Annotation of OpenXM/src/asir-doc/parts/builtin/poly.texi, Revision 1.4
1.4 ! noro 1: @comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/poly.texi,v 1.3 2002/09/03 01:50:59 noro Exp $
1.2 noro 2: \BJP
1.1 noro 3: @node $BB?9`<0$*$h$SM-M}<0$N1i;;(B,,, $BAH$_9~$_H!?t(B
4: @section $BB?9`<0(B, $BM-M}<0$N1i;;(B
1.2 noro 5: \E
6: \BEG
7: @node Polynomials and rational expressions,,, Built-in Function
8: @section operations with polynomials and rational expressions
9: \E
1.1 noro 10:
11: @menu
12: * var::
13: * vars::
14: * uc::
15: * coef::
16: * deg mindeg::
17: * nmono::
18: * ord::
19: * sdiv sdivm srem sremm sqr sqrm::
20: * tdiv::
21: * %::
22: * subst psubst::
23: * diff::
24: * res::
25: * fctr sqfr::
26: * modfctr::
27: * ufctrhint::
28: * ptozp::
29: * prim cont::
30: * gcd gcdz::
31: * red::
32: @end menu
33:
1.2 noro 34: \JP @node var,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
35: \EG @node var,,, Polynomials and rational expressions
1.1 noro 36: @subsection @code{var}
37: @findex var
38:
39: @table @t
40: @item var(@var{rat})
1.2 noro 41: \JP :: @var{rat} $B$N<gJQ?t(B.
42: \EG :: Main variable (indeterminate) of @var{rat}.
1.1 noro 43: @end table
44:
45: @table @var
46: @item return
1.2 noro 47: \JP $BITDj85(B
48: \EG indeterminate
1.1 noro 49: @item rat
1.2 noro 50: \JP $BM-M}<0(B
51: \EG rational expression
1.1 noro 52: @end table
53:
54: @itemize @bullet
1.2 noro 55: \BJP
1.1 noro 56: @item
57: $B<gJQ?t$K4X$7$F$O(B, @xref{Asir $B$G;HMQ2DG=$J7?(B}.
58: @item
59: $B%G%U%)%k%H$NJQ?t=g=x$O<!$N$h$&$K$J$C$F$$$k(B.
60:
61: @code{x}, @code{y}, @code{z}, @code{u}, @code{v}, @code{w}, @code{p}, @code{q}, @code{r}, @code{s}, @code{t}, @code{a}, @code{b}, @code{c}, @code{d}, @code{e},
62: @code{f}, @code{g}, @code{h}, @code{i}, @code{j}, @code{k}, @code{l}, @code{m}, @code{n}, @code{o},$B0J8e$OJQ?t$N8=$l$?=g(B.
1.2 noro 63: \E
64: \BEG
65: @item
1.3 noro 66: See @ref{Types in Asir} for main variable.
1.2 noro 67: @item
68: Indeterminates (variables) are ordered by default as follows.
69:
70: @code{x}, @code{y}, @code{z}, @code{u}, @code{v}, @code{w}, @code{p}, @code{q},
71: @code{r}, @code{s}, @code{t}, @code{a}, @code{b}, @code{c}, @code{d}, @code{e},
72: @code{f}, @code{g}, @code{h}, @code{i}, @code{j}, @code{k}, @code{l}, @code{m},
73: @code{n}, @code{o}. The other variables will be ordered after the above noted variables
74: so that the first comer will be ordered prior to the followers.
75: \E
1.1 noro 76: @end itemize
77:
78: @example
79: [0] var(x^2+y^2+a^2);
80: x
81: [1] var(a*b*c*d*e);
82: a
83: [2] var(3/abc+2*xy/efg);
84: abc
85: @end example
86:
87: @table @t
1.2 noro 88: \JP @item $B;2>H(B
89: \EG @item References
1.1 noro 90: @fref{ord}, @fref{vars}.
91: @end table
92:
1.2 noro 93: \JP @node vars,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
94: \EG @node vars,,, Polynomials and rational expressions
1.1 noro 95: @subsection @code{vars}
96: @findex vars
97:
98: @table @t
99: @item vars(@var{obj})
1.2 noro 100: \JP :: @var{obj} $B$K4^$^$l$kJQ?t$N%j%9%H(B.
101: \EG :: A list of variables (indeterminates) in an expression @var{obj}.
1.1 noro 102: @end table
103:
104: @table @var
105: @item return
1.2 noro 106: \JP $B%j%9%H(B
107: \EG list
1.1 noro 108: @item obj
1.2 noro 109: \JP $BG$0U(B
110: \EG arbitrary
1.1 noro 111: @end table
112:
113: @itemize @bullet
1.2 noro 114: \BJP
1.1 noro 115: @item
116: $BM?$($i$l$?<0$K4^$^$l$kJQ?t$N%j%9%H$rJV$9(B.
117: @item
118: $BJQ?t=g=x$N9b$$$b$N$+$i=g$KJB$Y$k(B.
1.2 noro 119: \E
120: \BEG
121: @item
122: Returns a list of variables (indeterminates) contained in a given expression.
123: @item
124: Lists variables according to the variable ordering.
125: \E
1.1 noro 126: @end itemize
127:
128: @example
129: [0] vars(x^2+y^2+a^2);
130: [x,y,a]
131: [1] vars(3/abc+2*xy/efg);
132: [abc,xy,efg]
133: [2] vars([x,y,z]);
134: [x,y,z]
135: @end example
136:
137: @table @t
1.2 noro 138: \JP @item $B;2>H(B
139: \EG @item References
1.1 noro 140: @fref{var}, @fref{uc}, @fref{ord}.
141: @end table
142:
1.2 noro 143: \JP @node uc,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
144: \EG @node uc,,, Polynomials and rational expressions
1.1 noro 145: @subsection @code{uc}
146: @findex uc
147:
148: @table @t
149: @item uc()
1.2 noro 150: \JP :: $BL$Dj78?tK!$N$?$a$NITDj85$r@8@.$9$k(B.
151: \EG :: Create a new indeterminate for an undermined coeficient.
1.1 noro 152: @end table
153:
154: @table @var
155: @item return
1.2 noro 156: \JP @code{vtype} $B$,(B 1 $B$NITDj85(B
157: \EG indeterminate with its @code{vtype} 1.
1.1 noro 158: @end table
159:
160: @itemize @bullet
1.2 noro 161: \BJP
1.1 noro 162: @item
163: @code{uc()} $B$r<B9T$9$k$?$S$K(B, @code{_0}, @code{_1}, @code{_2},... $B$H$$$&(B
164: $BITDj85$r@8@.$9$k(B.
165: @item
166: @code{uc()} $B$G@8@.$5$l$?ITDj85$O(B, $BD>@\%-!<%\!<%I$+$iF~NO$9$k$3$H$,$G$-$J$$(B.
167: $B$3$l$O(B, $B%W%m%0%i%`Cf$GL$Dj78?t$r<+F0@8@.$9$k>l9g(B, $BF~NO$J$I$K4^$^$l$k(B
168: $BITDj85$HF10l$N$b$N$,@8@.$5$l$k$3$H$rKI$0$?$a$G$"$k(B.
169: @item
170: $BDL>o$NITDj85(B (@code{vtype} $B$,(B 0) $B$N<+F0@8@.$K$O(B @code{rtostr()},
171: @code{strtov()} $B$rMQ$$$k(B.
172: @item
173: @code{uc()} $B$G@8@.$5$l$?ITDj85$NITDj85$H$7$F$N7?(B (@code{vtype}) $B$O(B 1 $B$G$"$k(B.
1.3 noro 174: (@xref{$BITDj85$N7?(B}.)
1.2 noro 175: \E
176: \BEG
177: @item
178: At every evaluation of command @code{uc()}, a new indeterminate in
179: the sequence of indeterminates @code{_0}, @code{_1}, @code{_2}, @dots{}
180: is created successively.
181: @item
182: Indeterminates created by @code{uc()} cannot be input on the keyboard.
183: By this property, you are free, no matter how many indeterminates you
184: will create dynamically by a program, from collision of created names
185: with indeterminates input from the keyboard or from program files.
186: @item
187: Functions, @code{rtostr()} and @code{strtov()}, are used to create
188: ordinary indeterminates (indeterminates having 0 for their @code{vtype}).
189: @item
190: Kernel sub-type of indeterminates created by @code{uc()} is 1.
191: (@code{vtype(uc())}=1)
192: \E
1.1 noro 193: @end itemize
194:
195: @example
196: [0] A=uc();
197: _0
198: [1] B=uc();
199: _1
200: [2] (uc()+uc())^2;
201: _2^2+2*_3*_2+_3^2
202: [3] (A+B)^2;
203: _0^2+2*_1*_0+_1^2
204: @end example
205:
206: @table @t
1.2 noro 207: \JP @item $B;2>H(B
208: \EG @item References
1.1 noro 209: @fref{vtype}, @fref{rtostr}, @fref{strtov}.
210: @end table
211:
1.2 noro 212: \JP @node coef,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
213: \EG @node coef,,, Polynomials and rational expressions
1.1 noro 214: @subsection @code{coef}
215: @findex coef
216:
217: @table @t
218: @item coef(@var{poly},@var{deg}[,@var{var}])
1.2 noro 219: \JP :: @var{poly} $B$N(B @var{var} ($B>JN,;~$O<gJQ?t(B) $B$K4X$9$k(B @var{deg} $B<!$N78?t(B.
220: \BEG
221: :: The coefficient of a polynomial @var{poly} at degree @var{deg}
222: with respect to the variable @var{var} (main variable if unspecified).
223: \E
1.1 noro 224: @end table
225:
226: @table @var
227: @item return
1.2 noro 228: \JP $BB?9`<0(B
229: \EG polynomial
1.1 noro 230: @item poly
1.2 noro 231: \JP $BB?9`<0(B
232: \EG polynomial
1.1 noro 233: @item var
1.2 noro 234: \JP $BITDj85(B
235: \EG indeterminate
1.1 noro 236: @item deg
1.2 noro 237: \JP $B<+A3?t(B
238: \EG non-negative integer
1.1 noro 239: @end table
240:
241: @itemize @bullet
1.2 noro 242: \BJP
1.1 noro 243: @item
244: @var{poly} $B$N(B @var{var} $B$K4X$9$k(B @var{deg} $B<!$N78?t$r=PNO$9$k(B.
245: @item
246: @var{var} $B$O(B, $B>JN,$9$k$H<gJQ?t(B @t{var}(@var{poly}) $B$@$H$_$J$5$l$k(B.
247: @item
248: @var{var} $B$,<gJQ?t$G$J$$;~(B, @var{var} $B$,<gJQ?t$N>l9g$KHf3S$7$F(B
249: $B8zN($,Mn$A$k(B.
1.2 noro 250: \E
251: \BEG
252: @item
253: The coefficient of a polynomial @var{poly} at degree @var{deg}
254: with respect to the variable @var{var}.
255: @item
256: The default value for @var{var} is the main variable, i.e.,
257: @t{var(@var{poly})}.
258: @item
259: For multi-variate polynomials, access to coefficients depends on
260: the specified indeterminates. For example, taking coef for the main
261: variable is much faster than for other variables.
262: \E
1.1 noro 263: @end itemize
264:
265: @example
266: [0] A = (x+y+z)^3;
267: x^3+(3*y+3*z)*x^2+(3*y^2+6*z*y+3*z^2)*x+y^3+3*z*y^2+3*z^2*y+z^3
268: [1] coef(A,1,y);
269: 3*x^2+6*z*x+3*z^2
270: [2] coef(A,0);
271: y^3+3*z*y^2+3*z^2*y+z^3
272: @end example
273:
274: @table @t
1.2 noro 275: \JP @item $B;2>H(B
276: \EG @item References
1.1 noro 277: @fref{var}, @fref{deg mindeg}.
278: @end table
279:
1.2 noro 280: \JP @node deg mindeg,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
281: \EG @node deg mindeg,,, Polynomials and rational expressions
1.1 noro 282: @subsection @code{deg}, @code{mindeg}
283: @findex deg
284: @findex mindeg
285:
286: @table @t
287: @item deg(@var{poly},@var{var})
1.2 noro 288: \JP :: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:G9b<!?t(B.
289: \EG :: The degree of a polynomial @var{poly} with respect to variable.
1.1 noro 290: @item mindeg(@var{poly},@var{var})
1.2 noro 291: \JP :: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:GDc<!?t(B.
292: \BEG
293: :: The least exponent of the terms with non-zero coefficients in
294: a polynomial @var{poly} with respect to the variable @var{var}.
295: In this manual, this quantity is sometimes referred to the minimum
296: degree of a polynomial for short.
297: \E
1.1 noro 298: @end table
299:
300: @table @var
301: @item return
1.2 noro 302: \JP $B<+A3?t(B
303: \EG non-negative integer
1.1 noro 304: @item poly
1.2 noro 305: \JP $BB?9`<0(B
306: \EG polynomial
1.1 noro 307: @item var
1.2 noro 308: \JP $BITDj85(B
309: \EG indeterminate
1.1 noro 310: @end table
311:
312: @itemize @bullet
1.2 noro 313: \BJP
1.1 noro 314: @item
315: $BM?$($i$l$?B?9`<0$NJQ?t(B @var{var} $B$K4X$9$k:G9b<!?t(B, $B:GDc<!?t$r=PNO$9$k(B.
316: @item
317: $BJQ?t(B @var{var} $B$r>JN,$9$k$3$H$O=PMh$J$$(B.
1.2 noro 318: \E
319: \BEG
320: @item
321: The least exponent of the terms with non-zero coefficients in
322: a polynomial @var{poly} with respect to the variable @var{var}.
323: In this manual, this quantity is sometimes referred to the minimum
324: degree of a polynomial for short.
325: @item
326: Variable @var{var} must be specified.
327: \E
1.1 noro 328: @end itemize
329:
330: @example
331: [0] deg((x+y+z)^10,x);
332: 10
333: [1] deg((x+y+z)^10,w);
334: 0
335: [75] mindeg(x^2+3*x*y,x);
336: 1
337: @end example
338:
1.2 noro 339: \JP @node nmono,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
340: \EG @node nmono,,,Polynomials and rational expressions
1.1 noro 341: @subsection @code{nmono}
342: @findex nmono
343:
344: @table @t
345: @item nmono(@var{rat})
1.2 noro 346: \JP :: @var{rat} $B$NC19`<0$N9`?t(B.
347: \EG :: Number of monomials in rational expression @var{rat}.
1.1 noro 348: @end table
349:
350: @table @var
351: @item return
1.2 noro 352: \JP $B<+A3?t(B
353: \EG non-negative integer
1.1 noro 354: @item rat
1.2 noro 355: \JP $BM-M}<0(B
356: \EG rational expression
1.1 noro 357: @end table
358:
359: @itemize @bullet
1.2 noro 360: \BJP
1.1 noro 361: @item
362: $BB?9`<0$rE83+$7$?>uBV$G$N(B 0 $B$G$J$$78?t$r;}$DC19`<0$N9`?t$r5a$a$k(B.
363: @item
364: $BM-M}<0$N>l9g$O(B, $BJ,;R$HJ,Jl$N9`?t$NOB$,JV$5$l$k(B.
365: @item
1.3 noro 366: $BH!?t7A<0(B (@ref{$BITDj85$N7?(B}) $B$O(B, $B0z?t$,2?$G$"$C$F$bC19`$H$_$J$5$l$k(B. (1 $B8D$NITDj85$HF1$8(B. )
1.2 noro 367: \E
368: \BEG
369: @item
370: Number of monomials with non-zero number coefficients in the full
371: expanded form of the given polynomial.
372: @item
373: For a rational expression, the sum of the numbers of monomials
374: of the numerator and denominator.
375: @item
376: A function form is regarded as a single indeterminate no matter how
377: complex arguments it has.
378: \E
1.1 noro 379: @end itemize
380:
381: @example
382: [0] nmono((x+y)^10);
383: 11
384: [1] nmono((x+y)^10/(x+z)^10);
385: 22
386: [2] nmono(sin((x+y)^10));
387: 1
388: @end example
389:
390: @table @t
1.2 noro 391: \JP @item $B;2>H(B
392: \EG @item References
1.1 noro 393: @fref{vtype}.
394: @end table
395:
1.2 noro 396: \JP @node ord,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
397: \EG @node ord,,, Polynomials and rational expressions
1.1 noro 398: @subsection @code{ord}
399: @findex ord
400:
401: @table @t
402: @item ord([@var{varlist}])
1.2 noro 403: \JP :: $BJQ?t=g=x$N@_Dj(B
404: \EG :: It sets the ordering of indeterminates (variables).
1.1 noro 405: @end table
406:
407: @table @var
408: @item return
1.2 noro 409: \JP $BJQ?t$N%j%9%H(B
410: \EG list of indeterminates
1.1 noro 411: @item varlist
1.2 noro 412: \JP $BJQ?t$N%j%9%H(B
413: \EG list of indeterminates
1.1 noro 414: @end table
415:
416: @itemize @bullet
1.2 noro 417: \BJP
1.1 noro 418: @item
419: $B0z?t$,$"$k$H$-(B, $B0z?t$NJQ?t%j%9%H$r@hF,$K=P$7(B, $B;D$j$NJQ?t$,$=$N8e$K(B
420: $BB3$/$h$&$KJQ?t=g=x$r@_Dj$9$k(B. $B0z?t$N$"$k$J$7$K4X$o$i$:(B, @code{ord()}
421: $B$N=*N;;~$K$*$1$kJQ?t=g=x%j%9%H$rJV$9(B.
422:
423: @item
424: $B$3$NH!?t$K$h$kJQ?t=g=x$NJQ99$r9T$C$F$b(B, $B4{$K%W%m%0%i%`JQ?t$J$I$K(B
425: $BBeF~$5$l$F$$$k<0$NFbIt7A<0$O?7$7$$=g=x$K=>$C$F$OJQ99$5$l$J$$(B.
426: $B=>$C$F(B, $B$3$NH!?t$K$h$k=g=x$NJQ99$O(B, @b{Asir} $B$N5/F0D>8e(B,
427: $B$"$k$$$O(B, $B?7$?$JJQ?t$,8=$l$?;~E@$K9T$o$l$k(B
428: $B$Y$-$G$"$k(B. $B0[$J$kJQ?t=g=x$N$b$H$G@8@.$5$l$?<0$I$&$7$N1i;;(B
429: $B$,9T$o$l$?>l9g(B, $BM=4|$;$L7k2L$,@8$:$k$3$H$b$"$jF@$k(B.
1.2 noro 430: \E
431: \BEG
432: @item
433: When an argument is given,
434: this function rearranges the ordering of variables (indeterminates)
435: so that the indeterminates in the argument @var{varlist} precede
436: and the other indeterminates follow in the system's variable ordering.
437: Regardless of the existence of an argument, it always returns the
438: final variable ordering.
439:
440: @item
441: Note that no change will be made to the variable ordering of internal
442: forms of objects which already exists in the system, no matter what
443: reordering you specify. Therefore, the reordering should be limited to
444: the time just after starting @b{Asir}, or to the time when one has
445: decided himself to start a totally new computation which has no relation
446: with the previous results.
447: Note that unexpected results may be obtained from operations between
448: objects which are created under different variable ordering.
449: \E
1.1 noro 450: @end itemize
451:
452: @example
453: [0] ord();
454: [x,y,z,u,v,w,p,q,r,s,t,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,_x,_y,_z,_u,_v,_w,_p,
455: _q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,exp(_x),(_x)^(_y),
456: log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),(-_x^2+1)^(-1/2),cosh(_x),sinh(_x),
457: tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]
458: [1] ord([dx,dy,dz,a,b,c]);
459: [dx,dy,dz,a,b,c,x,y,z,u,v,w,p,q,r,s,t,d,e,f,g,h,i,j,k,l,m,n,o,_x,_y,_z,_u,_v,
460: _w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,exp(_x),
461: (_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),(-_x^2+1)^(-1/2),
462: cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]
463: @end example
464:
1.2 noro 465: \JP @node sdiv sdivm srem sremm sqr sqrm,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
466: \EG @node sdiv sdivm srem sremm sqr sqrm,,, Polynomials and rational expressions
1.1 noro 467: @subsection @code{sdiv}, @code{sdivm}, @code{srem}, @code{sremm}, @code{sqr}, @code{sqrm}
468: @findex sdiv
469: @findex sdivm
470: @findex srem
471: @findex sremm
472: @findex sqr
473: @findex sqrm
474:
475: @table @t
476: @item sdiv(@var{poly1},@var{poly2}[,@var{v}])
477: @itemx sdivm(@var{poly1},@var{poly2},@var{mod}[,@var{v}])
1.2 noro 478: \JP :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>&$r5a$a$k(B.
479: \BEG
480: :: Quotient of @var{poly1} divided by @var{poly2} provided that the
481: division can be performed within polynomial arithmetic over the
482: rationals.
483: \E
1.1 noro 484: @item srem(@var{poly1},@var{poly2}[,@var{v}])
485: @item sremm(@var{poly1},@var{poly2},@var{mod}[,@var{v}])
1.2 noro 486: \JP :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>jM>$r5a$a$k(B.
487: \BEG
488: :: Remainder of @var{poly1} divided by @var{poly2} provided that the
489: division can be performed within polynomial arithmetic over the
490: rationals.
491: \E
1.1 noro 492: @item sqr(@var{poly1},@var{poly2}[,@var{v}])
493: @item sqrm(@var{poly1},@var{poly2},@var{mod}[,@var{v}])
1.2 noro 494: \BJP
1.1 noro 495: :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>&(B, $B>jM>$r(B
496: $B5a$a$k(B.
1.2 noro 497: \E
498: \BEG
499: :: Quotient and remainder of @var{poly1} divided by @var{poly2} provided
500: that the division can be performed within polynomial arithmetic over
501: the rationals.
502: \E
1.1 noro 503: @end table
504:
505: @table @var
506: @item return
1.2 noro 507: \JP @code{sdiv()}, @code{sdivm()}, @code{srem()}, @code{sremm()} : $BB?9`<0(B, @code{sqr()}, @code{sqrm()} : @code{[$B>&(B,$B>jM>(B]} $B$J$k%j%9%H(B
508: \EG @code{sdiv()}, @code{sdivm()}, @code{srem()}, @code{sremm()} : polynomial @code{sqr()}, @code{sqrm()} : a list @code{[quotient,remainder]}
1.1 noro 509: @item poly1 poly2
1.2 noro 510: \JP $BB?9`<0(B
511: \EG polynomial
1.1 noro 512: @item v
1.2 noro 513: \JP $BITDj85(B
514: \EG indeterminate
1.1 noro 515: @item mod
1.2 noro 516: \JP $BAG?t(B
517: \EG prime
1.1 noro 518: @end table
519:
520: @itemize @bullet
1.2 noro 521: \BJP
1.1 noro 522: @item
523: @var{poly1} $B$r(B @var{poly2} $B$N<gJQ?t(B @t{var}(@var{poly2})
524: ( $B0z?t(B @var{v} $B$,$"$k>l9g$K$O(B @var{v}) $B$K4X$9$kB?9`<0$H8+$F(B,
525: @var{poly2} $B$G(B, $B3d$j;;$r9T$&(B.
526: @item
527: @code{sdivm()}, @code{sremm()}, @code{sqrm()} $B$O(B GF(@var{mod}) $B>e$G7W;;$9$k(B.
528: @item
529: $BB?9`<0$N=|;;$O(B, $B<g78?t$I$&$7$N3d;;$K$h$jF@$i$l$?>&$H(B, $B<gJQ?t$NE,Ev$JQQ$N(B
530: $B@Q$r(B @var{poly2} $B$K3]$1$F(B, @var{poly1} $B$+$i0z$/$H$$$&A`:n$r(B
531: @var{poly1} $B$N<!?t$,(B @var{poly2} $B$N<!?t$h$j>.$5$/$J$k$^$G7+$jJV$7$F(B
532: $B9T$&(B. $B$3$NA`:n$,(B, $BB?9`<0$NHO0OFb$G9T$o$l$k$?$a$K$O(B, $B3F%9%F%C%W$K$*$$$F(B
533: $B<g78?t$I$&$7$N=|;;$,(B, $BB?9`<0$H$7$F$N@0=|$G$"$kI,MW$,$"$k(B. $B$3$l$,(B, $B!V=|;;(B
534: $B$,:G8e$^$G<B9T$G$-$k!W$3$H$N0UL#$G$"$k(B.
535: @item
536: $BE57?E*$J>l9g$H$7$F(B, @var{poly2} $B$N<g78?t$,(B, $BM-M}?t$G$"$k>l9g(B, $B$"$k$$$O(B,
537: @var{poly2} $B$,(B @var{poly1} $B$N0x;R$G$"$k$3$H$,$o$+$C$F$$$k>l9g$J$I(B
538: $B$,$"$k(B.
539: @item
540: @code{sqr()} $B$O>&$H>jM>$rF1;~$K5a$a$?$$;~$KMQ$$$k(B.
541: @item
542: $B@0?t=|;;$N>&(B, $B>jM>$O(B @code{idiv}, @code{irem} $B$rMQ$$$k(B.
543: @item
544: $B78?t$KBP$9$k>jM>1i;;$O(B @code{%} $B$rMQ$$$k(B.
1.2 noro 545: \E
546: \BEG
547: @item
548: Regarding @var{poly1} as an uni-variate polynomial in the main variable
549: of @var{poly2},
550: i.e. @t{var(@var{poly2})} (@var{v} if specified), @code{sdiv()} and
551: @code{srem()} compute
552: the polynomial quotient and remainder of @var{poly1} divided by @var{poly2}.
553: @item @code{sdivm()}, @code{sremm()}, @code{sqrm()} execute the same
554: operation over GF(@var{mod}).
555: @item
556: Division operation of polynomials is performed by the following steps:
557: (1) obtain the quotient of leading coefficients; let it be Q;
558: (2) remove the leading term of @var{poly1} by subtracting, from
559: @var{poly1}, the product of Q with some powers of main variable
560: and @var{poly2}; obtain a new @var{poly1};
561: (3) repeat the above step until the degree of @var{poly1} become smaller
562: than that of @var{poly2}.
563: For fulfillment, by operating in polynomials, of this procedure, the
564: divisions at step (1) in every repetition must be an exact division of
565: polynomials. This is the true meaning of what we say
566: ``division can be performed within polynomial arithmetic
567: over the rationals.''
568: @item
569: There are typical cases where the division is possible:
570: leading coefficient of @var{poly2} is a rational number;
571: @var{poly2} is a factor of @var{poly1}.
572: @item
573: Use @code{sqr()} to get both the quotient and remainder at once.
574: @item
575: Use @code{idiv()}, @code{irem()} for integer quotient.
576: @item
577: For remainder operation on all integer coefficients, use @code{%}.
578: \E
1.1 noro 579: @end itemize
580:
581: @example
582: [0] sdiv((x+y+z)^3,x^2+y+a);
583: x+3*y+3*z
584: [1] srem((x+y+z)^2,x^2+y+a);
585: (2*y+2*z)*x+y^2+(2*z-1)*y+z^2-a
586: [2] X=(x+y+z)*(x-y-z)^2;
587: x^3+(-y-z)*x^2+(-y^2-2*z*y-z^2)*x+y^3+3*z*y^2+3*z^2*y+z^3
588: [3] Y=(x+y+z)^2*(x-y-z);
589: x^3+(y+z)*x^2+(-y^2-2*z*y-z^2)*x-y^3-3*z*y^2-3*z^2*y-z^3
590: [4] G=gcd(X,Y);
591: x^2-y^2-2*z*y-z^2
592: [5] sqr(X,G);
593: [x-y-z,0]
594: [6] sqr(Y,G);
595: [x+y+z,0]
596: [7] sdiv(y*x^3+x+1,y*x+1);
597: divsp: cannot happen
598: return to toplevel
599: @end example
600:
601: @table @t
1.2 noro 602: \JP @item $B;2>H(B
603: \EG @item References
1.1 noro 604: @fref{idiv irem}, @fref{%}.
605: @end table
606:
1.2 noro 607: \JP @node tdiv,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
608: \EG @node tdiv,,, Polynomials and rational expressions
1.1 noro 609: @subsection @code{tdiv}
610: @findex tdiv
611:
612: @table @t
613: @item tdiv(@var{poly1},@var{poly2})
1.2 noro 614: \JP :: @var{poly1} $B$,(B @var{poly2} $B$G3d$j@Z$l$k$+$I$&$+D4$Y$k(B.
615: \EG :: Tests whether @var{poly2} divides @var{poly1}.
1.1 noro 616: @end table
617:
618: @table @var
619: @item return
1.2 noro 620: \JP $B3d$j@Z$l$k$J$i$P>&(B, $B3d$j@Z$l$J$1$l$P(B 0
621: \EG Quotient if @var{poly2} divides @var{poly1}, 0 otherwise.
1.1 noro 622: @item poly1 poly2
1.2 noro 623: \JP $BB?9`<0(B
624: \EG polynomial
1.1 noro 625: @end table
626:
627: @itemize @bullet
1.2 noro 628: \BJP
1.1 noro 629: @item
630: @var{poly2} $B$,(B @var{poly1} $B$rB?9`<0$H$7$F3d$j@Z$k$+$I$&$+D4$Y$k(B.
631: @item
632: $B$"$kB?9`<0$,4{Ls0x;R$G$"$k$3$H$O$o$+$C$F$$$k$,(B, $B$=$N=EJ#EY$,$o$+$i$J$$(B
633: $B>l9g$K(B, @code{tdiv()} $B$r7+$jJV$78F$V$3$H$K$h$j=EJ#EY$,$o$+$k(B.
1.2 noro 634: \E
635: \BEG
636: @item
637: Tests whether @var{poly2} divides @var{poly1} in polynomial ring.
638: @item
639: One application of this function: Consider the case where a polynomial
640: is certainly an irreducible factor of the other polynomial, but
641: the multiplicity of the factor is unknown. Application of @code{tdiv()}
642: to the polynomials repeatedly yields the multiplicity.
643: \E
1.1 noro 644: @end itemize
645:
646: @example
647: [11] Y=(x+y+z)^5*(x-y-z)^3;
648: x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6+(-6*y^3-18*z*y^2-18*z^2*y-6*z^3)*x^5
649: +(6*y^5+30*z*y^4+60*z^2*y^3+60*z^3*y^2+30*z^4*y+6*z^5)*x^3+(2*y^6+12*z*y^5
650: +30*z^2*y^4+40*z^3*y^3+30*z^4*y^2+12*z^5*y+2*z^6)*x^2+(-2*y^7-14*z*y^6
651: -42*z^2*y^5-70*z^3*y^4-70*z^4*y^3-42*z^5*y^2-14*z^6*y-2*z^7)*x-y^8-8*z*y^7
652: -28*z^2*y^6-56*z^3*y^5-70*z^4*y^4-56*z^5*y^3-28*z^6*y^2-8*z^7*y-z^8
653: [12] for(I=0,F=x+y+z,T=Y; T=tdiv(T,F); I++);
654: [13] I;
655: 5
656: @end example
657:
658: @table @t
1.2 noro 659: \JP @item $B;2>H(B
660: \EG @item References
1.1 noro 661: @fref{sdiv sdivm srem sremm sqr sqrm}.
662: @end table
663:
1.2 noro 664: \JP @node %,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
665: \EG @node %,,, Polynomials and rational expressions
1.1 noro 666: @subsection @code{%}
667: @findex %
668:
669: @table @t
670: @item @var{poly} % @var{m}
1.2 noro 671: \JP :: $B@0?t$K$h$k>jM>(B
672: \EG :: integer remainder to all integer coefficients of the polynomial.
1.1 noro 673: @end table
674:
675: @table @var
676: @item return
1.2 noro 677: \JP $B@0?t$^$?$OB?9`<0(B
678: \EG integer or polynomial
1.1 noro 679: @item poly
1.2 noro 680: \JP $B@0?t$^$?$O@0?t78?tB?9`<0(B
681: \EG integer or polynomial with integer coefficients
1.1 noro 682: @item m
1.2 noro 683: \JP $B@0?t(B
684: \EG intger
1.1 noro 685: @end table
686:
687: @itemize @bullet
1.2 noro 688: \BJP
1.1 noro 689: @item
690: @var{poly} $B$N3F78?t$r(B @var{m} $B$G3d$C$?>jM>$GCV$-49$($?B?9`<0$rJV$9(B.
691: @item
692: $B7k2L$N78?t$OA4$F@5$N@0?t$H$J$k(B.
693: @item
694: @var{poly} $B$O@0?t$G$b$h$$(B. $B$3$N>l9g(B, $B7k2L$,@5$K@55,2=$5$l$k$3$H$r=|$1$P(B
695: @code{irem()} $B$HF1MM$KMQ$$$k$3$H$,$G$-$k(B.
696: @item
697: @var{poly} $B$N78?t(B, @var{m} $B$H$b@0?t$G$"$kI,MW$,$"$k$,(B, $B%A%'%C%/$O9T$J$o$l$J$$(B.
1.2 noro 698: \E
699: \BEG
700: @item
701: Returns a polynomial whose coefficients are remainders of the
702: coefficients of the input polynomial divided by @var{m}.
703: @item
704: The resulting coefficients are all normalized to non-negative integers.
705: @item
706: An integer is allowed for @var{poly}. This can be used for an
707: alternative for @code{irem()} except that the result is normalized to
708: a non-negative integer.
709: @item
710: Coefficients of @var{poly} and @var{m} must all be integers, though the
711: type checking is not done.
712: \E
1.1 noro 713: @end itemize
714:
715: @example
716: [0] (x+2)^5 % 3;
717: x^5+x^4+x^3+2*x^2+2*x+2
718: [1] (x-2)^5 % 3;
719: x^5+2*x^4+x^3+x^2+2*x+1
720: [2] (-5) % 4;
721: 3
722: [3] irem(-5,4);
723: -1
724: @end example
725:
726: @table @t
1.2 noro 727: \JP @item $B;2>H(B
728: \EG @item References
1.1 noro 729: @fref{idiv irem}.
730: @end table
731:
1.2 noro 732: \JP @node subst psubst,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
733: \EG @node subst psubst,,, Polynomials and rational expressions
1.1 noro 734: @subsection @code{subst}, @code{psubst}
735: @findex subst
736: @findex psubst
737:
738: @table @t
739: @item subst(@var{rat}[,@var{varn},@var{ratn}]*)
740: @item psubst(@var{rat}[,@var{var},@var{rat}]*)
1.2 noro 741: \BJP
1.1 noro 742: :: @var{rat} $B$N(B @var{varn} $B$K(B @var{ratn} $B$rBeF~(B
1.4 ! noro 743: (@var{n}=1,2,... $B$G:8$+$i1&$K=g<!BeF~$9$k(B).
1.2 noro 744: \E
745: \BEG
746: :: Substitute @var{ratn} for @var{varn} in expression @var{rat}.
1.4 ! noro 747: (@var{n}=1,2,@dots{}.
1.2 noro 748: Substitution will be done successively from left to right
749: if arguments are repeated.)
750: \E
1.1 noro 751: @end table
752:
753: @table @var
754: @item return
1.2 noro 755: \JP $BM-M}<0(B
756: \EG rational expression
1.4 ! noro 757: @item rat ratn
1.2 noro 758: \JP $BM-M}<0(B
759: \EG rational expression
1.1 noro 760: @item varn
1.2 noro 761: \JP $BITDj85(B
762: \EG indeterminate
1.1 noro 763: @end table
764:
765: @itemize @bullet
1.2 noro 766: \BJP
1.1 noro 767: @item
768: $BM-M}<0$NFCDj$NITDj85$K(B, $BDj?t$"$k$$$OB?9`<0(B, $BM-M}<0$J$I$rBeF~$9$k$N$KMQ$$$k(B.
769: @item
770: @t{subst}(@var{rat},@var{var1},@var{rat1},@var{var2},@var{rat2},...) $B$O(B,
771: @t{subst}(@t{subst}(@var{rat},@var{var1},@var{rat1}),@var{var2},@var{rat2},...)
772: $B$HF1$80UL#$G$"$k(B.
773: @item
774: $BF~NO$N:8B&$+$i=g$KBeF~$r7+$jJV$9$?$a$K(B, $BF~NO$N=g$K$h$C$F7k2L$,JQ$o$k$3$H$,$"$k(B.
775: @item
776: @code{subst()} $B$O(B, @code{sin()} $B$J$I$NH!?t$N0z?t$KBP$7$F$bBeF~$r9T$&(B.
777: @code{psubst()} $B$O(B, $B$3$N$h$&$JH!?t$r0l$D$NFHN)$7$?ITDj85$H8+$J$7$F(B, $B$=(B
778: $B$N0z?t$K$OBeF~$O9T$o$J$$(B. (partial substitution $B$N$D$b$j(B)
779: @item
780: @b{Asir} $B$G$O(B, $BM-M}<0$NLsJ,$O<+F0E*$K$O9T$o$J$$$?$a(B,
781: $BM-M}<0$NBeF~$O(B, $B;W$o$L7W;;;~4V$NA}Bg$r0z$-5/$3$9>l9g$,$"$k(B.
782: $BM-M}<0$rBeF~$9$k>l9g$K$O(B, $BLdBj$K1~$8$?FH<+$NH!?t$r=q$$$F(B,
783: $B$J$k$Y$/J,Jl(B, $BJ,;R$,Bg$-$/$J$i$J$$$h$&$KG[N8$9$k$3$H$b$7$P$7$PI,MW$H$J$k(B.
784: @item
785: $BJ,?t$rBeF~$9$k>l9g$bF1MM$G$"$k(B.
1.2 noro 786: \E
787: \BEG
788: @item
789: Substitutes rational expressions for specified kernels in a rational
790: expression.
791: @item
792: @t{subst}(@var{rat},@var{var1},@var{rat1},@var{var2},@var{rat2},@dots{})
793: has the same effect as
794: @t{subst}(@t{subst}(@var{rat},@var{var1},@var{rat1}),@var{var2},@var{rat2},@dots{}).
795: @item
796: Note that repeated substitution is done from left to right successively.
797: You may get different result by changing the specification order.
798: @item
799: Ordinary @code{subst()} performs
800: substitution at all levels of a scalar algebraic expression creeping
801: into arguments of function forms recursively.
802: Function @code{psubst()} regards such a function form as an independent
803: indeterminate, and does not attempt to apply substitution to its
804: arguments. (The name comes after Partial SUBSTitution.)
805: @item
806: Since @b{Asir} does not reduce common divisors of a rational expression
807: automatically, substitution of a rational expression to an expression
808: may cause unexpected increase of computation time.
809: Thus, it is often necessary to write a special function to meet the
810: individual problem so that the denominator and the numerator do not
811: become too large.
812: @item
813: The same applies to substitution by rational numbers.
814: \E
1.1 noro 815: @end itemize
816:
817: @example
818: [0] subst(x^3-3*y*x^2+3*y^2*x-y^3,y,2);
819: x^3-6*x^2+12*x-8
820: [1] subst(@@@@,x,-1);
821: -27
822: [2] subst(x^3-3*y*x^2+3*y^2*x-y^3,y,2,x,-1);
823: -27
824: [3] subst(x*y^3,x,y,y,x);
825: x^4
826: [4] subst(x*y^3,y,x,x,y);
827: y^4
828: [5] subst(x*y^3,x,t,y,x,t,y);
829: y*x^3
830: [6] subst(x*sin(x),x,t);
831: sint(t)*t
832: [7] psubst(x*sin(x),x,t);
833: sin(x)*t
834: @end example
835:
1.2 noro 836: \JP @node diff,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
837: \EG @node diff,,, Polynomials and rational expressions
1.1 noro 838: @subsection @code{diff}
839: @findex diff
840:
841: @table @t
842: @item diff(@var{rat}[,@var{varn}]*)
843: @item diff(@var{rat},@var{varlist})
1.2 noro 844: \JP :: @var{rat} $B$r(B @var{varn} $B$"$k$$$O(B @var{varlist} $B$NCf$NJQ?t$G=g<!HyJ,$9$k(B.
845: \BEG
846: :: Differentiate @var{rat} successively by @var{var}'s for the first
847: form, or by variables in @var{varlist} for the second form.
848: \E
1.1 noro 849: @end table
850:
851: @table @var
852: @item return
1.2 noro 853: \JP $B<0(B
854: \EG expression
1.1 noro 855: @item rat
1.2 noro 856: \JP $BM-M}<0(B ($B=iEyH!?t$r4^$s$G$b$h$$(B)
857: \EG rational expression which contains elementary functions.
1.1 noro 858: @item varn
1.2 noro 859: \JP $BITDj85(B
860: \EG indeterminate
1.1 noro 861: @item varlist
1.2 noro 862: \JP $BITDj85$N%j%9%H(B
863: \EG list of indeterminates
1.1 noro 864: @end table
865:
866: @itemize @bullet
1.2 noro 867: \BJP
1.1 noro 868: @item
869: $BM?$($i$l$?=iEyH!?t$r(B @var{varn} $B$"$k$$$O(B @var{varlist} $B$NCf$NJQ?t$G(B
870: $B=g<!HyJ,$9$k(B.
871: @item
872: $B:8B&$NITDj85$h$j(B, $B=g$KHyJ,$7$F$$$/(B. $B$D$^$j(B, @t{diff}(@var{rat},@t{x,y}) $B$O(B,
873: @t{diff}(@t{diff}(@var{rat},@t{x}),@t{y}) $B$HF1$8$G$"$k(B.
1.2 noro 874: \E
875: \BEG
876: @item
877: Differentiate @var{rat} successively by @var{var}'s for the first
878: form, or by variables in @var{varlist} for the second form.
879: @item
880: differentiation is performed by the specified indeterminates (variables)
881: from left to right.
882: @t{diff}(@var{rat},@t{x,y}) is the same as
883: @t{diff}(@t{diff}(@var{rat},@t{x}),@t{y}).
884: \E
1.1 noro 885: @end itemize
886:
887: @example
888: [0] diff((x+2*y)^2,x);
889: 2*x+4*y
890: [1] diff((x+2*y)^2,x,y);
891: 4
892: [2] diff(x/sin(log(x)+1),x);
893: (sin(log(x)+1)-cos(log(x)+1))/(sin(log(x)+1)^2)
894: [3] diff(sin(x),[x,x,x,x]);
895: sin(x)
896: @end example
897:
1.2 noro 898: \JP @node res,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
899: \EG @node res,,, Polynomials and rational expressions
1.1 noro 900: @subsection @code{res}
901: @findex res
902:
903: @table @t
904: @item res(@var{var},@var{poly1},@var{poly2}[,@var{mod}])
1.2 noro 905: \JP :: @var{var} $B$K4X$9$k(B @var{poly1} $B$H(B @var{poly2} $B$N=*7k<0(B.
906: \EG :: Resultant of @var{poly1} and @var{poly2} with respect to @var{var}.
1.1 noro 907: @end table
908:
909: @table @var
910: @item return
1.2 noro 911: \JP $BB?9`<0(B
912: \EG polynomial
1.1 noro 913: @item var
1.2 noro 914: \JP $BITDj85(B
915: \EG indeterminate
1.4 ! noro 916: @item poly1 poly2
1.2 noro 917: \JP $BB?9`<0(B
918: \EG polynomial
1.1 noro 919: @item mod
1.2 noro 920: \JP $BAG?t(B
921: \EG prime
1.1 noro 922: @end table
923:
924: @itemize @bullet
1.2 noro 925: \BJP
1.1 noro 926: @item
927: $BFs$D$NB?9`<0(B @var{poly1} $B$H(B @var{poly2} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k(B
928: $B=*7k<0$r5a$a$k(B.
929: @item
930: $BItJ,=*7k<0%"%k%4%j%:%`$K$h$k(B.
931: @item
932: $B0z?t(B @var{mod} $B$,$"$k;~(B, GF(@var{mod}) $B>e$G$N7W;;$r9T$&(B.
1.2 noro 933: \E
934: \BEG
935: @item
936: Resultant of two polynomials @var{poly1} and @var{poly2}
937: with respect to @var{var}.
938: @item
939: Sub-resultant algorithm is used to compute the resultant.
940: @item
941: The computation is done over GF(@var{mod}) if @var{mod} is specified.
942: \E
1.1 noro 943: @end itemize
944:
945: @example
946: [0] res(t,(t^3+1)*x+1,(t^3+1)*y+t);
947: -x^3-x^2-y^3
948: @end example
949:
1.2 noro 950: \JP @node fctr sqfr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
951: \EG @node fctr sqfr,,, Polynomials and rational expressions
1.1 noro 952: @subsection @code{fctr}, @code{sqfr}
953: @findex fctr
954: @findex sqfr
955:
956: @table @t
957: @item fctr(@var{poly})
1.2 noro 958: \JP :: @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B.
959: \EG :: Factorize polynomial @var{poly} over the rationals.
1.1 noro 960: @item sqfr(@var{poly})
1.2 noro 961: \JP :: @var{poly} $B$rL5J?J}J,2r$9$k(B.
962: \EG :: Gets a square-free factorization of polynomial @var{poly}.
1.1 noro 963: @end table
964:
965: @table @var
966: @item return
1.2 noro 967: \JP $B%j%9%H(B
968: \EG list
1.1 noro 969: @item poly
1.2 noro 970: \JP $BM-M}?t78?t$NB?9`<0(B
971: \EG polynomial with rational coefficients
1.1 noro 972: @end table
973:
974: @itemize @bullet
1.2 noro 975: \BJP
1.1 noro 976: @item
977: $BM-M}?t78?t$NB?9`<0(B @var{poly} $B$r0x?tJ,2r$9$k(B. @code{fctr()} $B$O4{Ls0x;RJ,2r(B,
978: @code{sqfr()} $B$OL5J?J}0x;RJ,2r(B.
979: @item
980: $B7k2L$O(B [[@b{$B?t78?t(B},1],[@b{$B0x;R(B},@b{$B=EJ#EY(B}],...] $B$J$k%j%9%H(B.
981: @item
982: @b{$B?t78?t(B} $B$H(B $BA4$F$N(B @b{$B0x;R(B}^@b{$B=EJ#EY(B} $B$N@Q$,(B @var{poly} $B$HEy$7$$(B.
983: @item
984: @b{$B?t78?t(B} $B$O(B, (@var{poly}/@b{$B?t78?t(B}) $B$,(B, $B@0?t78?t$G(B, $B78?t$N(B GCD $B$,(B 1 $B$H$J$k(B
985: $B$h$&$JB?9`<0$K$J$k$h$&$KA*$P$l$F$$$k(B. (@code{ptozp()} $B;2>H(B)
1.2 noro 986: \E
987: \BEG
988: @item
989: Factorizes polynomial @var{poly} over the rationals.
990: @code{fctr()} for irreducible factorization;
991: @code{sqfr()} for square-free factorization.
992: @item
993: The result is represented by a list, whose elements are a pair
994: represented as
995:
996: [[@b{num},1],[@b{factor},@b{multiplicity}],...].
997: @item
998: Products of all @b{factor}^@b{multiplicity} and @b{num} is equal to
999: @var{poly}.
1000: @item
1001: The number @b{num} is determined so that (@var{poly}/@b{num}) is an
1002: integral polynomial and its content (GCD of all coefficients) is 1.
1003: (@xref{ptozp}.)
1004: \E
1.1 noro 1005: @end itemize
1006:
1007: @example
1008: [0] fctr(x^10-1);
1009: [[1,1],[x-1,1],[x+1,1],[x^4+x^3+x^2+x+1,1],[x^4-x^3+x^2-x+1,1]]
1010: [1] fctr(x^3+y^3+(z/3)^3-x*y*z);
1011: [[1/27,1],[9*x^2+(-9*y-3*z)*x+9*y^2-3*z*y+z^2,1],[3*x+3*y+z,1]]
1012: [2] A=(a+b+c+d)^2;
1013: a^2+(2*b+2*c+2*d)*a+b^2+(2*c+2*d)*b+c^2+2*d*c+d^2
1014: [3] fctr(A);
1015: [[1,1],[a+b+c+d,2]]
1016: [4] A=(x+1)*(x^2-y^2)^2;
1017: x^5+x^4-2*y^2*x^3-2*y^2*x^2+y^4*x+y^4
1018: [5] sqfr(A);
1019: [[1,1],[x+1,1],[-x^2+y^2,2]]
1020: [6] fctr(A);
1021: [[1,1],[x+1,1],[-x-y,2],[x-y,2]]
1022: @end example
1023:
1024: @table @t
1.2 noro 1025: \JP @item $B;2>H(B
1026: \EG @item References
1.1 noro 1027: @fref{ufctrhint}.
1028: @end table
1029:
1.2 noro 1030: \JP @node ufctrhint,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
1031: \EG @node ufctrhint,,, Polynomials and rational expressions
1.1 noro 1032: @subsection @code{ufctrhint}
1033: @findex ufctrhint
1034:
1035: @table @t
1036: @item ufctrhint(@var{poly},@var{hint})
1.2 noro 1037: \JP :: $B<!?t>pJs$rMQ$$$?(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B
1038: \BEG
1039: :: Factorizes uni-variate polynomial @var{poly} over the rational number
1040: field when the degrees of its factors are known to be some integer
1041: multiples of @var{hint}.
1042: \E
1.1 noro 1043: @end table
1044:
1045: @table @var
1046: @item return
1.2 noro 1047: \JP $B%j%9%H(B
1048: \EG list
1.1 noro 1049: @item poly
1.2 noro 1050: \JP $BM-M}?t78?t$N(B 1 $BJQ?tB?9`<0(B
1051: \EG uni-variate polynomial with rational coefficients
1.1 noro 1052: @item hint
1.2 noro 1053: \JP $B<+A3?t(B
1054: \EG non-negative integer
1.1 noro 1055: @end table
1056:
1057: @itemize @bullet
1.2 noro 1058: \BJP
1.1 noro 1059: @item
1060: $B3F4{Ls0x;R$N<!?t$,(B @var{hint} $B$NG\?t$G$"$k$3$H$,$o$+$C$F$$$k>l9g$K(B
1061: @var{poly} $B$N4{Ls0x;RJ,2r$r(B @code{fctr()} $B$h$j8zN(NI$/9T$&(B.
1062: @var{poly} $B$,(B, @var{d} $B<!$N3HBgBN>e$K$*$1$k(B
1.3 noro 1063: $B$"$kB?9`<0$N%N%k%`(B (@ref{$BBe?tE*?t$K4X$9$k1i;;(B}) $B$GL5J?J}$G$"$k>l9g(B,
1.1 noro 1064: $B3F4{Ls0x;R$N<!?t$O(B @var{d} $B$NG\?t$H$J$k(B. $B$3$N$h$&$J>l9g$K(B
1065: $BMQ$$$i$l$k(B.
1.2 noro 1066: \E
1067: \BEG
1068: @item
1069: By any reason, if the degree of all the irreducible factors of @var{poly}
1070: is known to be some multiples of @var{hint}, factors can be computed
1071: more efficiently by the knowledge than @code{fctr()}.
1072: @item
1073: When @var{hint} is 1, @code{ufctrhint()} is the same as @code{fctr()} for
1074: uni-variate polynomials.
1075: An typical application where @code{ufctrhint()} is effective:
1.3 noro 1076: Consider the case where @var{poly} is a norm (@ref{Algebraic numbers})
1.2 noro 1077: of a certain polynomial over an extension field with its extension
1078: degree @var{d}, and it is square free; Then, every irreducible factor
1079: has a degree that is a multiple of @var{d}.
1080: \E
1.1 noro 1081: @end itemize
1082:
1083: @example
1084: [10] A=t^9-15*t^6-87*t^3-125;
1085: t^9-15*t^6-87*t^3-125
1086: 0msec
1087: [11] N=res(t,subst(A,t,x-2*t),A);
1088: -x^81+1215*x^78-567405*x^75+139519665*x^72-19360343142*x^69+1720634125410*x^66
1089: -88249977024390*x^63-4856095669551930*x^60+1999385245240571421*x^57
1090: -15579689952590251515*x^54+15956967531741971462865*x^51
1091: ...
1092: +140395588720353973535526123612661444550659875*x^6
1093: +10122324287343155430042768923500799484375*x^3
1094: +139262743444407310133459021182733314453125
1095: 980msec + gc : 250msec
1096: [12] sqfr(N);
1097: [[-1,1],[x^81-1215*x^78+567405*x^75-139519665*x^72+19360343142*x^69
1098: -1720634125410*x^66+88249977024390*x^63+4856095669551930*x^60
1099: -1999385245240571421*x^57+15579689952590251515*x^54
1100: ...
1101: -10122324287343155430042768923500799484375*x^3
1102: -139262743444407310133459021182733314453125,1]]
1103: 20msec
1104: [13] fctr(N);
1105: [[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],
1106: [x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3
1107: +296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],
1108: [x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3+31524548679,1],
1109: [x^18+10773*x^12+2784051*x^6+307546875,1]]
1110: 167.050sec + gc : 1.890sec
1111: [14] ufctrhint(N,9);
1112: [[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],
1113: [x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3
1114: +296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],
1115: [x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3+31524548679,1],
1116: [x^18+10773*x^12+2784051*x^6+307546875,1]]
1117: 119.340sec + gc : 1.300sec
1118: @end example
1119:
1120: @table @t
1.2 noro 1121: \JP @item $B;2>H(B
1122: \EG @item References
1.1 noro 1123: @fref{fctr sqfr}.
1124: @end table
1125:
1.2 noro 1126: \JP @node modfctr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
1127: \EG @node modfctr,,, Polynomials and rational expressions
1.1 noro 1128: @subsection @code{modfctr}
1129: @findex modfctr
1130:
1131: @table @t
1132: @item modfctr(@var{poly},@var{mod})
1.2 noro 1133: \JP :: $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B
1134: \EG :: Univariate factorizer over small finite fields
1.1 noro 1135: @end table
1136:
1137: @table @var
1138: @item return
1.2 noro 1139: \JP $B%j%9%H(B
1140: \EG list
1.1 noro 1141: @item poly
1.2 noro 1142: \JP $B@0?t78?t$N(B 1 $BJQ?tB?9`<0(B
1143: \EG univariate polynomial with integer coefficients
1.1 noro 1144: @item mod
1.2 noro 1145: \JP $B<+A3?t(B
1146: \EG non-negative integer
1.1 noro 1147: @end table
1148:
1149: @itemize @bullet
1.2 noro 1150: \BJP
1.1 noro 1151: @item
1152: 2^31 $BL$K~$N<+A3?t(B @var{mod} $B$rI8?t$H$9$kAGBN>e$G0lJQ?tB?9`<0(B
1153: @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B.
1154: @item
1155: $B7k2L$O(B [[@b{$B?t78?t(B},1],[@b{$B0x;R(B},@b{$B=EJ#EY(B}],...] $B$J$k%j%9%H(B.
1156: @item
1157: @b{$B?t78?t(B} $B$H(B $BA4$F$N(B @b{$B0x;R(B}^@b{$B=EJ#EY(B} $B$N@Q$,(B @var{poly} $B$HEy$7$$(B.
1.2 noro 1158: @item
1159: $BBg$-$J0L?t$r;}$DM-8BBN>e$N0x?tJ,2r$K$O(B @code{fctr_ff} $B$rMQ$$$k(B.
1160: (@ref{$BM-8BBN$K4X$9$k1i;;(B},@pxref{fctr_ff}).
1161: \E
1162: \BEG
1163: @item
1164: This function factorizes a univarate polynomial @var{poly} over
1165: the finite prime field of characteristic @var{mod}, where
1166: @var{mod} must be smaller than 2^31.
1167: @item
1168: The result is represented by a list, whose elements are a pair
1169: represented as
1170:
1171: [[@b{num},1],[@b{factor},@b{multiplicity}],...].
1172: @item
1173: Products of all @b{factor}^@b{multiplicity} and @b{num} is equal to
1174: @var{poly}.
1175: @item
1176: To factorize polynomials over large finite fields, use
1177: @code{fctr_ff} (@pxref{Finite fields},@ref{fctr_ff}).
1178: \E
1.1 noro 1179: @end itemize
1180:
1181: @example
1182: [0] modfctr(x^10+x^2+1,2147483647);
1183: [[1,1],[x+1513477736,1],[x+2055628767,1],[x+91854880,1],
1184: [x+634005911,1],[x+1513477735,1],[x+634005912,1],
1185: [x^4+1759639395*x^2+2045307031,1]]
1186: @end example
1187:
1188: @table @t
1.2 noro 1189: \JP @item $B;2>H(B
1190: \EG @item References
1.1 noro 1191: @fref{fctr sqfr}.
1192: @end table
1193:
1.2 noro 1194: \JP @node ptozp,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
1195: \EG @node ptozp,,, Polynomials and rational expressions
1.1 noro 1196: @subsection @code{ptozp}
1197: @findex ptozp
1198:
1199: @table @t
1200: @item ptozp(@var{poly})
1.2 noro 1201: \JP :: @var{poly} $B$rM-M}?tG\$7$F@0?t78?tB?9`<0$K$9$k(B.
1202: \BEG
1203: :: Converts a polynomial @var{poly} with rational coefficients into
1204: an integral polynomial such that GCD of all its coefficients is 1.
1205: \E
1.1 noro 1206: @end table
1207:
1208: @table @var
1209: @item return
1.2 noro 1210: \JP $BB?9`<0(B
1211: \EG polynomial
1.1 noro 1212: @item poly
1.2 noro 1213: \JP $BB?9`<0(B
1214: \EG polynomial
1.1 noro 1215: @end table
1216:
1217: @itemize @bullet
1.2 noro 1218: \BJP
1.1 noro 1219: @item
1220: $BM?$($i$l$?B?9`<0(B @var{poly} $B$KE,Ev$JM-M}?t$r3]$1$F(B, $B@0?t78?t$+$D(B
1221: $B78?t$N(B GCD $B$,(B 1 $B$K$J$k$h$&$K$9$k(B.
1222: @item
1223: $BJ,?t$N;MB'1i;;$O(B, $B@0?t$N1i;;$KHf3S$7$FCY$$$?$a(B, $B<o!9$NB?9`<01i;;(B
1224: $B$NA0$K(B, $BB?9`<0$r@0?t78?t$K$7$F$*$/$3$H$,K>$^$7$$(B.
1225: @item
1226: $BM-M}<0$rLsJ,$9$k(B @code{red()} $B$GJ,?t78?tM-M}<0$rLsJ,$7$F$b(B,
1227: $BJ,;RB?9`<0$N78?t$OM-M}?t$N$^$^$G$"$j(B, $BM-M}<0$NJ,;R$r5a$a$k(B
1228: @code{nm()} $B$G$O(B, $BJ,?t78?tB?9`<0$O(B, $BJ,?t78?t$N$^$^$N7A$G=PNO$5$l$k$?$a(B,
1229: $BD>$A$K@0?t78?tB?9`<0$rF@$k;v$O=PMh$J$$(B.
1.2 noro 1230: \E
1231: \BEG
1232: @item
1233: Converts the given polynomial by multiplying some rational number
1234: into an integral polynomial such that GCD of all its coefficients is 1.
1235: @item
1236: In general, operations on polynomials can be
1237: performed faster for integer coefficients than for rational number
1238: coefficients. Therefore, this function is conveniently used to improve
1239: efficiency.
1240: @item
1241: Function @code{red} does not convert rational coefficients of the
1242: numerator.
1243: You cannot obtain an integral polynomial by direct use of the function
1244: @code{nm()}. The function @code{nm()} returns the numerator of its
1245: argument, and a polynomial with rational coefficients is
1246: the numerator of itself and will be returned as it is.
1247: \E
1.1 noro 1248: @end itemize
1249:
1250: @example
1251: [0] ptozp(2*x+5/3);
1252: 6*x+5
1253: [1] nm(2*x+5/3);
1254: 2*x+5/3
1255: @end example
1256:
1257: @table @t
1.2 noro 1258: \JP @item $B;2>H(B
1259: \EG @item References
1.1 noro 1260: @fref{nm dn}.
1261: @end table
1262:
1.2 noro 1263: \JP @node prim cont,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
1264: \EG @node prim cont,,, Polynomials and rational expressions
1.1 noro 1265: @subsection @code{prim}, @code{cont}
1266: @findex prim
1267:
1268: @table @t
1269: @item prim(@var{poly}[,@var{v}])
1.2 noro 1270: \JP :: @var{poly} $B$N86;OE*ItJ,(B (primitive part).
1271: \EG :: Primitive part of @var{poly}.
1.1 noro 1272: @item cont(@var{poly}[,@var{v}])
1.2 noro 1273: \JP :: @var{poly} $B$NMFNL(B (content).
1274: \EG :: Content of @var{poly}.
1.1 noro 1275: @end table
1276:
1277: @table @var
1278: @item return poly
1.2 noro 1279: \JP $BM-M}?t78?tB?9`<0(B
1280: \EG polynomial over the rationals
1.1 noro 1281: @item v
1.2 noro 1282: \JP $BITDj85(B
1283: \EG indeterminate
1.1 noro 1284: @end table
1285:
1286: @itemize @bullet
1.2 noro 1287: \BJP
1.1 noro 1288: @item
1289: @var{poly} $B$N<gJQ?t(B ($B0z?t(B @var{v} $B$,$"$k>l9g$K$O(B @var{v})
1290: $B$K4X$9$k86;OE*ItJ,(B, $BMFNL$r5a$a$k(B.
1.2 noro 1291: \E
1292: \BEG
1293: @item
1294: The primitive part and the content of a polynomial @var{poly}
1295: with respect to its main variable (@var{v} if specified).
1296: \E
1.1 noro 1297: @end itemize
1298:
1299: @example
1300: [0] E=(y-z)*(x+y)*(x-z)*(2*x-y);
1301: (2*y-2*z)*x^3+(y^2-3*z*y+2*z^2)*x^2+(-y^3+z^2*y)*x+z*y^3-z^2*y^2
1302: [1] prim(E);
1303: 2*x^3+(y-2*z)*x^2+(-y^2-z*y)*x+z*y^2
1304: [2] cont(E);
1305: y-z
1306: [3] prim(E,z);
1307: (y-z)*x-z*y+z^2
1308: @end example
1309:
1310: @table @t
1.2 noro 1311: \JP @item $B;2>H(B
1312: \EG @item References
1.1 noro 1313: @fref{var}, @fref{ord}.
1314: @end table
1315:
1.2 noro 1316: \JP @node gcd gcdz,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
1317: \EG @node gcd gcdz,,, Polynomials and rational expressions
1.1 noro 1318: @subsection @code{gcd}, @code{gcdz}
1319: @findex gcd
1320:
1321: @table @t
1322: @item gcd(@var{poly1},@var{poly2}[,@var{mod}])
1323: @item gcdz(@var{poly1},@var{poly2})
1.2 noro 1324: \JP :: @var{poly1} $B$H(B @var{poly2} $B$N(B gcd.
1325: \EG :: The polynomial greatest common divisor of @var{poly1} and @var{poly2}.
1.1 noro 1326: @end table
1327:
1328: @table @var
1329: @item return
1.2 noro 1330: \JP $BB?9`<0(B
1331: \EG polynomial
1.4 ! noro 1332: @item poly1 poly2
1.2 noro 1333: \JP $BB?9`<0(B
1334: \EG polynomial
1.1 noro 1335: @item mod
1.2 noro 1336: \JP $BAG?t(B
1337: \EG prime
1.1 noro 1338: @end table
1339:
1340: @itemize @bullet
1.2 noro 1341: \BJP
1.1 noro 1342: @item
1343: $BFs$D$NB?9`<0$N:GBg8xLs<0(B (GCD) $B$r5a$a$k(B.
1344: @item
1345: @code{gcd()} $B$OM-M}?tBN>e$NB?9`<0$H$7$F$N(B GCD $B$rJV$9(B.
1346: $B$9$J$o$A(B, $B7k2L$O@0?t78?t$G(B, $B$+$D78?t$N(B GCD
1347: $B$,(B 1 $B$K$J$k$h$&$JB?9`<0(B, $B$^$?$O(B, $B8_$$$KAG$N>l9g$O(B 1 $B$rJV$9(B.
1348: @item
1349: @code{gcdz()} $B$O(B @var{poly1}, @var{poly2} $B$H$b$K@0?t78?t$N>l9g$K(B,
1350: $B@0?t4D>e$NB?9`<0$H$7$F$N(B GCD $B$rJV$9(B.
1351: $B$9$J$o$A(B, @code{gcd()} $B$NCM$K(B, $B78?tA4BN$N@0?t(B GCD$B$NCM$r3]$1$?$b$N$rJV$9(B.
1352: @item
1353: $B0z?t(B @var{mod} $B$,$"$k;~(B, @code{gcd()} $B$O(B GF(@var{mod}) $B>e$G$N(B GCD $B$rJV$9(B.
1354: @item
1355: @code{gcd()}, @code{gcdz()} Extended Zassenhaus $B%"%k%4%j%:%`$K$h$k(B.
1356: $BM-8BBN>e$N(B GCD $B$O(B PRS $B%"%k%4%j%:%`$K$h$C$F$$$k$?$a(B, $BBg$-$JLdBj(B,
1357: GCD $B$,(B 1 $B$N>l9g$J$I$K$*$$$F8zN($,0-$$(B.
1.2 noro 1358: \E
1359: \BEG
1360: @item
1361: Functions @code{gcd()} and @code{gcdz()} return the greatest common divisor
1362: (GCD) of the given two polynomials.
1363: @item
1364: Function @code{gcd()} returns an integral polynomial GCD over the
1365: rational number field. The coefficients are normalized such that
1366: their GCD is 1. It returns 1 in case that the given polynomials are
1367: mutually prime.
1368: @item
1369: Function @code{gcdz()} works for arguments of integral polynomials,
1370: and returns a polynomial GCD over the integer ring, that is,
1371: it returns @code{gcd()} multiplied by the contents of all coefficients
1372: of the two input polynomials.
1373: @item
1374: @code{gcd()} computes the GCD over GF(@var{mod}) if @var{mod} is specified.
1375: @item
1376: Polynomial GCD is computed by an improved algorithm based
1377: on Extended Zassenhaus algorithm.
1378: @item
1379: GCD over a finite field is computed by PRS algorithm and it may not be
1380: efficient for large inputs and co-prime inputs.
1381: \E
1.1 noro 1382: @end itemize
1383:
1384: @example
1385: [0] gcd(12*(x^2+2*x+1)^2,18*(x^2+(y+1)*x+y)^3);
1386: x^3+3*x^2+3*x+1
1387: [1] gcdz(12*(x^2+2*x+1)^2,18*(x^2+(y+1)*x+y)^3);
1388: 6*x^3+18*x^2+18*x+6
1389: [2] gcd((x+y)*(x-y)^2,(x+y)^2*(x-y));
1390: x^2-y^2
1391: [3] gcd((x+y)*(x-y)^2,(x+y)^2*(x-y),2);
1392: x^3+y*x^2+y^2*x+y^3
1393: @end example
1394:
1395: @table @t
1.2 noro 1396: \JP @item $B;2>H(B
1397: \EG @item References
1.1 noro 1398: @fref{igcd igcdcntl}.
1399: @end table
1400:
1.2 noro 1401: \JP @node red,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
1402: \EG @node red,,, Polynomials and rational expressions
1.1 noro 1403: @subsection @code{red}
1404: @findex red
1405:
1406: @table @t
1407: @item red(@var{rat})
1.2 noro 1408: \JP :: @var{rat} $B$rLsJ,$7$?$b$N(B.
1409: \EG :: Reduced form of @var{rat} by canceling common divisors.
1.1 noro 1410: @end table
1411:
1412: @table @var
1413: @item return
1.2 noro 1414: \JP $BM-M}<0(B
1415: \EG rational expression
1.1 noro 1416: @item rat
1.2 noro 1417: \JP $BM-M}<0(B
1418: \EG rational expression
1.1 noro 1419: @end table
1420:
1421: @itemize @bullet
1.2 noro 1422: \BJP
1.1 noro 1423: @item
1424: @b{Asir} $B$OM-M}?t$NLsJ,$r>o$K<+F0E*$K9T$&(B.
1425: $B$7$+$7(B, $BM-M}<0$K$D$$$F$ODLJ,$O9T$&$,(B,
1426: $BLsJ,$O%f!<%6!<$,;XDj$7$J$$8B$j9T$o$J$$(B.
1427: $B$3$NLsJ,$r9T$&%3%^%s%I$,(B @t{red} $B$G$"$k(B.
1428: @item
1429: EZGCD $B$K$h$j(B @var{rat} $B$NJ,;R(B, $BJ,Jl$rLsJ,$9$k(B.
1430: @item
1431: $B=PNO$5$l$kM-M}<0$NJ,Jl$NB?9`<0$O(B, $B3F78?t$N(B GCD $B$,(B 1 $B$N(B
1432: $B@0?t78?tB?9`<0$G$"$k(B.
1433: $BJ,;R$K$D$$$F$O@0?t78?tB?9`<0$H$J$k$H$O8B$i$J$$(B.
1434: @item
1435: GCD $B$OBgJQ=E$$1i;;$J$N$G(B, $BB>$NJ}K!$G=|$1$k6&DL0x;R$O2DG=$J8B$j=|$/$N$,(B
1436: $BK>$^$7$$(B. $B$^$?(B, $BJ,Jl(B, $BJ,;R$,Bg$-$/$J$C$F$+$i$N$3$NH!?t$N8F$S=P$7$O(B,
1437: $BHs>o$K;~4V$,3]$+$k>l9g$,B?$$(B. $BM-M}<01i;;$r9T$&>l9g$O(B, $B$"$kDxEY(B
1438: $BIQHK$K(B, $BLsJ,$r9T$&I,MW$,$"$k(B.
1.2 noro 1439: \E
1440: \BEG
1441: @item
1442: @b{Asir} automatically performs cancellation of common divisors of rational numb
1443: ers.
1444: But, without an explicit command, it does not cancel common polynomial divisors
1445: of rational expressions.
1446: (Reduction of rational expressions to a common denominator will be always done.)
1447: Use command @t{red()} to perform this cancellation.
1448: @item
1449: Cancel the common divisors of the numerator and the denominator of
1450: a rational expression @var{rat} by computing their GCD.
1451: @item
1452: The denominator polynomial of the result is an integral polynomial
1453: which has no common divisors in its coefficients,
1454: while the numerator may have rational coefficients.
1455: @item
1456: Since GCD computation is a very hard operation, it is desirable to
1457: detect and remove by any means common divisors as far as possible.
1458: Furthermore, a call to this function after swelling of the denominator
1459: and the numerator shall usually take a very long time. Therefore,
1460: often, to some extent, reduction of common divisors is inevitable for
1461: operations of rational expressions.
1462: \E
1.1 noro 1463: @end itemize
1464:
1465: @example
1466: [0] (x^3-1)/(x-1);
1467: (x^3-1)/(x-1)
1468: [1] red((x^3-1)/(x-1));
1469: x^2+x+1
1470: [2] red((x^3+y^3+z^3-3*x*y*z)/(x+y+z));
1471: x^2+(-y-z)*x+y^2-z*y+z^2
1472: [3] red((3*x*y)/(12*x^2+21*y^3*x));
1473: (y)/(4*x+7*y^3)
1474: [4] red((3/4*x^2+5/6*x)/(2*y*x+4/3*x));
1475: (9/8*x+5/4)/(3*y+2)
1476: @end example
1477:
1478: @table @t
1.2 noro 1479: \JP @item $B;2>H(B
1480: \EG @item References
1.1 noro 1481: @fref{nm dn}, @fref{gcd gcdz}, @fref{ptozp}.
1482: @end table
1483:
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