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Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.14 and 1.15

-version 1.14, 2004/09/14 01:32:34
+version 1.15, 2004/09/14 02:28:20
 Line 1
 Line 1
 Line 1
- @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.13 2004/09/13 09:23:30 noro Exp $
+ @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.14 2004/09/14 01:32:34 noro Exp $
  \BJP
  @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
  @chapter $B%0%l%V%J4pDl$N7W;;(B
 Line 1464  Computation of the global b function is implemented as
 Line 1464  Computation of the global b function is implemented as
 Line 1464  Computation of the global b function is implemented as
  * tolexm minipolym::
  * dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main::
  * dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main::
+ * nd_gr nd_gr_trace nd_f4 nd_weyl_gr nd_weyl_gr_trace::
  * dp_gr_flags dp_gr_print::
  * dp_ord::
  * dp_ptod::
-Line 2299  except for lack of the argument for controlling homoge
+Line 2300  except for lack of the argument for controlling homoge
 Line 2299  except for lack of the argument for controlling homoge
 Line 2300  except for lack of the argument for controlling homoge
  @fref{dp_ord},
  @fref{dp_gr_flags dp_gr_print},
  @fref{gr hgr gr_mod},
+ \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
+ \EG @fref{Controlling Groebner basis computations}
+ @end table
+ \JP @node nd_gr nd_gr_trace nd_f4 nd_weyl_gr nd_weyl_gr_trace,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node nd_gr nd_gr_trace nd_f4 nd_weyl_gr nd_weyl_gr_trace,,, Functions for Groebner basis computation
+ @subsection @code{nd_gr}, @code{nd_gr_trace}, @code{nd_f4}, @code{nd_weyl_gr}, @code{nd_weyl_gr_trace}
+ @findex nd_gr
+ @findex nd_gr_trace
+ @findex nd_f4
+ @findex nd_weyl_gr
+ @findex nd_weyl_gr_trace
+ @table @t
+ @item nd_gr(@var{plist},@var{vlist},@var{p},@var{order})
+ @itemx nd_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
+ @itemx nd_f4(@var{plist},@var{vlist},@var{modular},@var{order})
+ @item nd_weyl_gr(@var{plist},@var{vlist},@var{p},@var{order})
+ @itemx nd_weyl_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
+ \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
+ \EG :: Groebner basis computation (built-in functions)
+ @end table
+ @table @var
+ @item return
+ \JP $B%j%9%H(B
+ \EG list
+ @item plist  vlist
+ \JP $B%j%9%H(B
+ \EG list
+ @item order
+ \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+ \EG number, list or matrix
+ @item homo
+ \JP $B%U%i%0(B
+ \EG flag
+ @item modular
+ \JP $B%U%i%0$^$?$OAG?t(B
+ \EG flag or prime
+ @end table
+ \BJP
+ @itemize @bullet
+ @item
+ $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;AH$_9~$_4X?t$N?7<BAu$G$"$k(B.
+ @item @code{nd_gr} $B$O(B, @code{p} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B Buchberger
+ $B%"%k%4%j%:%`$r<B9T$9$k(B. @code{p} $B$,(B 2 $B0J>e$N<+A3?t$N$H$-(B, GF(p) $B>e$N(B
+ Buchberger $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @item @code{nd_gr_trace} $B$OM-M}?tBN>e$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @code{p} $B$,(B 0 $B$^$?$O(B 1 $B$N$H$-(B, $B<+F0E*$KA*$P$l$?AG?t$rMQ$$$F(B, $B@.8y$9$k(B
+ $B$^$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @code{p} $B$,(B 2 $B0J>e$N$H$-(B, trace $B$O(BGF(p) $B>e$G7W;;$5$l$k(B. trace $B%"%k%4%j%:%`(B
+ $B$,<:GT$7$?>l9g(B 0 $B$,JV$5$l$k(B. @code{p} $B$,Ii$N>l9g(B, $B%0%l%V%J4pDl%A%'%C%/$O(B
+ $B9T$o$J$$(B. $B$3$N>l9g(B, @code{p} $B$,(B -1 $B$J$i$P<+F0E*$KA*$P$l$?AG?t$,(B,
+ $B$=$l0J30$O;XDj$5$l$?AG?t$rMQ$$$F%0%l%V%J4pDl8uJd$N7W;;$,9T$o$l$k(B.
+ @item
+ @code{nd_f4} $B$O(B, $BM-8BBN>e$N(B F4 $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @item
+ @code{nd_weyl_gr}, @code{nd_weyl_gr_trace} $B$O(B Weyl $BBe?tMQ$G$"$k(B.
+ @item
+ $B$$$:$l$N4X?t$b(B, $BM-M}4X?tBN>e$N7W;;$OL$BP1~$G$"$k(B.
+ @item
+ $B0lHL$K(B @code{dp_gr_main}, @code{dp_gr_mod_main} $B$h$j9bB.$G$"$k$,(B,
+ $BFC$KM-8BBN>e$N>l9g82Cx$G$"$k(B.
+ @end itemize
+ \E
+ \BEG
+ @itemize @bullet
+ @item
+ These functions are new implementations for computing Groebner bases.
+ @item @code{nd_gr} executes Buchberger algorithm over the rationals
+ if  @code{p} is 0, and that over GF(p) if @code{p} is a prime.
+ @item @code{nd_gr_trace} executes the trace algorithm over the rationals.
+ If @code{p} is 0 or 1, the trace algorithm is executed until it succeeds
+ by using automatically chosen primes.
+ If @code{p} a positive prime,
+ the trace is comuted over GF(p).
+ If the trace algorithm fails 0 is returned.
+ If @code{p} is negative,
+ the Groebner basis check and ideal-membership check are omitted.
+ In this case, an automatically chosen prime if @code{p} is 1,
+ otherwise the specified prime is used to compute a Groebner basis
+ candidate.
+ @item
+ @code{nd_f4} executes F4 algorithm over a finite field.
+ @item
+ @code{nd_weyl_gr}, @code{nd_weyl_gr_trace} are for Weyl algebra computation.
+ @item
+ Each function cannot handle rational function coefficient cases.
+ @item
+ In general these functions are more efficient than
+ @code{dp_gr_main}, @code{dp_gr_mod_main}, especially over finite fields.
+ @end itemize
+ \E
+ @example
+ [38] load("cyclic")$
+ [49] C=cyclic(7)$
+ [50] V=vars(C)$
+ [51] cputime(1)$
+ [52] dp_gr_mod_main(C,V,0,31991,0)$
+.06sec + gc : 0.313sec(26.4sec)
+ [53] nd_gr(C,V,31991,0)$
+ ndv_alloc=1477188
+.737sec + gc : 0.1837sec(5.921sec)
+ [54] dp_f4_mod_main(C,V,31991,0)$
+.51sec + gc : 0.7109sec(4.221sec)
+ [55] nd_f4(C,V,31991,0)$
+.906sec + gc : 0.126sec(2.032sec)
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{dp_ord},
+ @fref{dp_gr_flags dp_gr_print},
  \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
  \EG @fref{Controlling Groebner basis computations}
  @end table

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