version 1.4, 2012/06/11 05:23:52 |
version 1.8, 2019/09/13 05:21:33 |
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/*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v 1.3 2010/02/06 00:50:32 takayama Exp $ */ |
/*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v 1.7 2019/09/09 23:39:52 takayama Exp $ */ |
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/*&C |
/*&C |
@c DO NOT EDIT THIS FILE |
@c DO NOT EDIT THIS FILE |
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@findex sm1.gb |
@findex sm1.gb |
@findex sm1.gb_d |
@findex sm1.gb_d |
@table @t |
@table @t |
@item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r}) |
@item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r},needBack=@var{n},ring_var=@{r}) |
:: computes the Grobner basis of @var{f} in the ring of differential |
:: computes the Grobner basis of @var{f} in the ring of differential |
operators with the variable @var{v}. |
operators with the variable @var{v}. |
@item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p}) |
@item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p}) |
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Each polynomial is expressed as a string temporally for now. |
Each polynomial is expressed as a string temporally for now. |
When the optional variable @var{r} is set to one, |
When the optional variable @var{r} is set to one, |
the polynomials are dehomogenized (,i.e., h is set to 1). |
the polynomials are dehomogenized (,i.e., h is set to 1). |
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@item If you want to have a reduced basis or compute the initial form ideal exactly, |
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execute sm1.auto_reduce(1) before executing this function. |
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@item When the needBack option @var{n} is 1, it returns the answer is a different format as [groebner basis,initial, gb,1,all,[groebner basis, backward transformation]] |
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@item The default value of ring_var is ring_var_for_asir. The server ox_sm1 saves the ring structure used to the global this ring_var. See also reduction. |
@end itemize |
@end itemize |
*/ |
*/ |
/*&ja |
/*&ja |
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@findex sm1.gb |
@findex sm1.gb |
@findex sm1.gb_d |
@findex sm1.gb_d |
@table @t |
@table @t |
@item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r}) |
@item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r},needBack=@var{n},ring_var=@var{r}) |
:: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. |
:: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. |
@item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p}) |
@item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p}) |
:: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. $B7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B. |
:: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. $B7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B. |
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$B$$$^$N$H$3$m$3$NB?9`<0$O(B, $BJ8;zNs$GI=8=$5$l$k(B. |
$B$$$^$N$H$3$m$3$NB?9`<0$O(B, $BJ8;zNs$GI=8=$5$l$k(B. |
$B%*%W%7%g%J%kJQ?t(B @var{r} $B$,%;%C%H$5$l$F$$$k$H$-$O(B, |
$B%*%W%7%g%J%kJQ?t(B @var{r} $B$,%;%C%H$5$l$F$$$k$H$-$O(B, |
$BLa$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). |
$BLa$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). |
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@item Reduced $B%0%l%V%J!<4pDl$^$?$O(B in_w $B$r7W;;$7$?$$$H$-$O(B, $B$3$N4X?t$N<B9T$NA0$K(B |
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sm1.auto_reduce(1) $B$r<B9T$7$F$*$/$3$H(B. |
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@item needBack $B%*%W%7%g%s$,(B 1 $B$N;~$O(B, $BB>$N>l9g$H$O0[$J$k7A<0(B |
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[groebner basis, initial, gb,1,all, [groebner basis, backward transformation]] |
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$B$GEz$($rLa$9(B. (sm1 $B$N(B getAttribute $B$r;2>H(B) |
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@item ring_var $B%*%W%7%g%s$N4{DjCM$O(B ring_var_for_asir $B$G$"$k(B. sm1 $B$O$3$NBg0hJQ?tL>$G7W;;$KMQ$$$?(B ring $B9=B$BN$rJ]B8$9$k(B. reduction $B$r;2>H(B. |
@end itemize |
@end itemize |
*/ |
*/ |
/*&C |
/*&C |
Line 706 $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$ |
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Line 716 $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$ |
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,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]] |
,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]] |
@end example |
@end example |
*/ |
*/ |
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/*&C |
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@example |
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[1834] sm1.gb([[dx^2-x,dx],[x]] | needBack=1); |
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[[dx,dx^2-x,1],[dx,dx^2,1],gb,1,all,[[dx,dx^2-x,1],[[0,1],[1,0],[-dx,dx^2-x]]]] |
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@end example |
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*/ |
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/*&en |
/*&en |
@table @t |
@table @t |
@item Reference |
@item Reference |
@code{sm1.reduction}, @code{sm1.rat_to_p} |
@code{sm1.auto_reduce}, @code{sm1.reduction}, @code{sm1.rat_to_p} |
@end table |
@end table |
*/ |
*/ |
/*&ja |
/*&ja |
@table @t |
@table @t |
@item $B;2>H(B |
@item $B;2>H(B |
@code{sm1.reduction}, @code{sm1.rat_to_p} |
@code{sm1.auto_reduce}, @code{sm1.reduction}, @code{sm1.rat_to_p} |
@end table |
@end table |
*/ |
*/ |
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Line 1147 the inputs @var{f} and @var{g} are left ideals of D. |
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Line 1163 the inputs @var{f} and @var{g} are left ideals of D. |
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@findex sm1.reduction |
@findex sm1.reduction |
@table @t |
@table @t |
@item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p}) |
@item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p}) |
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@item sm1.reduction([@var{f},@var{g},@var{v}]|proc=@var{p}) |
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@item sm1.reduction([@var{f},@var{g}]|proc=@var{p},ring_var=@var{r}) |
:: |
:: |
@end table |
@end table |
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Line 1176 in lower order terms. |
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Line 1194 in lower order terms. |
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@item The functions |
@item The functions |
sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G) |
sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G) |
are for distributed polynomials. |
are for distributed polynomials. |
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@item When the arguments are two, the function mod_reduction is called. It uses the ring |
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structure saved in the global variable ring_var in the ox_sm1 server. |
@end itemize |
@end itemize |
*/ |
*/ |
/*&ja |
/*&ja |
Line 1184 are for distributed polynomials. |
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Line 1204 are for distributed polynomials. |
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@findex sm1.reduction |
@findex sm1.reduction |
@table @t |
@table @t |
@item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p}) |
@item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p}) |
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@item sm1.reduction([@var{f},@var{g},@var{v}]|proc=@var{p}) |
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@item sm1.reduction([@var{f},@var{g}]|proc=@var{p},ring_var=@var{r}) |
:: |
:: |
@end table |
@end table |
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Line 1215 r/c0 $B$,(B normal form $B$G$"$k(B. |
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Line 1237 r/c0 $B$,(B normal form $B$G$"$k(B. |
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@item $BH!?t(B |
@item $BH!?t(B |
sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_d(P,F,G) |
sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_d(P,F,G) |
$B$O(B, $BJ,;6B?9`<0MQ$G$"$k(B. |
$B$O(B, $BJ,;6B?9`<0MQ$G$"$k(B. |
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@item $B0z?t$,#2$D$N;~$O(B mod_reduction $B4X?t$,8F$P$l$k(B. $B$3$l$O(B ox_sm1 $B$NBg0hJQ?t(B |
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ring_var $BJQ?t$KJ]B8$5$l$?(B ring $B$K$*$$$F4JLs$r9T$&(B. auto_reduce(1) $B$,<+F0$G%;%C%H$5$l$k(B. |
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gb $B$r;2>H(B. |
@end itemize |
@end itemize |
*/ |
*/ |
/*&C |
/*&C |
Line 1223 sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_ |
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Line 1248 sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_ |
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[x^2+y^2-4,1,[0,0],[y^4-4*y^2+1,x+y^3-4*y]] |
[x^2+y^2-4,1,[0,0],[y^4-4*y^2+1,x+y^3-4*y]] |
[260] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y],[[x,1]]]); |
[260] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y],[[x,1]]]); |
[0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]] |
[0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]] |
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[1837] XM_debug=0$ S=sm1.syz([ [x^2-1,x^3-1,x^4-1],[x]])$ |
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[1838] sm1.auto_reduce(1); |
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1 |
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[1839] S0=sm1.gb([S[0],[x]]); |
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[[[-x^2-x-1,x+1,0],[x^2+1,0,-1]],[[0,x,0],[0,0,-1]]] |
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[1840] sm1.reduction([ [-x^4-x^3-x^2-x,x^3+x^2+x+1,-1], S0[0]]); |
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[[0,0,0],-1,[[x^2+1,0,0],[1,0,0]],[[-x^2-x-1,x+1,0],[x^2+1,0,-1]]] |
@end example |
@end example |
*/ |
*/ |
/*&en |
/*&en |