Annotation of OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw, Revision 1.2
1.2 ! takayama 1: /*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v 1.1 2005/04/13 23:50:17 takayama Exp $ */
1.1 takayama 2:
3: /*&C
4: @c DO NOT EDIT THIS FILE
5: */
6: /*&C
7: @node SM1 Functions,,, Top
8:
9: */
10: /*&ja
11: @chapter SM1 $BH!?t(B
12:
13: $B$3$N@a$G$O(B sm1 $B$N(B ox $B%5!<%P(B @code{ox_sm1_forAsir}
14: $B$H$N%$%s%?%U%'!<%94X?t$r2r@b$9$k(B.
15: $B$3$l$i$N4X?t$O%U%!%$%k(B @file{sm1.rr} $B$GDj5A$5$l$F$$$k(B.
16: @file{sm1.rr} $B$O(B @file{$(OpenXM_HOME)/lib/asir-contrib} $B$K$"$k(B.
17: $B%7%9%F%`(B @code{sm1} $B$OHyJ,:nMQAG4D$G7W;;$9$k$?$a$N%7%9%F%`$G$"$k(B.
18: $B7W;;Be?t4v2?$N$$$m$$$m$JITJQNL$N7W;;$,HyJ,:nMQAG$N7W;;$K5"Ce$9$k(B.
19: @code{sm1} $B$K$D$$$F$NJ8=q$O(B @code{OpenXM/doc/kan96xx} $B$K$"$k(B.
20:
21: $B$H$3$KCG$j$,$J$$$+$.$j$3$N@a$N$9$Y$F$N4X?t$O(B,
22: $BM-M}?t78?t$N<0$rF~NO$H$7$F$&$1$D$1$J$$(B.
23: $B$9$Y$F$NB?9`<0$N78?t$O@0?t$G$J$$$H$$$1$J$$(B.
24:
25: @tex
26: $B6u4V(B
27: $X:={\bf C} \setminus \{ 0, 1 \} = {\bf C} \setminus V(x(x-1))$
28: $B$N%I%i!<%`%3%[%b%m%872C#$N<!85$r7W;;$7$F$_$h$&(B.
29: $X$ $B$OJ?LL$KFs$D$N7j$r$"$1$?6u4V$G$"$k$N$G(B, $BE@(B $x=0$, $x=1$ $B$N$^$o$j$r(B
30: $B$^$o$kFs$D$N%k!<%W$,(B1$B<!85$N%[%b%m%8!<72$N6u4V$r$O$k(B.
31: $B$7$?$,$C$F(B, 1$B<!85%I%i!<%`%3%[%b%m%872$N<!85$O(B $2$ $B$G$"$k(B.
32: @code{sm1} $B$O(B $0$ $B<!85$N%3%[%b%m%872$N<!85$*$h$S(B $1$ $B<!85$N%3%[%b%m%872$N(B
33: $B<!85$rEz$($k(B.
34: @end tex
35: */
36: /*&en
37: @chapter SM1 Functions
38:
39: This chapter describes interface functions for
40: sm1 ox server @code{ox_sm1_forAsir}.
41: These interface functions are defined in the file @file{sm1.rr}.
42: The file @file{sm1.rr} is @*
43: at @file{$(OpenXM_HOME)/lib/asir/contrib-packages}.
44: The system @code{sm1} is a system to compute in the ring of differential
45: operators.
46: Many constructions of invariants
47: in the computational algebraic geometry reduce
48: to constructions in the ring of differential operators.
49: Documents on @code{sm1} are in
50: the directory @code{OpenXM/doc/kan96xx}.
51:
52: All the coefficients of input polynomials should be
53: integers for most functions in this section.
54: Other functions accept rational numbers as inputs
55: and it will be explicitely noted in each explanation
56: of these functions.
57:
58:
59:
60: @tex
61: Let us evaluate the dimensions of the de Rham cohomology groups
62: of
63: $X:={\bf C} \setminus \{ 0, 1 \} = {\bf C} \setminus V(x(x-1))$.
64: The space $X$ is a two punctured plane, so two loops that encircles the
65: points $x=0$ and $x=1$ respectively spans the first homology group.
66: Hence, the dimension of the first de Rham cohomology group is $2$.
67: @code{sm1} answers the dimensions of the 0th and the first
68: cohomology groups.
69: @end tex
70: */
71: /*&C
72: @example
73:
74: [283] sm1.deRham([x*(x-1),[x]]);
75: [1,2]
76: @end example
77: */
78: /*&C
79: @noindent
80: The author of @code{sm1} : Nobuki Takayama, @code{takayama@@math.sci.kobe-u.ac.jp} @*
81: The author of sm1 packages : Toshinori Oaku, @code{oaku@@twcu.ac.jp} @*
82: Reference: [SST] Saito, M., Sturmfels, B., Takayama, N.,
83: Grobner Deformations of Hypergeometric Differential Equations,
84: 1999, Springer.
85: @url{http://www.math.kobe-u.ac.jp/KAN}
86: */
87:
88: /*&C
89: @menu
90: * ox_sm1_forAsir::
91: * sm1.start::
92: * sm1.sm1::
93: * sm1.push_int0::
94: * sm1.gb::
95: * sm1.deRham::
96: * sm1.hilbert::
97: * sm1.genericAnn::
98: * sm1.wTensor0::
99: * sm1.reduction::
100: * sm1.xml_tree_to_prefix_string::
101: * sm1.syz::
102: * sm1.mul::
103: * sm1.distraction::
104: * sm1.gkz::
1.2 ! takayama 105: * sm1.mgkz::
1.1 takayama 106: * sm1.appell1::
107: * sm1.appell4::
108: * sm1.rank::
109: * sm1.auto_reduce::
110: * sm1.slope::
111: * sm1.ahg::
112: * sm1.bfunction::
113: * sm1.generalized_bfunction::
114: * sm1.restriction::
115: * sm1.saturation::
116: @end menu
117: */
118:
119: /*&ja
120: @section @code{ox_sm1_forAsir} $B%5!<%P(B
121: */
122: /*&en
123: @section @code{ox_sm1_forAsir} Server
124: */
125:
126: /*&en
127: @node ox_sm1_forAsir,,, SM1 Functions
128: @subsection @code{ox_sm1_forAsir}
129: @findex ox_sm1_forAsir
130: @table @t
131: @item ox_sm1_forAsir
132: :: @code{sm1} server for @code{asir}.
133: @end table
134: @itemize @bullet
135: @item
136: @code{ox_sm1_forAsir} is the @code{sm1} server started from asir
137: by the command @code{sm1.start}.
138: In the standard setting, @*
139: @code{ox_sm1_forAsir} =
140: @file{$(OpenXM_HOME)/lib/sm1/bin/ox_sm1}
141: +
142: @file{$(OpenXM_HOME)/lib/sm1/callsm1.sm1} (macro file) @*
143: +
144: @file{$(OpenXM_HOME)/lib/sm1/callsm1b.sm1} (macro file) @*
145: The macro files @file{callsm1.sm1} and @file{callsm1b.sm1}
146: are searched from
147: current directory, @code{$(LOAD_SM1_PATH)},
148: @code{$(OpenXM_HOME)/lib/sm1},
149: @code{/usr/local/lib/sm1}
150: in this order.
151: @item Note for programmers: See the files
152: @file{$(OpenXM_HOME)/src/kxx/oxserver00.c},
153: @file{$(OpenXM_HOME)/src/kxx/sm1stackmachine.c}
154: to build your own server by reading @code{sm1} macros.
155: @end itemize
156: */
157: /*&ja
158: @node ox_sm1_forAsir,,, SM1 Functions
159: @subsection @code{ox_sm1_forAsir}
160: @findex ox_sm1_forAsir
161: @table @t
162: @item ox_sm1_forAsir
163: :: @code{asir} $B$N$?$a$N(B @code{sm1} $B%5!<%P(B.
164: @end table
165: @itemize @bullet
166: @item
167: $B%5!<%P(B @code{ox_sm1_forAsir} $B$O(B @code{asir} $B$h$j%3%^%s%I(B
168: @code{sm1.start} $B$G5/F0$5$l$k(B @code{sm1} $B%5!<%P$G$"$k(B.
169:
170: $BI8=`E*@_Dj$G$O(B, @*
171: @code{ox_sm1_forAsir} =
172: @file{$(OpenXM_HOME)/lib/sm1/bin/ox_sm1}
173: +
174: @file{$(OpenXM_HOME)/lib/sm1/callsm1.sm1} (macro file) @*
175: +
176: @file{$(OpenXM_HOME)/lib/sm1/callsm1b.sm1} (macro file) @*
177: $B$G$"$j(B, $B$3$l$i$N%^%/%m%U%!%$%k$O(B, $B0lHL$K$O(B
178: current directory, @code{$(LOAD_SM1_PATH)},
179: @code{$(OpenXM_HOME)/lib/sm1},
180: @code{/usr/local/lib/sm1}
181: $B$N=gHV$G$5$,$5$l$k(B.
182: @item $B%W%m%0%i%^!<$N$?$a$N%N!<%H(B:
183: @code{sm1} $B%^%/%m$rFI$_9~$s$G<+J,FH<+$N%5!<%P$r:n$k$K$O(B
184: $B<!$N%U%!%$%k$b8+$h(B
185: @file{$(OpenXM_HOME)/src/kxx/oxserver00.c},
186: @file{$(OpenXM_HOME)/src/kxx/sm1stackmachine.c}
187: @end itemize
188: */
189:
190:
191: /*&ja
192: @section $BH!?t0lMw(B
193: */
194: /*&en
195: @section Functions
196: */
197:
198: /*&en
199: @c sort-sm1.start
200: @node sm1.start,,, SM1 Functions
201: @subsection @code{sm1.start}
202: @findex sm1.start
203: @table @t
204: @item sm1.start()
205: :: Start @code{ox_sm1_forAsir} on the localhost.
206: @end table
207:
208: @table @var
209: @item return
210: Integer
211: @end table
212:
213: @itemize @bullet
214: @item Start @code{ox_sm1_forAsir} on the localhost.
215: It returns the descriptor of @code{ox_sm1_forAsir}.
216: @item Set @code{Xm_noX = 1} to start @code{ox_sm1_forAsir}
217: without a debug window.
218: @item You might have to set suitable orders of variable by the command
219: @code{ord}. For example,
220: when you are working in the
221: ring of differential operators on the variable @code{x} and @code{dx}
222: (@code{dx} stands for
223: @tex $\partial/\partial x$
224: @end tex
225: ),
226: @code{sm1} server assumes that
227: the variable @code{dx} is collected to the right and the variable
228: @code{x} is collected to the left in the printed expression.
229: In the example below, you must not use the variable @code{cc}
230: for computation in @code{sm1}.
231: @item The variables from @code{a} to @code{z} except @code{d} and @code{o}
232: and @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},
233: @code{z0}, ..., @code{z20} can be used as variables for ring of
234: differential operators in default. (cf. @code{Sm1_ord_list} in @code{sm1}).
235: @item The descriptor is stored in @code{static Sm1_proc}.
236: The descriptor can be obtained by the function
237: @code{sm1.get_Sm1_proc()}.
238: @end itemize
239: */
240: /*&ja
241: @c sort-sm1.start
242: @node sm1.start,,, SM1 Functions
243: @subsection @code{sm1.start}
244: @findex sm1.start
245: @table @t
246: @item sm1.start()
247: :: localhost $B$G(B @code{ox_sm1_forAsir} $B$r%9%?!<%H$9$k(B.
248: @end table
249:
250: @table @var
251: @item return
252: $B@0?t(B
253: @end table
254:
255: @itemize @bullet
256: @item localhost $B$G(B @code{ox_sm1_forAsir} $B$r%9%?!<%H$9$k(B.
257: $B%5!<%P(B @code{ox_sm1_forAsir} $B$N<1JLHV9f$rLa$9(B.
258: @item @code{Xm_noX = 1} $B$H$*$/$H%5!<%P(B @code{ox_sm1_forAsir} $B$r%G%P%C%0MQ$N(B
259: $B%&%#%s%I%&$J$7$K5/F0$G$-$k(B.
260: @item $B%3%^%s%I(B @code{ord} $B$rMQ$$$FJQ?t=g=x$r@5$7$/@_Dj$7$F$*$/I,MW$,(B
261: $B$"$k(B.
262: $B$?$H$($P(B,
263: $BJQ?t(B @code{x} $B$H(B @code{dx} $B>e$NHyJ,:nMQAG4D(B
264: (@code{dx} $B$O(B
265: @tex $\partial/\partial x$
266: @end tex
267: $B$KBP1~(B)
268: $B$G7W;;$7$F$$$k$H$-(B,
269: @code{sm1} $B%5!<%P$O<0$r0u:~$7$?$H$-(B,
270: $BJQ?t(B @code{dx} $B$O1&B&$K=8$a$lJQ?t(B
271: @code{x} $B$O:8B&$K$"$D$a$i$l$F$$$k$H2>Dj$7$F$$$k(B.
272: $B<!$NNc$G$O(B, $BJQ?t(B @code{cc} $B$r(B @code{sm1} $B$G$N7W;;$N$?$a$KMQ$$$F$O$$$1$J$$(B.
273: @item @code{a} $B$h$j(B @code{z} $B$N$J$+$G(B, @code{d} $B$H(B @code{o} $B$r=|$$$?$b$N(B,
274: $B$=$l$+$i(B, @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},
275: @code{z0}, ..., @code{z20} $B$O(B, $B%G%U%)!<%k%H$GHyJ,:nMQAG4D$NJQ?t$H$7$F(B
276: $B;H$($k(B (cf. @code{Sm1_ord_list} in @code{sm1}).
277: @item $B<1JLHV9f$O(B @code{static Sm1_proc} $B$K3JG<$5$l$k(B.
278: $B$3$N<1JLHV9f$O4X?t(B @code{sm1.get_Sm1_proc()} $B$G$H$j$@$9$3$H$,$G$-$k(B.
279: @end itemize
280: */
281: /*&C
282: @example
283: [260] ord([da,a,db,b]);
284: [da,a,db,b,dx,dy,dz,x,y,z,dt,ds,t,s,u,v,w,
285: ......... omit ..................
286: ]
287: [261] a*da;
288: a*da
289: [262] cc*dcc;
290: dcc*cc
291: [263] sm1.mul(da,a,[a]);
292: a*da+1
293: [264] sm1.mul(a,da,[a]);
294: a*da
295: @end example
296: */
297: /*&en
298: @table @t
299: @item Reference
300: @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
301: @code{ord}
302: @end table
303: */
304: /*&ja
305: @table @t
306: @item $B;2>H(B
307: @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
308: @code{ord}
309: @end table
310: */
311:
312:
313:
314: /*&en
315: @c sort-sm1
316: @node sm1.sm1,,, SM1 Functions
317: @subsection @code{sm1.sm1}
318: @findex sm1.sm1
319: @table @t
320: @item sm1.sm1(@var{p},@var{s})
321: :: ask the @code{sm1} server to execute the command string @var{s}.
322: @end table
323:
324: @table @var
325: @item return
326: Void
327: @item p
328: Number
329: @item s
330: String
331: @end table
332:
333: @itemize @bullet
334: @item It asks the @code{sm1} server of the descriptor number @var{p}
335: to execute the command string @var{s}.
336: (In the next example, the descriptor number is 0.)
337: @end itemize
338: */
339: /*&ja
340: @node sm1.sm1,,, SM1 Functions
341: @subsection @code{sm1.sm1}
342: @findex sm1.sm1
343: @table @t
344: @item sm1.sm1(@var{p},@var{s})
345: :: $B%5!<%P(B @code{sm1} $B$K%3%^%s%INs(B @var{s} $B$r<B9T$7$F$/$l$k$h$&$K$?$N$`(B.
346: @end table
347:
348: @table @var
349: @item return
350: $B$J$7(B
351: @item p
352: $B?t(B
353: @item s
354: $BJ8;zNs(B
355: @end table
356:
357: @itemize @bullet
358: @item $B<1JLHV9f(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B
359: $B%3%^%s%INs(B @var{s} $B$r<B9T$7$F$/$l$k$h$&$KMj$`(B.
360: ($B<!$NNc$G$O(B, $B<1JLHV9f(B 0)
361: @end itemize
362: */
363: /*&C
364: @example
365: [261] sm1.sm1(0," ( (x-1)^2 ) . ");
366: 0
367: [262] ox_pop_string(0);
368: x^2-2*x+1
369: [263] sm1.sm1(0," [(x*(x-1)) [(x)]] deRham ");
370: 0
371: [264] ox_pop_string(0);
372: [1 , 2]
373: @end example
374: */
375:
376: /*&ja
377: @table @t
378: @item $B;2>H(B
379: @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
380: @end table
381: */
382: /*&en
383: @table @t
384: @item Reference
385: @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
386: @end table
387: */
388:
389:
390: /*&en
391: @c sort-sm1.push_int0
392: @node sm1.push_int0,,, SM1 Functions
393: @subsection @code{sm1.push_int0}
394: @findex sm1.push_int0
395: @table @t
396: @item sm1.push_int0(@var{p},@var{f})
397: :: push the object @var{f} to the server with the descriptor number @var{p}.
398: @end table
399:
400: @table @var
401: @item return
402: Void
403: @item p
404: Number
405: @item f
406: Object
407: @end table
408:
409: @itemize @bullet
410: @item When @code{type(@var{f})} is 2 (recursive polynomial),
411: @var{f} is converted to a string (type == 7)
412: and is sent to the server by @code{ox_push_cmo}.
413: @item When @code{type(@var{f})} is 0 (zero),
414: it is translated to the 32 bit integer zero
415: on the server.
416: Note that @code{ox_push_cmo(@var{p},0)} sends @code{CMO_NULL} to the server.
417: In other words, the server does not get the 32 bit integer 0 nor
418: the bignum 0.
419: @item @code{sm1} integers are classfied into the 32 bit integer and
420: the bignum.
421: When @code{type(@var{f})} is 1 (number), it is translated to the
422: 32 bit integer on the server.
423: Note that @code{ox_push_cmo(@var{p},1234)} send the bignum 1234 to the
424: @code{sm1} server.
425: @item In other cases, @code{ox_push_cmo} is called without data conversion.
426: @end itemize
427: */
428: /*&ja
429: @c sort-sm1.push_int0
430: @node sm1.push_int0,,, SM1 Functions
431: @subsection @code{sm1.push_int0}
432: @findex sm1.push_int0
433: @table @t
434: @item sm1.push_int0(@var{p},@var{f})
435: :: $B%*%V%8%'%/%H(B @var{f} $B$r<1JL;R(B @var{p} $B$N%5!<%P$XAw$k(B.
436: @end table
437:
438: @table @var
439: @item return
440: $B$J$7(B
441: @item p
442: $B?t(B
443: @item f
444: $B%*%V%8%'%/%H(B
445: @end table
446:
447: @itemize @bullet
448: @item @code{type(@var{f})} $B$,(B 2 ($B:F5"B?9`<0(B) $B$N$H$-(B,
449: @var{f} $B$OJ8;zNs(B (type == 7) $B$KJQ49$5$l$F(B,
450: @code{ox_push_cmo} $B$rMQ$$$F%5!<%P$XAw$i$l$k(B.
451: @item @code{type(@var{f})} $B$,(B 0 (zero) $B$N$H$-$O(B,
452: $B%5!<%P>e$G$O(B, 32 bit $B@0?t$H2r<a$5$l$k(B.
453: $B$J$*(B @code{ox_push_cmo(P,0)} $B$O%5!<%P$KBP$7$F(B @code{CMO_NULL}
454: $B$r$*$/$k$N$G(B, $B%5!<%PB&$G$O(B, 32 bit $B@0?t$r<u$1<h$k$o$1$G$O$J$$(B.
455: @item @code{sm1} $B$N@0?t$O(B, 32 bit $B@0?t$H(B bignum $B$K$o$1$k$3$H$,$G$-$k(B.
456: @code{type(@var{f})} $B$,(B 1 ($B?t(B)$B$N$H$-(B, $B$3$N4X?t$O(B 32 bit integer $B$r%5!<%P$K(B
457: $B$*$/$k(B.
458: @code{ox_push_cmo(@var{p},1234)} $B$O(B bignum $B$N(B 1234 $B$r(B
459: @code{sm1} $B%5!<%P$K$*$/$k$3$H$KCm0U$7$h$&(B.
460: @item $B$=$NB>$N>l9g$K$O(B @code{ox_push_cmo} $B$r%G!<%?7?$NJQ49$J$7$K8F$S=P$9(B.
461: @end itemize
462: */
463: /*&C
464: @example
465: [219] P=sm1.start();
466: 0
467: [220] sm1.push_int0(P,x*dx+1);
468: 0
469: [221] A=ox_pop_cmo(P);
470: x*dx+1
471: [223] type(A);
472: 7 (string)
473: @end example
474:
475: @example
476: [271] sm1.push_int0(0,[x*(x-1),[x]]);
477: 0
478: [272] ox_execute_string(0," deRham ");
479: 0
480: [273] ox_pop_cmo(0);
481: [1,2]
482: @end example
483: */
484: /*&en
485: @table @t
486: @item Reference
487: @code{ox_push_cmo}
488: @end table
489: */
490: /*&ja
491: @table @t
492: @item Reference
493: @code{ox_push_cmo}
494: @end table
495: */
496:
497:
498:
499: /*&en
500: @c sort-sm1.gb
501: @node sm1.gb,,, SM1 Functions
502: @subsection @code{sm1.gb}
503: @findex sm1.gb
504: @findex sm1.gb_d
505: @table @t
506: @item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
507: :: computes the Grobner basis of @var{f} in the ring of differential
508: operators with the variable @var{v}.
509: @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
510: :: computes the Grobner basis of @var{f} in the ring of differential
511: operators with the variable @var{v}.
512: The result will be returned as a list of distributed polynomials.
513: @end table
514:
515: @table @var
516: @item return
517: List
518: @item p, q, r
519: Number
520: @item f, v, w
521: List
522: @end table
523:
524: @itemize @bullet
525: @item
526: It returns the Grobner basis of the set of polynomials @var{f}
527: in the ring of deferential operators with the variables @var{v}.
528: @item
529: The weight vectors are given by @var{w}, which can be omitted.
530: If @var{w} is not given,
531: the graded reverse lexicographic order will be used to compute Grobner basis.
532: @item
533: The return value of @code{sm1.gb}
534: is the list of the Grobner basis of @var{f} and the initial
535: terms (when @var{w} is not given) or initial ideal (when @var{w} is given).
536: @item
537: @code{sm1.gb_d} returns the results by a list of distributed polynomials.
538: Monomials in each distributed polynomial are ordered in the given order.
539: The return value consists of
540: [variable names, order matrix, grobner basis in districuted polynomials,
541: initial monomials or initial polynomials].
542: @item
543: When a non-term order is given, the Grobner basis is computed in
544: the homogenized Weyl algebra (See Section 1.2 of the book of SST).
545: The homogenization variable h is automatically added.
546: @item
547: When the optional variable @var{q} is set, @code{sm1.gb} returns,
548: as the third return value, a list of
549: the Grobner basis and the initial ideal
550: with sums of monomials sorted by the given order.
551: Each polynomial is expressed as a string temporally for now.
552: When the optional variable @var{r} is set to one,
553: the polynomials are dehomogenized (,i.e., h is set to 1).
554: @end itemize
555: */
556: /*&ja
557: @c sort-sm1.gb
558: @node sm1.gb,,, SM1 Functions
559: @subsection @code{sm1.gb}
560: @findex sm1.gb
561: @findex sm1.gb_d
562: @table @t
563: @item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
564: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
565: @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
566: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. $B7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.
567: @end table
568:
569: @table @var
570: @item return
571: $B%j%9%H(B
572: @item p, q, r
573: $B?t(B
574: @item f, v, w
575: $B%j%9%H(B
576: @end table
577:
578: @itemize @bullet
579: @item
580: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
581: @item
582: Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
583: $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
584: $B%V%l%V%J4pDl$r7W;;$9$k(B.
585: @item
586: @code{sm1.gb} $B$NLa$jCM$O(B @var{f} $B$N%0%l%V%J4pDl$*$h$S%$%K%7%c%k%b%N%_%"%k(B
587: ( @var{w} $B$,$J$$$H$-(B ) $B$^$?$O(B $B%$%K%7%!%kB?9`<0(B ( @var{w} $B$,M?$($i$?$H$-(B)
588: $B$N%j%9%H$G$"$k(B.
589: @item
590: @code{sm1.gb_d} $B$O7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.
591: $BB?9`<0$NCf$K8=$l$k%b%N%_%"%k$O%0%l%V%J4pDl$r7W;;$9$k$H$-$KM?$($i$?=g=x$G%=!<%H$5$l$F$$$k(B.
592: $BLa$jCM$O(B
593: [$BJQ?tL>$N%j%9%H(B, $B=g=x$r$-$a$k9TNs(B, $B%0%l%V%J4pDl(B, $B%$%K%7%c%k%b%N%_%"%k$^$?$O%$%K%7%!%kB?9`<0(B]
594: $B$G$"$k(B.
595: @item
596: Term order $B$G$J$$=g=x$,M?$($i$l$?>l9g$O(B, $BF1<!2=%o%$%kBe?t$G%0%l%V%J4pDl$,7W;;$5$l$k(B (SST $B$NK\$N(B Section 1.2 $B$r8+$h(B).
597: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
598: @item $B%*%W%7%g%J%kJQ?t(B @var{q} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
599: 3 $BHVL\$NLa$jCM$H$7$F(B, $B%0%l%V%J4pDl$*$h$S%$%K%7%!%k$N%j%9%H$,(B
600: $BM?$($i$l$?=g=x$G%=!<%H$5$l$?%b%N%_%"%k$NOB$H$7$FLa$5$l$k(B.
601: $B$$$^$N$H$3$m$3$NB?9`<0$O(B, $BJ8;zNs$GI=8=$5$l$k(B.
602: $B%*%W%7%g%J%kJQ?t(B @var{r} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
603: $BLa$jB?9`<0$O(B dehomogenize $B$5$l$k(B ($B$9$J$o$A(B h $B$K(B 1 $B$,BeF~$5$l$k(B).
604: @end itemize
605: */
606: /*&C
607: @example
608: [293] sm1.gb([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
609: [[x*dx+y*dy-1,y^2*dy^2+2],[x*dx,y^2*dy^2]]
610: @end example
611: */
612: /*&en
613: In the example above,
614: @tex the set $\{ x \partial_x + y \partial_y -1,
615: y^2 \partial_y^2+2\}$
616: is the Gr\"obner basis of the input with respect to the
617: graded reverse lexicographic order such that
618: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$.
619: The set $\{x \partial_x, y^2 \partial_y\}$ is the leading monomials
620: (the initial monominals) of the Gr\"obner basis.
621: @end tex
622: */
623: /*&ja
624: $B>e$NNc$K$*$$$F(B,
625: @tex $B=89g(B $\{ x \partial_x + y \partial_y -1,
626: y^2 \partial_y^2+2\}$
627: $B$O(B
628: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$
629: $B$G$"$k$h$&$J(B
630: graded reverse lexicographic order $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
631: $B=89g(B $\{x \partial_x, y^2 \partial_y\}$ $B$O%0%l%V%J4pDl$N3F85$K(B
632: $BBP$9$k(B leading monomial (initial monomial) $B$G$"$k(B.
633: @end tex
634: */
635: /*&C
636: @example
637: [294] sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]);
638: [[dx+dy^3-4*dy,-dy^4+4*dy^2-1],[dx,-dy^4]]
639: @end example
640: */
641: /*&en
642: In the example above, two monomials
643: @tex
644: $m = x^a y^b \partial_x^c \partial_y^d$ and
645: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
646: are firstly compared by the weight vector
647: {\tt (dx,dy,x,y) = (50,2,1,0)}
648: (i.e., $m$ is larger than $m'$ if $50c+2d+a > 50c'+2d'+a'$)
649: and when the comparison is tie, then these are
650: compared by the reverse lexicographic order
651: (i.e., if $50c+2d+a = 50c'+2d'+a'$, then use the reverse lexicogrpahic order).
652: @end tex
653: */
654: /*&ja
655: $B>e$NNc$K$*$$$FFs$D$N%b%N%_%"%k(B
656: @tex
657: $m = x^a y^b \partial_x^c \partial_y^d$ $B$*$h$S(B
658: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
659: $B$O:G=i$K(B weight vector
660: {\tt (dx,dy,x,y) = (50,2,1,0)} $B$rMQ$$$FHf3S$5$l$k(B
661: ($B$D$^$j(B $m$ $B$O(B $50c+2d+a > 50c'+2d'+a'$ $B$N$H$-(B
662: $m'$ $B$h$jBg$-$$(B )
663: $B<!$K$3$NHf3S$G>!Ii$,$D$+$J$$$H$-$O(B reverse lexicographic order $B$GHf3S$5$l$k(B
664: ($B$D$^$j(B $50c+2d+a = 50c'+2d'+a'$ $B$N$H$-(B reverse lexicographic order $B$GHf3S(B
665: $B$5$l$k(B).
666: @end tex
667: */
668: /*&C
669: @example
670: [294] F=sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);
671: map(print,F[2][0])$
672: map(print,F[2][1])$
673: @end example
674: */
675: /*&C
676: @example
677: [595]
678: sm1.gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],
679: [x,y],[[dx,1,x,-1],[dy,1]]]);
680:
681: [[x*dx^2+(y*dy-h^2)*dx-h^3,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx-h^3*dy],
682: [x*dx^2+(y*dy-h^2)*dx,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx]]
683:
684: [596]
685: sm1.gb_d([["dx (x dx +y dy-2)-1","dy (x dx + y dy -2)-1"],
686: "x,y",[[dx,1,x,-1],[dy,1]]]);
687: [[[e0,x,y,H,E,dx,dy,h],
688: [[0,-1,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],
689: [0,1,1,1,1,1,1,0],[0,0,0,0,0,0,-1,0],[0,0,0,0,0,-1,0,0],
690: [0,0,0,0,-1,0,0,0],[0,0,0,-1,0,0,0,0],[0,0,-1,0,0,0,0,0],
691: [0,0,0,0,0,0,0,1]]],
692: [[(1)*<<0,0,1,0,0,1,1,0>>+(1)*<<0,1,0,0,0,2,0,0>>+(-1)*<<0,0,0,0,0,1,0,2>>+(-1)*
693: <<0,0,0,0,0,0,0,3>>,(1)*<<0,0,1,0,0,0,2,0>>+(1)*<<0,1,0,0,0,1,1,0>>+(-1)*<<0,0,0
694: ,0,0,0,1,2>>+(-1)*<<0,0,0,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>+(-1)*<<0,0,0,0,0,0
695: ,1,3>>],
696: [(1)*<<0,0,1,0,0,1,1,0>>+(1)*<<0,1,0,0,0,2,0,0>>+(-1)*<<0,0,0,0,0,1,0,2>>,(1)*<
697: <0,0,1,0,0,0,2,0>>+(1)*<<0,1,0,0,0,1,1,0>>+(-1)*<<0,0,0,0,0,0,1,2>>+(-1)*<<0,0,0
698: ,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]]
699: @end example
700: */
701:
702: /*&en
703: @table @t
704: @item Reference
705: @code{sm1.reduction}, @code{sm1.rat_to_p}
706: @end table
707: */
708: /*&ja
709: @table @t
710: @item $B;2>H(B
711: @code{sm1.reduction}, @code{sm1.rat_to_p}
712: @end table
713: */
714:
715:
716:
717: /*&en
718: @c sort-sm1.deRham
719: @node sm1.deRham,,, SM1 Functions
720: @subsection @code{sm1.deRham}
721: @findex sm1.deRham
722: @table @t
723: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
724: :: ask the server to evaluate the dimensions of the de Rham cohomology groups
725: of C^n - (the zero set of @var{f}=0).
726: @end table
727:
728: @table @var
729: @item return
730: List
731: @item p
732: Number
733: @item f
734: String or polynomial
735: @item v
736: List
737: @end table
738:
739: @itemize @bullet
740: @item It returns the dimensions of the de Rham cohomology groups
741: of X = C^n \ V(@var{f}).
742: In other words, it returns
743: [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)].
744: @item @var{v} is a list of variables. n = @code{length(@var{v})}.
745: @item
746: @code{sm1.deRham} requires huge computer resources.
747: For example, @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
748: is already very hard.
749: @item
750: To efficiently analyze the roots of b-function, @code{ox_asir} should be used
751: from @code{ox_sm1_forAsir}.
752: It is recommended to load the communication module for @code{ox_asir}
753: by the command @*
754: @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
755: This command is automatically executed when @code{ox_sm1_forAsir} is started.
756: @item If you make an interruption to the function @code{sm1.deRham}
757: by @code{ox_reset(sm1.get_Sm1_proc());}, the server might get out of the standard
758: mode. So, it is strongly recommended to execute the command
759: @code{ox_shutdown(sm1.get_Sm1_proc());} to interrupt and restart the server.
760: @end itemize
761: */
762: /*&ja
763: @c sort-sm1.deRham
764: @node sm1.deRham,,, SM1 Functions
765: @subsection @code{sm1.deRham}
766: @findex sm1.deRham
767: @table @t
768: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
769: :: $B6u4V(B C^n - (the zero set of @var{f}=0) $B$N%I%i!<%`%3%[%b%m%872$N<!85$r7W;;$7$F$/$l$k$h$&$K%5!<%P$KMj$`(B.
770: @end table
771:
772: @table @var
773: @item return
774: $B%j%9%H(B
775: @item p
776: $B?t(B
777: @item f
778: $BJ8;zNs(B $B$^$?$O(B $BB?9`<0(B
779: @item v
780: $B%j%9%H(B
781: @end table
782:
783: @itemize @bullet
784: @item $B$3$NH!?t$O6u4V(B X = C^n \ V(@var{f}) $B$N%I%i!<%`%3%[%b%m%872$N<!85$r7W;;$9$k(B.
785: $B$9$J$o$A(B,
786: [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)]
787: $B$rLa$9(B.
788: @item @var{v} $B$OJQ?t$N%j%9%H(B. n = @code{length(@var{v})} $B$G$"$k(B.
789: @item
790: @code{sm1.deRham} $B$O7W;;5!$N;q8;$rBgNL$K;HMQ$9$k(B.
791: $B$?$H$($P(B @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
792: $B$N7W;;$9$i$9$G$KHs>o$KBgJQ$G$"$k(B.
793: @item
794: b-$B4X?t$N:,$r8zN($h$/2r@O$9$k$K$O(B, @code{ox_asir} $B$,(B @code{ox_sm1_forAsir}
795: $B$h$j;HMQ$5$l$k$Y$-$G$"$k(B. $B%3%^%s%I(B @*
796: @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
797: $B$rMQ$$$F(B, @code{ox_asir} $B$H$NDL?.%b%8%e!<%k$r$"$i$+$8$a%m!<%I$7$F$*$/$H$h$$(B.
798: $B$3$N%3%^%s%I$O(B @code{ox_asir_forAsir} $B$N%9%?!<%H;~$K<+F0E*$K<B9T$5$l$F$$$k(B.
799: @item
800: @code{sm1.deRham} $B$r(B @code{ox_reset(sm1.get_Sm1_proc());} $B$GCfCG$9$k$H(B,
801: $B0J8e(B sm1 $B%5!<%P$,HsI8=`%b!<%I$KF~$jM=4|$7$J$$F0:n$r$9$k>l9g(B
802: $B$,$"$k$N$G(B, $B%3%^%s%I(B @code{ox_shutdown(sm1.get_Sm1_proc());} $B$G(B, @code{ox_sm1_forAsir}
803: $B$r0l;~(B shutdown $B$7$F%j%9%?!<%H$7$?J}$,0BA4$G$"$k(B.
804: @end itemize
805: */
806: /*&C
807: @example
808: [332] sm1.deRham([x^3-y^2,[x,y]]);
809: [1,1,0]
810: [333] sm1.deRham([x*(x-1),[x]]);
811: [1,2]
812: @end example
813: */
814: /*&en
815: @table @t
816: @item Reference
817: @code{sm1.start}, @code{deRham} (sm1 command)
818: @item Algorithm:
819: Oaku, Takayama, An algorithm for de Rham cohomology groups of the
820: complement of an affine variety via D-module computation,
821: Journal of pure and applied algebra 139 (1999), 201--233.
822: @end table
823: */
824: /*&ja
825: @table @t
826: @item $B;2>H(B
827: @code{sm1.start}, @code{deRham} (sm1 command)
828: @item Algorithm:
829: Oaku, Takayama, An algorithm for de Rham cohomology groups of the
830: complement of an affine variety via D-module computation,
831: Journal of pure and applied algebra 139 (1999), 201--233.
832: @end table
833: */
834:
835:
836:
837:
838: /*&en
839: @c sort-sm1.hilbert
840: @node sm1.hilbert,,, SM1 Functions
841: @subsection @code{sm1.hilbert}
842: @findex sm1.hilbert
843: @findex hilbert_polynomial
844: @table @t
845: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
846: :: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
847: @item hilbert_polynomial(@var{f},@var{v})
848: :: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
849: @end table
850:
851: @table @var
852: @item return
853: Polynomial
854: @item p
855: Number
856: @item f, v
857: List
858: @end table
859:
860: @itemize @bullet
861: @item It returns the Hilbert polynomial h(k) of the set of polynomials
862: @var{f}
863: with respect to the set of variables @var{v}.
864: @item
865: h(k) = dim_Q F_k/I \cap F_k where F_k the set of polynomials of which
866: degree is less than or equal to k and I is the ideal generated by the
867: set of polynomials @var{f}.
868: @item
869: Note for sm1.hilbert:
870: For an efficient computation, it is preferable that
871: the set of polynomials @var{f} is a set of monomials.
872: In fact, this function firstly compute a Grobner basis of @var{f}, and then
873: compute the Hilbert polynomial of the initial monomials of the basis.
874: If the input @var{f} is already a Grobner
875: basis, a Grobner basis is recomputed in this function,
876: which is a waste of time and Grobner basis computation in the ring of
877: polynomials in @code{sm1} is slower than in @code{asir}.
878: @end itemize
879: */
880: /*&ja
881: @c sort-sm1.hilbert
882: @node sm1.hilbert,,, SM1 Functions
883: @subsection @code{sm1.hilbert}
884: @findex sm1.hilbert
885: @findex hilbert_polynomial
886: @table @t
887: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
888: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
889: @item hilbert_polynomial(@var{f},@var{v})
890: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
891: @end table
892:
893: @table @var
894: @item return
895: $BB?9`<0(B
896: @item p
897: $B?t(B
898: @item f, v
899: $B%j%9%H(B
900: @end table
901:
902: @itemize @bullet
903: @item $BB?9`<0$N=89g(B @var{f} $B$NJQ?t(B @var{v} $B$K$+$s$9$k%R%k%Y%k%HB?9`<0(B h(k)
904: $B$r7W;;$9$k(B.
905: @item
906: h(k) = dim_Q F_k/I \cap F_k $B$3$3$G(B F_k $B$O<!?t$,(B k $B0J2<$G$"$k$h$&$J(B
907: $BB?9`<0$N=89g$G$"$k(B. I $B$OB?9`<0$N=89g(B @var{f} $B$G@8@.$5$l$k%$%G%"%k$G$"$k(B.
908: @item
909: sm1.hilbert $B$K$+$s$9$k%N!<%H(B:
910: $B8zN($h$/7W;;$9$k$K$O(B @var{f} $B$O%b%N%_%"%k$N=89g$K$7$?J}$,$$$$(B.
911: $B<B:](B, $B$3$NH!?t$O$^$:(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$7(B, $B$=$l$+$i$=$N(B initial
912: monomial $BC#$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
913: $B$7$?$,$C$F(B, $BF~NO(B @var{f} $B$,$9$G$K%0%l%V%J4pDl$@$H$3$NH!?t$N$J$+$G$b$&0lEY(B
914: $B%0%l%V%J4pDl$N7W;;$,$*$3$J$o$l$k(B. $B$3$l$O;~4V$NL5BL$G$"$k$7(B, @code{sm1} $B$N(B
915: $BB?9`<0%0%l%V%J4pDl7W;;$O(B @code{asir} $B$h$jCY$$(B.
916: @end itemize
917: */
918:
919: /*&C
920: @example
921:
922: [346] load("katsura")$
923: [351] A=hilbert_polynomial(katsura(5),[u0,u1,u2,u3,u4,u5]);
924: 32
925:
926: @end example
927:
928: @example
929: [279] load("katsura")$
930: [280] A=gr(katsura(5),[u0,u1,u2,u3,u4,u5],0)$
931: [281] dp_ord();
932: 0
933: [282] B=map(dp_ht,map(dp_ptod,A,[u0,u1,u2,u3,u4,u5]));
934: [(1)*<<1,0,0,0,0,0>>,(1)*<<0,0,0,2,0,0>>,(1)*<<0,0,1,1,0,0>>,(1)*<<0,0,2,0,0,0>>,
935: (1)*<<0,1,1,0,0,0>>,(1)*<<0,2,0,0,0,0>>,(1)*<<0,0,0,1,1,1>>,(1)*<<0,0,0,1,2,0>>,
936: (1)*<<0,0,1,0,2,0>>,(1)*<<0,1,0,0,2,0>>,(1)*<<0,1,0,1,1,0>>,(1)*<<0,0,0,0,2,2>>,
937: (1)*<<0,0,1,0,1,2>>,(1)*<<0,1,0,0,1,2>>,(1)*<<0,1,0,1,0,2>>,(1)*<<0,0,0,0,3,1>>,
938: (1)*<<0,0,0,0,4,0>>,(1)*<<0,0,0,0,1,4>>,(1)*<<0,0,0,1,0,4>>,(1)*<<0,0,1,0,0,4>>,
939: (1)*<<0,1,0,0,0,4>>,(1)*<<0,0,0,0,0,6>>]
940: [283] C=map(dp_dtop,B,[u0,u1,u2,u3,u4,u5]);
941: [u0,u3^2,u3*u2,u2^2,u2*u1,u1^2,u5*u4*u3,u4^2*u3,u4^2*u2,u4^2*u1,u4*u3*u1,
942: u5^2*u4^2,u5^2*u4*u2,u5^2*u4*u1,u5^2*u3*u1,u5*u4^3,u4^4,u5^4*u4,u5^4*u3,
943: u5^4*u2,u5^4*u1,u5^6]
944: [284] sm1.hilbert([C,[u0,u1,u2,u3,u4,u5]]);
945: 32
946: @end example
947: */
948:
949: /*&en
950: @table @t
951: @item Reference
952: @code{sm1.start}, @code{sm1.gb}, @code{longname}
953: @end table
954: */
955: /*&ja
956: @table @t
957: @item $B;2>H(B
958: @code{sm1.start}, @code{sm1.gb}, @code{longname}
959: @end table
960: */
961:
962:
963: /*&en
964: @c sort-sm1.genericAnn
965: @node sm1.genericAnn,,, SM1 Functions
966: @subsection @code{sm1.genericAnn}
967: @findex sm1.genericAnn
968: @table @t
969: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
970: :: It computes the annihilating ideal for @var{f}^s.
971: @var{v} is the list of variables. Here, s is @var{v}[0] and
972: @var{f} is a polynomial in the variables @code{rest}(@var{v}).
973: @end table
974:
975: @table @var
976: @item return
977: List
978: @item p
979: Number
980: @item f
981: Polynomial
982: @item v
983: List
984: @end table
985:
986: @itemize @bullet
987: @item This function computes the annihilating ideal for @var{f}^s.
988: @var{v} is the list of variables. Here, s is @var{v}[0] and
989: @var{f} is a polynomial in the variables @code{rest}(@var{v}).
990: @end itemize
991: */
992: /*&ja
993: @c sort-sm1.genericAnn
994: @node sm1.genericAnn,,, SM1 Functions
995: @subsection @code{sm1.genericAnn}
996: @findex sm1.genericAnn
997: @table @t
998: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
999: :: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
1000: @var{v} $B$OJQ?t$N%j%9%H$G$"$k(B. $B$3$3$G(B, s $B$O(B @var{v}[0] $B$G$"$j(B,
1001: @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
1002: @end table
1003:
1004: @table @var
1005: @item return
1006: $B%j%9%H(B
1007: @item p
1008: $B?t(B
1009: @item f
1010: $BB?9`<0(B
1011: @item v
1012: $B%j%9%H(B
1013: @end table
1014:
1015: @itemize @bullet
1016: @item $B$3$NH!?t$O(B,
1017: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
1018: @var{v} $B$OJQ?t$N%j%9%H$G$"$k(B. $B$3$3$G(B, s $B$O(B @var{v}[0] $B$G$"$j(B,
1019: @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
1020: @end itemize
1021: */
1022: /*&C
1023: @example
1024: [595] sm1.genericAnn([x^3+y^3+z^3,[s,x,y,z]]);
1025: [-x*dx-y*dy-z*dz+3*s,z^2*dy-y^2*dz,z^2*dx-x^2*dz,y^2*dx-x^2*dy]
1026: @end example
1027: */
1028: /*&en
1029: @table @t
1030: @item Reference
1031: @code{sm1.start}
1032: @end table
1033: */
1034: /*&ja
1035: @table @t
1036: @item $B;2>H(B
1037: @code{sm1.start}
1038: @end table
1039: */
1040:
1041:
1042:
1043: /*&en
1044: @c sort-sm1.wTensor0
1045: @node sm1.wTensor0,,, SM1 Functions
1046: @subsection @code{sm1.wTensor0}
1047: @findex sm1.wTensor0
1048: @table @t
1049: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1050: :: It computes the D-module theoretic 0-th tensor product
1051: of @var{f} and @var{g}.
1052: @end table
1053:
1054: @table @var
1055: @item return
1056: List
1057: @item p
1058: Number
1059: @item f, g, v, w
1060: List
1061: @end table
1062:
1063: @itemize @bullet
1064: @item
1065: It returns the D-module theoretic 0-th tensor product
1066: of @var{f} and @var{g}.
1067: @item
1068: @var{v} is a list of variables.
1069: @var{w} is a list of weights. The integer @var{w}[i] is
1070: the weight of the variable @var{v}[i].
1071: @item
1072: @code{sm1.wTensor0} calls @code{wRestriction0} of @code{ox_sm1},
1073: which requires a generic weight
1074: vector @var{w} to compute the restriction.
1075: If @var{w} is not generic, the computation fails.
1076: @item Let F and G be solutions of @var{f} and @var{g} respectively.
1077: Intuitively speaking, the 0-th tensor product is a system of
1078: differential equations which annihilates the function FG.
1079: @item The answer is a submodule of a free module D^r in general even if
1080: the inputs @var{f} and @var{g} are left ideals of D.
1081: @end itemize
1082: */
1083:
1084: /*&ja
1085: @c sort-sm1.wTensor0
1086: @node sm1.wTensor0,,, SM1 Functions
1087: @subsection @code{sm1.wTensor0}
1088: @findex sm1.wTensor0
1089: @table @t
1090: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1091: :: @var{f} $B$H(B @var{g} $B$N(B D-module $B$H$7$F$N(B 0 $B<!%F%s%=%k@Q$r(B
1092: $B7W;;$9$k(B.
1093: @end table
1094:
1095: @table @var
1096: @item return
1097: $B%j%9%H(B
1098: @item p
1099: $B?t(B
1100: @item f, g, v, w
1101: $B%j%9%H(B
1102: @end table
1103:
1104: @itemize @bullet
1105: @item
1106: @var{f} $B$H(B @var{g} $B$N(B
1107: D-$B2C72$H$7$F$N(B 0 $B<!%F%s%=%k@Q$r7W;;$9$k(B.
1108: @item
1109: @var{v} $B$OJQ?t$N%j%9%H$G$"$k(B.
1110: @var{w} $B$O(B weight $B$N%j%9%H$G$"$k(B.
1111: $B@0?t(B @var{w}[i] $B$OJQ?t(B @var{v}[i] $B$N(B weight $B$G$"$k(B.
1112: @item
1113: @code{sm1.wTensor0} $B$O(B @code{ox_sm1} $B$N(B @code{wRestriction0}
1114: $B$r$h$s$G$$$k(B.
1115: @code{wRestriction0} $B$O(B, generic $B$J(B weight $B%Y%/%H%k(B @var{w}
1116: $B$r$b$H$K$7$F@)8B$r7W;;$7$F$$$k(B.
1117: Weight $B%Y%/%H%k(B @var{w} $B$,(B generic $B$G$J$$$H7W;;$,%(%i!<$GDd;_$9$k(B.
1118: @item F $B$*$h$S(B G $B$r(B @var{f} $B$H(B @var{g} $B$=$l$>$l$N2r$H$9$k(B.
1119: $BD>4QE*$K$$$($P(B, 0 $B<!$N%F%s%=%k@Q$O(B $B4X?t(B FG $B$N$_$?$9HyJ,J}Dx<07O$G$"$k(B.
1120: @item $BF~NO(B @var{f}, @var{g} $B$,(B D $B$N:8%$%G%"%k$G$"$C$F$b(B,
1121: $B0lHL$K(B, $B=PNO$O<+M32C72(B D^r $B$NItJ,2C72$G$"$k(B.
1122: @end itemize
1123: */
1124: /*&C
1125: @example
1126: [258] sm1.wTensor0([[x*dx -1, y*dy -4],[dx+dy,dx-dy^2],[x,y],[1,2]]);
1127: [[-y*x*dx-y*x*dy+4*x+y],[5*x*dx^2+5*x*dx+2*y*dy^2+(-2*y-6)*dy+3],
1128: [-25*x*dx+(-5*y*x-2*y^2)*dy^2+((5*y+15)*x+2*y^2+16*y)*dy-20*x-8*y-15],
1129: [y^2*dy^2+(-y^2-8*y)*dy+4*y+20]]
1130: @end example
1131: */
1132:
1133:
1134:
1135: /*&en
1136: @c sort-sm1.reduction
1137: @node sm1.reduction,,, SM1 Functions
1138: @subsection @code{sm1.reduction}
1139: @findex sm1.reduction
1140: @table @t
1141: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1142: ::
1143: @end table
1144:
1145: @table @var
1146: @item return
1147: List
1148: @item f
1149: Polynomial
1150: @item g, v, w
1151: List
1152: @item p
1153: Number (the process number of ox_sm1)
1154: @end table
1155:
1156: @itemize @bullet
1157: @item It reduces @var{f} by the set of polynomial @var{g}
1158: in the homogenized Weyl algebra; it applies the
1159: division algorithm to @var{f}. The set of variables is @var{v} and
1160: @var{w} is weight vectors to determine the order, which can be ommited.
1161: @code{sm1.reduction_noH} is for the Weyl algebra.
1162: @item The return value is of the form
1163: [r,c0,[c1,...,cm],[g1,...gm]] where @var{g}=[g1, ..., gm] and
1164: c0 f + c1 g1 + ... + cm gm = r.
1165: r/c0 is the normal form.
1166: @item The function reduction reduces reducible terms that appear
1167: in lower order terms.
1168: @item The functions
1169: sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G)
1170: are for distributed polynomials.
1171: @end itemize
1172: */
1173: /*&ja
1174: @node sm1.reduction,,, SM1 Functions
1175: @subsection @code{sm1.reduction}
1176: @findex sm1.reduction
1177: @table @t
1178: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1179: ::
1180: @end table
1181:
1182: @table @var
1183: @item return
1184: $B%j%9%H(B
1185: @item f
1186: $BB?9`<0(B
1187: @item g, v, w
1188: $B%j%9%H(B
1189: @item p
1190: $B?t(B (ox_sm1 $B$N%W%m%;%9HV9f(B)
1191: @end table
1192:
1193: @itemize @bullet
1194: @item $B$3$NH!?t$O(B @var{f} $B$r(B homogenized $B%o%$%kBe?t$K$*$$$F(B,
1195: $BB?9`<0=89g(B @var{g} $B$G4JC12=(B (reduce) $B$9$k(B; $B$D$^$j(B,
1196: $B$3$NH!?t$O(B, @var{f} $B$K3d;;%"%k%4%j%:%`$rE,MQ$9$k(B.
1197: $BJQ?t=89g$O(B @var{v} $B$G;XDj$9$k(B.
1198: @var{w} $B$O=g=x$r;XDj$9$k$?$a$N(B $B%&%(%$%H%Y%/%H%k$G$"$j(B,
1199: $B>JN,$7$F$b$h$$(B.
1200: @code{sm1.reduction_noH} $B$O(B, Weyl algebra $BMQ(B.
1201: @item $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
1202: [r,c0,[c1,...,cm],g] $B$3$3$G(B @var{g}=[g1, ..., gm] $B$G$"$j(B,
1203: c0 f + c1 g1 + ... + cm gm = r
1204: $B$,$J$j$?$D(B.
1205: r/c0 $B$,(B normal form $B$G$"$k(B.
1206: @item $B$3$NH!?t$O(B, $BDc<!9`$K$"$i$o$l$k(B reducible $B$J9`$b4JC12=$9$k(B.
1207: @item $BH!?t(B
1208: sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_d(P,F,G)
1209: $B$O(B, $BJ,;6B?9`<0MQ$G$"$k(B.
1210: @end itemize
1211: */
1212: /*&C
1213: @example
1214: [259] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y]]);
1215: [x^2+y^2-4,1,[0,0],[y^4-4*y^2+1,x+y^3-4*y]]
1216: [260] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y],[[x,1]]]);
1217: [0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]]
1218: @end example
1219: */
1220: /*&en
1221: @table @t
1222: @item Reference
1223: @code{sm1.start}, @code{d_true_nf}
1224: @end table
1225: */
1226: /*&ja
1227: @table @t
1228: @item $B;2>H(B
1229: @code{sm1.start}, @code{d_true_nf}
1230: @end table
1231: */
1232:
1233:
1234: /*&en
1235: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
1236: @subsection @code{sm1.xml_tree_to_prefix_string}
1237: @findex sm1.xml_tree_to_prefix_string
1238: @table @t
1239: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1240: :: Translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
1241: @end table
1242:
1243: @table @var
1244: @item return
1245: String
1246: @item p
1247: Number
1248: @item s
1249: String
1250: @end table
1251:
1252: @itemize @bullet
1253: @item It translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
1254: @item This function should be moved to om_* in a future.
1255: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} returns CMO_TREE.
1256: asir has not yet understood this CMO.
1257: @item @code{java} execution environment is required.
1258: (For example, @code{/usr/local/jdk1.1.8/bin} should be in the
1259: command search path.)
1260: @end itemize
1261: */
1262: /*&ja
1263: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
1264: @subsection @code{sm1.xml_tree_to_prefix_string}
1265: @findex sm1.xml_tree_to_prefix_string
1266: @table @t
1267: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1268: :: XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
1269: @end table
1270:
1271: @table @var
1272: @item return
1273: String
1274: @item p
1275: Number
1276: @item s
1277: String
1278: @end table
1279:
1280: @itemize @bullet
1281: @item XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
1282: @item $B$3$NH!?t$O(B om_* $B$K>-Mh0\$9$Y$-$G$"$k(B.
1283: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} $B$O(B CMO_TREE
1284: $B$rLa$9(B. @code{asir} $B$O$3$N(B CMO $B$r$^$@%5%]!<%H$7$F$$$J$$(B.
1285: @item @code{java} $B$N<B9T4D6-$,I,MW(B.
1286: ($B$?$H$($P(B, /usr/local/jdk1.1.8/bin $B$r%3%^%s%I%5!<%A%Q%9$KF~$l$k$J$I(B.)
1287: @end itemize
1288: */
1289: /*&C
1290: @example
1291: [263] load("om");
1292: 1
1293: [270] F=om_xml(x^4-1);
1294: control: wait OX
1295: Trying to connect to the server... Done.
1296: <OMOBJ><OMA><OMS name="plus" cd="basic"/><OMA>
1297: <OMS name="times" cd="basic"/><OMA>
1298: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>4</OMI></OMA>
1299: <OMI>1</OMI></OMA><OMA><OMS name="times" cd="basic"/><OMA>
1300: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>0</OMI></OMA>
1301: <OMI>-1</OMI></OMA></OMA></OMOBJ>
1302: [271] sm1.xml_tree_to_prefix_string(F);
1303: basic_plus(basic_times(basic_power(x,4),1),basic_times(basic_power(x,0),-1))
1304: @end example
1305: */
1306: /*&en
1307: @table @t
1308: @item Reference
1309: @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
1310: @end table
1311: */
1312: /*&ja
1313: @table @t
1314: @item $B;2>H(B
1315: @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
1316: @end table
1317: */
1318:
1319:
1320:
1321:
1322: /*&en
1323: @c sort-sm1.syz
1324: @node sm1.syz,,, SM1 Functions
1325: @subsection @code{sm1.syz}
1326: @findex sm1.syz
1327: @findex sm1.syz_d
1328: @table @t
1329: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1330: :: computes the syzygy of @var{f} in the ring of differential
1331: operators with the variable @var{v}.
1332: @end table
1333:
1334: @table @var
1335: @item return
1336: List
1337: @item p
1338: Number
1339: @item f, v, w
1340: List
1341: @end table
1342:
1343: @itemize @bullet
1344: @item
1345: The return values is of the form
1346: [@var{s},[@var{g}, @var{m}, @var{t}]].
1347: Here @var{s} is the syzygy of @var{f} in the ring of differential
1348: operators with the variable @var{v}.
1349: @var{g} is a Groebner basis of @var{f} with the weight vector @var{w},
1350: and @var{m} is a matrix that translates the input matrix @var{f} to the Gr\"obner
1351: basis @var{g}.
1352: @var{t} is the syzygy of the Gr\"obner basis @var{g}.
1353: In summary, @var{g} = @var{m} @var{f} and
1354: @var{s} @var{f} = 0 hold as matrices.
1355: @item
1356: The weight vectors are given by @var{w}, which can be omitted.
1357: If @var{w} is not given,
1358: the graded reverse lexicographic order will be used to compute Grobner basis.
1359: @item
1360: When a non-term order is given, the Grobner basis is computed in
1361: the homogenized Weyl algebra (See Section 1.2 of the book of SST).
1362: The homogenization variable h is automatically added.
1363: @end itemize
1364: */
1365: /*&ja
1366: @c sort-sm1.syz
1367: @node sm1.syz,,, SM1 Functions
1368: @subsection @code{sm1.syz}
1369: @findex sm1.syz
1370: @findex sm1.syz_d
1371: @table @t
1372: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1373: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N(B syzygy $B$r7W;;$9$k(B.
1374: @end table
1375:
1376: @table @var
1377: @item return
1378: $B%j%9%H(B
1379: @item p
1380: $B?t(B
1381: @item f, v, w
1382: $B%j%9%H(B
1383: @end table
1384:
1385: @itemize @bullet
1386: @item
1387: $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
1388: [@var{s},[@var{g}, @var{m}, @var{t}]].
1389: $B$3$3$G(B @var{s} $B$O(B @var{f} $B$N(B @var{v} $B$rJQ?t$H$9$kHyJ,:nMQAG4D$K$*$1$k(B
1390: syzygy $B$G$"$k(B.
1391: @var{g} $B$O(B @var{f} $B$N(B weight vector @var{w} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
1392: @var{m} $B$OF~NO9TNs(B @var{f} $B$r%0%l%V%J4pDl(B
1393: @var{g} $B$XJQ49$9$k9TNs$G$"$k(B.
1394: @var{t} $B$O%0%l%V%J4pDl(B @var{g} $B$N(B syzygy $B$G$"$k(B.
1395: $B$^$H$a$k$H(B, $B<!$NEy<0$,$J$j$?$D(B:
1396: @var{g} = @var{m} @var{f} ,
1397: @var{s} @var{f} = 0.
1398: @item
1399: Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
1400: $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
1401: $B%V%l%V%J4pDl$r7W;;$9$k(B.
1402: @item
1403: Term order $B$G$J$$=g=x$,M?$($i$l$?>l9g$O(B, $BF1<!2=%o%$%kBe?t$G%0%l%V%J4pDl$,7W;;$5$l$k(B (SST $B$NK\$N(B Section 1.2 $B$r8+$h(B).
1404: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
1405: @end itemize
1406: */
1407: /*&C
1408: @example
1409: [293] sm1.syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
1410: [[[y*x*dy*dx-2,-x*dx-y*dy+1]], generators of the syzygy
1411: [[[x*dx+y*dy-1],[y^2*dy^2+2]], grobner basis
1412: [[1,0],[y*dy,-1]], transformation matrix
1413: [[y*x*dy*dx-2,-x*dx-y*dy+1]]]]
1414: @end example
1415: */
1416: /*&C
1417: @example
1418: [294]sm1.syz([[x^2*dx^2+x*dx+y^2*dy^2+y*dy-4,x*y*dx*dy-1],[x,y],[[dx,-1,x,1]]]);
1419: [[[y*x*dy*dx-1,-x^2*dx^2-x*dx-y^2*dy^2-y*dy+4]], generators of the syzygy
1420: [[[x^2*dx^2+h^2*x*dx+y^2*dy^2+h^2*y*dy-4*h^4],[y*x*dy*dx-h^4], GB
1421: [h^4*x*dx+y^3*dy^3+3*h^2*y^2*dy^2-3*h^4*y*dy]],
1422: [[1,0],[0,1],[y*dy,-x*dx]], transformation matrix
1423: [[y*x*dy*dx-h^4,-x^2*dx^2-h^2*x*dx-y^2*dy^2-h^2*y*dy+4*h^4]]]]
1424: @end example
1425: */
1426:
1427:
1428:
1429: /*&en
1430: @node sm1.mul,,, SM1 Functions
1431: @subsection @code{sm1.mul}
1432: @findex sm1.mul
1433: @table @t
1434: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1435: :: ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
1436: @end table
1437:
1438: @table @var
1439: @item return
1440: Polynomial or List
1441: @item p
1442: Number
1443: @item f, g
1444: Polynomial or List
1445: @item v
1446: List
1447: @end table
1448:
1449: @itemize @bullet
1450: @item Ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
1451: @item @code{sm1.mul_h} is for homogenized Weyl algebra.
1452: @item BUG: @code{sm1.mul(p0*dp0,1,[p0])} returns
1453: @code{dp0*p0+1}.
1454: A variable order such that d-variables come after non-d-variables
1455: is necessary for the correct computation.
1456: @end itemize
1457: */
1458:
1459: /*&ja
1460: @node sm1.mul,,, SM1 Functions
1461: @subsection @code{sm1.mul}
1462: @findex sm1.mul
1463: @table @t
1464: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1465: :: sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
1466: $B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
1467: @end table
1468:
1469: @table @var
1470: @item return
1471: $BB?9`<0$^$?$O%j%9%H(B
1472: @item p
1473: $B?t(B
1474: @item f, g
1475: $BB?9`<0$^$?$O%j%9%H(B
1476: @item v
1477: $B%j%9%H(B
1478: @end table
1479:
1480: @itemize @bullet
1481: @item sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
1482: $B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
1483: @item @code{sm1.mul_h} $B$O(B homogenized Weyl $BBe?tMQ(B.
1484: @item BUG: @code{sm1.mul(p0*dp0,1,[p0])} $B$O(B
1485: @code{dp0*p0+1} $B$rLa$9(B.
1486: d$BJQ?t$,8e$m$K$/$k$h$&$JJQ?t=g=x$,$O$$$C$F$$$J$$$H(B, $B$3$N4X?t$O@5$7$$Ez$($rLa$5$J$$(B.
1487: @end itemize
1488: */
1489:
1490: /*&C
1491:
1492: @example
1493: [277] sm1.mul(dx,x,[x]);
1494: x*dx+1
1495: [278] sm1.mul([x,y],[1,2],[x,y]);
1496: x+2*y
1497: [279] sm1.mul([[1,2],[3,4]],[[x,y],[1,2]],[x,y]);
1498: [[x+2,y+4],[3*x+4,3*y+8]]
1499: @end example
1500:
1501: */
1502:
1503:
1504:
1505:
1506: /*&en
1507: @node sm1.distraction,,, SM1 Functions
1508: @subsection @code{sm1.distraction}
1509: @findex sm1.distraction
1510: @table @t
1511: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1512: :: ask the @code{sm1} server to compute the distraction of @var{f}.
1513: @end table
1514:
1515: @table @var
1516: @item return
1517: List
1518: @item p
1519: Number
1520: @item f
1521: Polynomial
1522: @item v,x,d,s
1523: List
1524: @end table
1525:
1526: @itemize @bullet
1527: @item It asks the @code{sm1} server of the descriptor number @var{p}
1528: to compute the distraction of @var{f} in the ring of differential
1529: operators with variables @var{v}.
1530: @item @var{x} is a list of x-variables and @var{d} is that of d-variables
1531: to be distracted. @var{s} is a list of variables to express the distracted @var{f}.
1532: @item Distraction is roughly speaking to replace x*dx by a single variable x.
1533: See Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations at page 68 for details.
1534: @end itemize
1535: */
1536:
1537: /*&ja
1538: @node sm1.distraction,,, SM1 Functions
1539:
1540: @subsection @code{sm1.distraction}
1541: @findex sm1.distraction
1542: @table @t
1543: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1544: :: @code{sm1} $B$K(B @var{f} $B$N(B distraction $B$r7W;;$7$F$b$i$&(B.
1545: @end table
1546:
1547: @table @var
1548: @item return
1549: $B%j%9%H(B
1550: @item p
1551: $B?t(B
1552: @item f
1553: $BB?9`<0(B
1554: @item v,x,d,s
1555: $B%j%9%H(B
1556: @end table
1557:
1558: @itemize @bullet
1559: @item $B<1JL;R(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B,
1560: @var{f} $B$N(B distraction $B$r(B @var{v} $B>e$NHyJ,:nMQAG4D$G7W;;$7$F$b$i$&(B.
1561: @item @var{x} , @var{d} $B$O(B, $B$=$l$>$l(B, distract $B$9$Y$-(B x $BJQ?t(B, d $BJQ?t$N(B
1562: $B%j%9%H(B. Distraction $B$7$?$i(B, @var{s} $B$rJQ?t$H$7$F7k2L$rI=$9(B.
1563: @item Distraction $B$H$$$&$N$O(B x*dx $B$r(B x $B$GCV$-49$($k$3$H$G$"$k(B.
1564: $B>\$7$/$O(B Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations $B$N(B page 68 $B$r8+$h(B.
1565: @end itemize
1566: */
1567:
1568: /*&C
1569:
1570: @example
1571: [280] sm1.distraction([x*dx,[x],[x],[dx],[x]]);
1572: x
1573: [281] sm1.distraction([dx^2,[x],[x],[dx],[x]]);
1574: x^2-x
1575: [282] sm1.distraction([x^2,[x],[x],[dx],[x]]);
1576: x^2+3*x+2
1577: [283] fctr(@@);
1578: [[1,1],[x+1,1],[x+2,1]]
1579: [284] sm1.distraction([x*dx*y+x^2*dx^2*dy,[x,y],[x],[dx],[x]]);
1580: (x^2-x)*dy+x*y
1581: @end example
1582: */
1583:
1584: /*&en
1585: @table @t
1586: @item Reference
1587: @code{distraction2(sm1)},
1588: @end table
1589: */
1590:
1591: /*&ja
1592: @table @t
1593: @item $B;2>H(B
1594: @code{distraction2(sm1)},
1595: @end table
1596: */
1597:
1598:
1599:
1600: /*&en
1601: @node sm1.gkz,,, SM1 Functions
1602: @subsection @code{sm1.gkz}
1603: @findex sm1.gkz
1604: @table @t
1605: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1606: :: Returns the GKZ system (A-hypergeometric system) associated to the matrix
1607: @var{A} with the parameter vector @var{B}.
1608: @end table
1609:
1610: @table @var
1611: @item return
1612: List
1613: @item p
1614: Number
1615: @item A, B
1616: List
1617: @end table
1618:
1619: @itemize @bullet
1620: @item Returns the GKZ hypergeometric system
1621: (A-hypergeometric system) associated to the matrix
1622: @end itemize
1623: */
1624:
1625: /*&ja
1626: @node sm1.gkz,,, SM1 Functions
1627: @subsection @code{sm1.gkz}
1628: @findex sm1.gkz
1629: @table @t
1630: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1631: :: $B9TNs(B @var{A} $B$H%Q%i%a!<%?(B @var{B} $B$KIU?o$7$?(B GKZ $B7O(B (A-hypergeometric system) $B$r$b$I$9(B.
1632: @end table
1633:
1634: @table @var
1635: @item return
1636: $B%j%9%H(B
1637: @item p
1638: $B?t(B
1639: @item A, B
1640: $B%j%9%H(B
1641: @end table
1642:
1643: @itemize @bullet
1644: @item $B9TNs(B @var{A} $B$H%Q%i%a!<%?(B @var{B} $B$KIU?o$7$?(B GKZ $B7O(B (A-hypergeometric system) $B$r$b$I$9(B.
1645: @end itemize
1646: */
1647:
1648: /*&C
1649:
1650: @example
1651:
1652: [280] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
1653: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1654: -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1655: [x1,x2,x3,x4]]
1656:
1657: @end example
1658:
1659: */
1660:
1.2 ! takayama 1661: /*&en
! 1662: @node sm1.mgkz,,, SM1 Functions
! 1663: @subsection @code{sm1.mgkz}
! 1664: @findex sm1.mgkz
! 1665: @table @t
! 1666: @item sm1.mgkz([@var{A},@var{W},@var{B}]|proc=@var{p})
! 1667: :: Returns the modified GKZ system (A-hypergeometric system) associated to the matrix
! 1668: @var{A} and the weight @var{w} with the parameter vector @var{B}.
! 1669: @end table
! 1670:
! 1671: @table @var
! 1672: @item return
! 1673: List
! 1674: @item p
! 1675: Number
! 1676: @item A, W, B
! 1677: List
! 1678: @end table
! 1679:
! 1680: @itemize @bullet
! 1681: @item Returns the modified GKZ hypergeometric system
! 1682: (A-hypergeometric system) associated to the matrix
! 1683: @item http://arxiv.org/abs/0707.0043
! 1684: @end itemize
! 1685: */
! 1686:
! 1687: /*&ja
! 1688: @node sm1.mgkz,,, SM1 Functions
! 1689: @subsection @code{sm1.mgkz}
! 1690: @findex sm1.mgkz
! 1691: @table @t
! 1692: @item sm1.mgkz([@var{A},@var{W},@var{B}]|proc=@var{p})
! 1693: :: $B9TNs(B @var{A}, weight @var{W} $B$H%Q%i%a!<%?(B @var{B} $B$KIU?o$7$?(B modified GKZ $B7O(B (A-hypergeometric system) $B$r$b$I$9(B.
! 1694: @end table
! 1695:
! 1696: @table @var
! 1697: @item return
! 1698: $B%j%9%H(B
! 1699: @item p
! 1700: $B?t(B
! 1701: @item A, W, B
! 1702: $B%j%9%H(B
! 1703: @end table
! 1704:
! 1705: @itemize @bullet
! 1706: @item $B9TNs(B @var{A}, weight vector @var{W} $B$H%Q%i%a!<%?(B @var{B} $B$KIU?o$7$?(B modified GKZ $B7O(B (A-hypergeometric system) $B$r$b$I$9(B.
! 1707: @item http://arxiv.org/abs/0707.0043
! 1708: @end itemize
! 1709: */
! 1710:
! 1711: /*&C
! 1712:
! 1713: @example
! 1714:
! 1715: [280] sm1.mgkz([ [[1,2,3]], [1,2,1], [a/2]]);
! 1716: [[6*x3*dx3+4*x2*dx2+2*x1*dx1-a,-x4*dx4+x3*dx3+2*x2*dx2+x1*dx1,
! 1717: -dx2+dx1^2,-x4^2*dx3+dx1*dx2],[x1,x2,x3,x4]]
! 1718:
! 1719: Modified A-hypergeometric system for
! 1720: A=(1,2,3), w=(1,2,1), beta=(a/2).
! 1721: @end example
! 1722:
! 1723: */
! 1724:
1.1 takayama 1725:
1726:
1727:
1728: /*&en
1729: @node sm1.appell1,,, SM1 Functions
1730: @subsection @code{sm1.appell1}
1731: @findex sm1.appell1
1732: @table @t
1733: @item sm1.appell1(@var{a}|proc=@var{p})
1734: :: Returns the Appell hypergeometric system F_1 or F_D.
1735: @end table
1736:
1737: @table @var
1738: @item return
1739: List
1740: @item p
1741: Number
1742: @item a
1743: List
1744: @end table
1745:
1746: @itemize @bullet
1747: @item Returns the hypergeometric system for the Lauricella function
1748: F_D(a,b1,b2,...,bn,c;x1,...,xn)
1749: where @var{a} =(a,c,b1,...,bn).
1750: When n=2, the Lauricella function is called the Appell function F_1.
1751: The parameters a, c, b1, ..., bn may be rational numbers.
1752: @item It does not call sm1 function appell1. As a concequence,
1753: when parameters are rational or symbolic, this function also works
1754: as well as integral parameters.
1755: @end itemize
1756: */
1757:
1758: /*&ja
1759: @node sm1.appell1,,, SM1 Functions
1760: @subsection @code{sm1.appell1}
1761: @findex sm1.appell1
1762: @table @t
1763: @item sm1.appell1(@var{a}|proc=@var{p})
1764: :: F_1 $B$^$?$O(B F_D $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
1765: @end table
1766:
1767: @table @var
1768: @item return
1769: $B%j%9%H(B
1770: @item p
1771: $B?t(B
1772: @item a
1773: $B%j%9%H(B
1774: @end table
1775:
1776: @itemize @bullet
1777: @item Appell $B$N4X?t(B F_1 $B$*$h$S(B $B$=$N(B n $BJQ?t2=$G$"$k(B Lauricella $B$N4X?t(B
1778: F_D(a,b1,b2,...,bn,c;x1,...,xn)
1779: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
1780: @var{a} =(a,c,b1,...,bn).
1781: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
1782: @item sm1 $B$N4X?t(B appell1 $B$r$h$V$o$1$G$J$$$N$G(B, $B%Q%i%a!<%?$,M-M}?t$dJ8;z<0$N>l9g$b(B
1783: $B@5$7$/F0$/(B.
1784: @end itemize
1785: */
1786:
1787: /*&C
1788:
1789: @example
1790:
1791: [281] sm1.appell1([1,2,3,4]);
1792: [[((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2+(-5*x1+2)*dx1-3,
1793: (-x2^2+x2)*dx2^2+((-x1*x2+x1)*dx1-6*x2+2)*dx2-4*x1*dx1-4,
1794: ((-x2+x1)*dx1+3)*dx2-4*dx1], equations
1795: [x1,x2]] the list of variables
1796:
1797: [282] sm1.gb(@@);
1798: [[((-x2+x1)*dx1+3)*dx2-4*dx1,((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2
1799: +(-5*x1+2)*dx1-3,(-x2^2+x2)*dx2^2+((-x2^2+x1)*dx1-3*x2+2)*dx2
1800: +(-4*x2-4*x1)*dx1-4,
1801: (x2^3+(-x1-1)*x2^2+x1*x2)*dx2^2+((-x1*x2+x1^2)*dx1+6*x2^2
1802: +(-3*x1-2)*x2+2*x1)*dx2-4*x1^2*dx1+4*x2-4*x1],
1803: [x1*dx1*dx2,-x1^2*dx1^2,-x2^2*dx1*dx2,-x1*x2^2*dx2^2]]
1804:
1805: [283] sm1.rank(sm1.appell1([1/2,3,5,-1/3]));
1806: 3
1807:
1808: [285] Mu=2$ Beta = 1/3$
1809: [287] sm1.rank(sm1.appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));
1810: 4
1811:
1812:
1813: @end example
1814:
1815: */
1816:
1817: /*&en
1818: @node sm1.appell4,,, SM1 Functions
1819: @subsection @code{sm1.appell4}
1820: @findex sm1.appell4
1821: @table @t
1822: @item sm1.appell4(@var{a}|proc=@var{p})
1823: :: Returns the Appell hypergeometric system F_4 or F_C.
1824: @end table
1825:
1826: @table @var
1827: @item return
1828: List
1829: @item p
1830: Number
1831: @item a
1832: List
1833: @end table
1834:
1835: @itemize @bullet
1836: @item Returns the hypergeometric system for the Lauricella function
1837: F_4(a,b,c1,c2,...,cn;x1,...,xn)
1838: where @var{a} =(a,b,c1,...,cn).
1839: When n=2, the Lauricella function is called the Appell function F_4.
1840: The parameters a, b, c1, ..., cn may be rational numbers.
1841: @item @item It does not call sm1 function appell4. As a concequence,
1842: when parameters are rational or symbolic, this function also works
1843: as well as integral parameters.
1844: @end itemize
1845: */
1846:
1847: /*&ja
1848: @node sm1.appell4,,, SM1 Functions
1849: @subsection @code{sm1.appell4}
1850: @findex sm1.appell4
1851: @table @t
1852: @item sm1.appell4(@var{a}|proc=@var{p})
1853: :: F_4 $B$^$?$O(B F_C $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
1854: @end table
1855:
1856: @table @var
1857: @item return
1858: $B%j%9%H(B
1859: @item p
1860: $B?t(B
1861: @item a
1862: $B%j%9%H(B
1863: @end table
1864:
1865: @itemize @bullet
1866: @item Appell $B$N4X?t(B F_4 $B$*$h$S(B $B$=$N(B n $BJQ?t2=$G$"$k(B Lauricella $B$N4X?t(B
1867: F_C(a,b,c1,c2,...,cn;x1,...,xn)
1868: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
1869: @var{a} =(a,b,c1,...,cn).
1870: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
1871: @item sm1 $B$N4X?t(B appell1 $B$r$h$V$o$1$G$J$$$N$G(B, $B%Q%i%a!<%?$,M-M}?t$dJ8;z<0$N>l9g$b(B
1872: $B@5$7$/F0$/(B.
1873: @end itemize
1874: */
1875:
1876: /*&C
1877:
1878: @example
1879:
1880: [281] sm1.appell4([1,2,3,4]);
1881: [[-x2^2*dx2^2+(-2*x1*x2*dx1-4*x2)*dx2+(-x1^2+x1)*dx1^2+(-4*x1+3)*dx1-2,
1882: (-x2^2+x2)*dx2^2+(-2*x1*x2*dx1-4*x2+4)*dx2-x1^2*dx1^2-4*x1*dx1-2],
1883: equations
1884: [x1,x2]] the list of variables
1885:
1886: [282] sm1.rank(@@);
1887: 4
1888:
1889: @end example
1890:
1891: */
1892:
1893:
1894:
1895:
1896: /*&en
1897: @node sm1.rank,,, SM1 Functions
1898: @subsection @code{sm1.rank}
1899: @findex sm1.rank
1900: @table @t
1901: @item sm1.rank(@var{a}|proc=@var{p})
1902: :: Returns the holonomic rank of the system of differential equations @var{a}.
1903: @end table
1904:
1905: @table @var
1906: @item return
1907: Number
1908: @item p
1909: Number
1910: @item a
1911: List
1912: @end table
1913:
1914: @itemize @bullet
1915: @item It evaluates the dimension of the space of holomorphic solutions
1916: at a generic point of the system of differential equations @var{a}.
1917: The dimension is called the holonomic rank.
1918: @item @var{a} is a list consisting of a list of differential equations and
1919: a list of variables.
1920: @item @code{sm1.rrank} returns the holonomic rank when @var{a} is regular
1921: holonomic. It is generally faster than @code{sm1.rank}.
1922: @end itemize
1923: */
1924:
1925: /*&ja
1926: @node sm1.rank,,, SM1 Functions
1927: @subsection @code{sm1.rank}
1928: @findex sm1.rank
1929: @table @t
1930: @item sm1.rank(@var{a}|proc=@var{p})
1931: :: $BHyJ,J}Dx<07O(B @var{a} $B$N(B holonomic rank $B$rLa$9(B.
1932: @end table
1933:
1934: @table @var
1935: @item return
1936: $B?t(B
1937: @item p
1938: $B?t(B
1939: @item a
1940: $B%j%9%H(B
1941: @end table
1942:
1943: @itemize @bullet
1944: @item $BHyJ,J}Dx<07O(B @var{a} $B$N(B, generic point $B$G$N@5B'2r$N<!85$r(B
1945: $BLa$9(B. $B$3$N<!85$r(B, holonomic rank $B$H8F$V(B.
1946: @item @var{a} $B$OHyJ,:nMQAG$N%j%9%H$HJQ?t$N%j%9%H$h$j$J$k(B.
1947: @item @var{a} $B$,(B regular holonomic $B$N$H$-$O(B @code{sm1.rrank}
1948: $B$b(B holonomic rank $B$rLa$9(B.
1949: $B$$$C$Q$s$K$3$N4X?t$NJ}$,(B @code{sm1.rank} $B$h$jAa$$(B.
1950: @end itemize
1951: */
1952:
1953: /*&C
1954:
1955: @example
1956:
1957: [284] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
1958: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1959: -dx1*dx4+dx2*dx3, -dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1960: [x1,x2,x3,x4]]
1961: [285] sm1.rrank(@@);
1962: 4
1963:
1964: [286] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [1,2]]);
1965: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1966: -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1967: [x1,x2,x3,x4]]
1968: [287] sm1.rrank(@@);
1969: 5
1970:
1971: @end example
1972:
1973: */
1974:
1975:
1976: /*&en
1977: @node sm1.auto_reduce,,, SM1 Functions
1978: @subsection @code{sm1.auto_reduce}
1979: @findex sm1.auto_reduce
1980: @table @t
1981: @item sm1.auto_reduce(@var{s}|proc=@var{p})
1982: :: Set the flag "AutoReduce" to @var{s}.
1983: @end table
1984:
1985: @table @var
1986: @item return
1987: Number
1988: @item p
1989: Number
1990: @item s
1991: Number
1992: @end table
1993:
1994: @itemize @bullet
1995: @item If @var{s} is 1, then all Grobner bases to be computed
1996: will be the reduced Grobner bases.
1997: @item If @var{s} is 0, then Grobner bases are not necessarily the reduced
1998: Grobner bases. This is the default.
1999: @end itemize
2000: */
2001:
2002: /*&ja
2003: @node sm1.auto_reduce,,, SM1 Functions
2004: @subsection @code{sm1.auto_reduce}
2005: @findex sm1.auto_reduce
2006: @table @t
2007: @item sm1.auto_reduce(@var{s}|proc=@var{p})
2008: :: $B%U%i%0(B "AutoReduce" $B$r(B @var{s} $B$K@_Dj(B.
2009: @end table
2010:
2011: @table @var
2012: @item $BLa$jCM(B
2013: $B?t(B
2014: @item p
2015: $B?t(B
2016: @item s
2017: $B?t(B
2018: @end table
2019:
2020: @itemize @bullet
2021: @item @var{s} $B$,(B 1 $B$N$H$-(B, $B0J8e7W;;$5$l$k%0%l%V%J4pDl$O$9$Y$F(B,
2022: reduced $B%0%l%V%J4pDl$H$J$k(B.
2023: @item @var{s} $B$,(B 0 $B$N$H$-(B, $B7W;;$5$l$k%0%l%V%J4pDl$O(B
2024: reduced $B%0%l%V%J4pDl$H$O$+$.$i$J$$(B. $B$3$A$i$,%G%U%)!<%k%H(B.
2025: @end itemize
2026: */
2027:
2028:
2029:
2030: /*&en
2031: @node sm1.slope,,, SM1 Functions
2032: @subsection @code{sm1.slope}
2033: @findex sm1.slope
2034: @table @t
2035: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
2036: :: Returns the slopes of differential equations @var{ii}.
2037: @end table
2038:
2039: @table @var
2040: @item return
2041: List
2042: @item p
2043: Number
2044: @item ii
2045: List (equations)
2046: @item v
2047: List (variables)
2048: @item f_filtration
2049: List (weight vector)
2050: @item v_filtration
2051: List (weight vector)
2052: @end table
2053:
2054: @itemize @bullet
2055: @item @code{sm1.slope} returns the (geometric) slopes
2056: of the system of differential equations @var{ii}
2057: along the hyperplane specified by
2058: the V filtration @var{v_filtration}.
2059: @item @var{v} is a list of variables.
2060: @item The return value is a list of lists.
2061: The first entry of each list is the slope and the second entry
2062: is the weight vector for which the microcharacteristic variety is
2063: not bihomogeneous.
2064: @end itemize
2065:
2066: @noindent
2067: Algorithm:
2068: see "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
2069: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
2070: Note that the signs of the slopes are negative, but the absolute values
2071: of the slopes are returned.
2072:
2073: */
2074:
2075: /*&ja
2076: @node sm1.slope,,, SM1 Functions
2077: @subsection @code{sm1.slope}
2078: @findex sm1.slope
2079: @table @t
2080: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
2081: :: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B slope $B$rLa$9(B.
2082: @end table
2083:
2084: @table @var
2085: @item return
2086: $B?t(B
2087: @item p
2088: $B?t(B
2089: @item ii
2090: $B%j%9%H(B ($BJ}Dx<0(B)
2091: @item v
2092: $B%j%9%H(B ($BJQ?t(B)
2093: @item f_filtration
2094: $B%j%9%H(B (weight vector)
2095: @item v_filtration
2096: $B%j%9%H(B (weight vector)
2097: @end table
2098:
2099: @itemize @bullet
2100: @item @code{sm1.slope} $B$O(B
2101: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B V filtration @var{v_filtration}
2102: $B$G;XDj$9$kD6J?LL$K1h$C$F$N(B (geomeric) slope $B$r7W;;$9$k(B.
2103: @item @var{v} $B$OJQ?t$N%j%9%H(B.
2104: @item $BLa$jCM$O(B, $B%j%9%H$r@.J,$H$9$k%j%9%H$G$"$k(B.
2105: $B@.J,%j%9%H$NBh(B 1 $BMWAG$,(B slope, $BBh(B 2 $BMWAG$O(B, $B$=$N(B weight vector $B$KBP1~$9$k(B
2106: microcharacteristic variety $B$,(B bihomogeneous $B$G$J$$(B.
2107: @end itemize
2108:
2109: @noindent
2110: Algorithm:
2111: "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
2112: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
2113: $B$r$_$h(B.
2114: Slope $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B, $B$3$N%W%m%0%i%`$O(B,
2115: Slope $B$N@dBPCM$rLa$9(B.
2116: */
2117:
2118: /*&C
2119:
2120: @example
2121:
2122: [284] A= sm1.gkz([ [[1,2,3]], [-3] ]);
2123:
2124:
2125: [285] sm1.slope(A[0],A[1],[0,0,0,1,1,1],[0,0,-1,0,0,1]);
2126:
2127: [286] A2 = sm1.gkz([ [[1,1,1,0],[2,-3,1,-3]], [1,0]]);
2128: (* This is an interesting example given by Laura Matusevich,
2129: June 9, 2001 *)
2130:
2131: [287] sm1.slope(A2[0],A2[1],[0,0,0,0,1,1,1,1],[0,0,0,-1,0,0,0,1]);
2132:
2133:
2134: @end example
2135:
2136: */
2137: /*&en
2138: @table @t
2139: @item Reference
2140: @code{sm.gb}
2141: @end table
2142: */
2143: /*&ja
2144: @table @t
2145: @item $B;2>H(B
2146: @code{sm.gb}
2147: @end table
2148: */
2149:
2150:
2151: /*&en
2152: @include sm1-auto.en
2153: */
2154:
2155: /*&ja
2156: @include sm1-auto.ja
2157: */
2158:
2159:
2160: end$
2161:
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