP $B@0?t(B
1102: \EG integer
1.1 noro 1103: @item n
1.2 noro 1104: \JP $B@0?t(B
1105: \EG integer
1.1 noro 1106: @end table
1107:
1108: @itemize @bullet
1.2 noro 1109: \BJP
1.1 noro 1110: @item
1111: $B0z?t$,$"$k>l9g(B, @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B.
1112: $B0z?t$N$"$k$J$7$K$+$+$o$i$:(B, $B0JA0$K@_Dj$5$l$F$$$?CM$rJV$9(B.
1113: @item
1.11 ! noro 1114: @b{bigfloat} $B$N7W;;$O(B @b{MPFR} $B%i%$%V%i%j$K$h$C$F9T$o$l$k(B.
1.1 noro 1115: @item
1116: @b{bigfloat} $B$G$N7W;;$KBP$7M-8z$G$"$k(B.
1117: @b{bigfloat} $B$N(B flag $B$r(B on $B$K$9$kJ}K!$O(B, @code{ctrl} $B$r;2>H(B.
1118: @item
1119: $B@_Dj$G$-$k7e?t$K>e8B$O$J$$$,(B, $B;XDj$7$?7e?t$K@_Dj$5$l$k$H$O(B
1120: $B8B$i$J$$(B. $BBg$-$a$NCM$r@_Dj$9$k$N$,0BA4$G$"$k(B.
1.2 noro 1121: \E
1122: \BEG
1123: @item
1.11 ! noro 1124: When an argument @var{n} is given, these functions
! 1125: set the precision for @b{bigfloat} operations to @var{n} bits or @var{n} digits.
! 1126: The return value is always the previous precision regardless of
1.2 noro 1127: the existence of an argument.
1128:
1129: @item
1.11 ! noro 1130: @b{Bigfloat} operations are done by @b{MPFR} library.
1.2 noro 1131: @item
1132: This is effective for computations in @b{bigfloat}.
1133: Refer to @code{ctrl()} for turning on the `@b{bigfloat} flag.'
1134: @item
1135: There is no upper limit for precision digits.
1136: It sets the precision to some digits around the specified precision.
1137: Therefore, it is safe to specify a larger value.
1138: \E
1.1 noro 1139: @end itemize
1140:
1141: @example
1142: [1] setprec();
1.11 ! noro 1143: 15
! 1144: [2] setprec(100);
! 1145: 15
! 1146: [3] setprec(100);
! 1147: 99
! 1148: [4] setbprec();
! 1149: 332
! 1150: @end example
! 1151:
! 1152: @table @t
! 1153: \JP @item $B;2>H(B
! 1154: @fref{ctrl}, @fref{eval deval}.
! 1155: @end table
! 1156:
! 1157: \JP @node setround,,, $B?t$N1i;;(B
! 1158: \EG @node setround,,, Numbers
! 1159: @subsection @code{setround}
! 1160: @findex setround
! 1161:
! 1162: @table @t
! 1163: @item setround([@var{mode}])
! 1164: \JP :: @b{bigfloat} $B$N4]$a%b!<%I$r(B @var{mode} $B$K@_Dj$9$k(B.
! 1165: \EG :: Sets the rounding mode @var{mode}.
! 1166: @end table
! 1167:
! 1168: @table @var
! 1169: @item return
! 1170: \JP $B@0?t(B
! 1171: \EG integer
! 1172: @item mode
! 1173: \JP $B@0?t(B
! 1174: \EG integer
! 1175: @end table
! 1176:
! 1177: @itemize @bullet
! 1178: \BJP
! 1179: @item
! 1180: $B0z?t$,$"$k>l9g(B, @b{bigfloat} $B$N4]$a%b!<%I$r(B @var{mode} $B$K@_Dj$9$k(B.
! 1181: $B0z?t$N$"$k$J$7$K$+$+$o$i$:(B, $B0JA0$K@_Dj$5$l$F$$$?CM$rJV$9(B.
! 1182: $B4]$a%b!<%I$N0UL#$O<!$N$H$*$j(B.
! 1183: @table @code
! 1184: @item 0
! 1185: Round to nearest
! 1186: @item 1
! 1187: Round toward 0
! 1188: @item 2
! 1189: Round toward +infinity
! 1190: @item 3
! 1191: Round toward -infinity
! 1192: @end table
! 1193: @item
! 1194: @b{bigfloat} $B$G$N7W;;$KBP$7M-8z$G$"$k(B.
! 1195: @b{bigfloat} $B$N(B flag $B$r(B on $B$K$9$kJ}K!$O(B, @code{ctrl} $B$r;2>H(B.
! 1196: \E
! 1197: \BEG
! 1198: @item
! 1199: When an argument @var{mode} is given, these functions
! 1200: set the rounding mode for @b{bigfloat} operations to @var{mode}.
! 1201: The return value is always the previous rounding mode regardless of
! 1202: the existence of an argument.
! 1203: The meanings of rounding modes are as follows
! 1204: @table @code
! 1205: @item 0
! 1206: Round to nearest
! 1207: @item 1
! 1208: Round toward 0
! 1209: @item 2
! 1210: Round toward +infinity
! 1211: @item 3
! 1212: Round toward -infinity
! 1213: @end table
! 1214:
! 1215: @item
! 1216: This is effective for computations in @b{bigfloat}.
! 1217: Refer to @code{ctrl()} for turning on the `@b{bigfloat} flag.'
! 1218: \E
! 1219: @end itemize
! 1220:
! 1221: @example
! 1222: [1] setprec();
! 1223: 15
1.1 noro 1224: [2] setprec(100);
1.11 ! noro 1225: 15
1.1 noro 1226: [3] setprec(100);
1.11 ! noro 1227: 99
! 1228: [4] setbprec();
! 1229: 332
1.1 noro 1230: @end example
1231:
1232: @table @t
1.2 noro 1233: \JP @item $B;2>H(B
1.11 ! noro 1234: @fref{ctrl}, @fref{eval deval}.
1.1 noro 1235: @end table
1236:
1.11 ! noro 1237:
1.2 noro 1238: \JP @node setmod,,, $B?t$N1i;;(B
1239: \EG @node setmod,,, Numbers
1.1 noro 1240: @subsection @code{setmod}
1241: @findex setmod
1242:
1243: @table @t
1244: @item setmod([@var{p}])
1.2 noro 1245: \JP :: $BM-8BBN$r(B GF(@var{p}) $B$K@_Dj$9$k(B.
1246: \EG :: Sets the ground field to GF(@var{p}).
1.1 noro 1247: @end table
1248:
1249: @table @var
1250: @item return
1.2 noro 1251: \JP $B@0?t(B
1252: \EG integer
1.1 noro 1253: @item n
1.2 noro 1254: \JP 2^27 $BL$K~$NAG?t(B
1255: \EG prime less than 2^27
1.1 noro 1256: @end table
1257:
1258: @itemize @bullet
1.2 noro 1259: \BJP
1.1 noro 1260: @item
1261: $BM-8BBN$r(B GF(@var{p}) $B$K@_Dj$9$k(B. $B@_DjCM$rJV$9(B.
1262: @item
1263: $BM-8BBN$N85$N7?$r;}$D?t$O(B, $B$=$l<+?H$O$I$NM-8BBN$KB0$9$k$+$N>pJs$r;}$?$:(B,
1264: $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 1265: @item
1266: $B0L?t$NBg$-$JM-8BBN$K4X$7$F$O(B @pxref{$BM-8BBN$K4X$9$k1i;;(B}.
1267: \E
1268: \BEG
1269: @item
1270: Sets the ground field to GF(@var{p}) and returns the value @var{p}.
1271: @item
1272: A member of a finite field does not have any information
1273: about the field and the arithmetic operations over GF(@var{p}) are applied
1274: with @var{p} set at the time.
1275: @item
1276: As for large finite fields, @pxref{Finite fields}.
1277: \E
1.1 noro 1278: @end itemize
1279:
1280: @example
1281: [0] A=dp_mod(dp_ptod(2*x,[x]),3,[]);
1282: (2)*<<1>>
1283: [1] A+A;
1284: addmi : invalid modulus
1285: return to toplevel
1286: [1] setmod(3);
1287: 3
1288: [2] A+A;
1289: (1)*<<1>>
1290: @end example
1291:
1292: @table @t
1.2 noro 1293: \JP @item $B;2>H(B
1294: \EG @item References
1295: \JP @fref{dp_mod dp_rat}, @fref{$B?t$N7?(B}.
1296: \EG @fref{dp_mod dp_rat}, @fref{Types of numbers}.
1.1 noro 1297: @end table
1298:
1.3 noro 1299: \JP @node ntoint32 int32ton,,, $B?t$N1i;;(B
1300: \EG @node ntoint32 int32ton,,, Numbers
1301: @subsection @code{ntoint32}, @code{int32ton}
1302: @findex ntoint32
1303: @findex int32ton
1304:
1305: @table @t
1306: @item ntoint32(@var{n})
1307: @itemx int32ton(@var{int32})
1308: \JP :: $BHsIi@0?t$HId9f$J$7(B 32bit $B@0?t$N4V$N7?JQ49(B.
1309: \EG :: Type-conversion between a non-negative integer and an unsigned 32bit integer.
1310: @end table
1311:
1312: @table @var
1313: @item return
1314: \JP $BId9f$J$7(B 32bit $B@0?t$^$?$OHsIi@0?t(B
1315: \EG unsigned 32bit integer or non-negative integer
1316: @item n
1317: \JP 2^32 $BL$K~$NHsIi@0?t(B
1318: \EG non-negative interger less than 2^32
1319: @item int32
1320: \JP $BId9f$J$7(B 32bit $B@0?t(B
1321: \EG unsigned 32bit integer
1322: @end table
1323:
1324: @itemize @bullet
1325: \BJP
1326: @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,
1327: $B$^$?$O$=$N5UJQ49$r9T$&(B.
1328: @item 32bit $B@0?t$O(B @b{OpenXM} $B$N4pK\9=@.MWAG$G$"$j(B, $B@0?t$r$=$N7?$GAw?.(B
1329: $B$9$kI,MW$,$"$k>l9g$KMQ$$$k(B.
1330: \E
1331: \BEG
1332: @item These functions do conversions between non-negative
1333: integers (the type id 1) and unsigned 32bit integers (the type id 10).
1334: @item An unsigned 32bit integer is a fundamental construct of @b{OpenXM}
1335: and one often has to send an integer to a server as an unsigned 32bit
1336: integer. These functions are used in such a case.
1337: \E
1338: @end itemize
1339:
1340: @table @t
1341: \JP @item $B;2>H(B
1342: \EG @item References
1343: \JP @fref{$BJ,;67W;;(B}, @fref{$B?t$N7?(B}.
1344: \EG @fref{Distributed computation}, @fref{Types of numbers}.
1345: @end table
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