===================================================================
RCS file: /home/cvs/OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v
retrieving revision 1.5
retrieving revision 1.10
diff -u -p -r1.5 -r1.10
--- OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw	2012/11/28 05:07:31	1.5
+++ OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw	2020/10/27 02:44:16	1.10
@@ -1,4 +1,4 @@
-/*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v 1.4 2012/06/11 05:23:52 takayama Exp $ */
+/*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v 1.9 2020/02/25 02:21:53 takayama Exp $ */
 
 /*&C
 @c DO NOT EDIT THIS FILE   
@@ -511,7 +511,7 @@ x*dx+1
 @findex sm1.gb
 @findex sm1.gb_d
 @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})
 ::  computes the Grobner basis of @var{f} in the ring of differential
 operators with the variable @var{v}.
 @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
@@ -561,6 +561,8 @@ List
    the polynomials are dehomogenized (,i.e., h is set to 1).
 @item If you want to have a reduced basis or compute the initial form ideal exactly, 
 execute sm1.auto_reduce(1) before executing this function.
+@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]]
+@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
 */
 /*&ja
@@ -570,7 +572,7 @@ execute sm1.auto_reduce(1) before executing this funct
 @findex sm1.gb
 @findex sm1.gb_d
 @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} 上の微分作用素環において @var{f} のグレブナ基底を計算する.
 @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
 ::  @var{v} 上の微分作用素環において @var{f} のグレブナ基底を計算する. 結果を分散多項式のリストで戻す.
@@ -613,6 +615,10 @@ execute sm1.auto_reduce(1) before executing this funct
     戻り多項式は dehomogenize される (すなわち h に 1 が代入される).
 @item Reduced グレブナー基底または in_w を計算したいときは, この関数の実行の前に
 sm1.auto_reduce(1) を実行しておくこと.
+@item needBack オプションが 1 の時は, 他の場合とは異なる形式
+[groebner basis, initial, gb,1,all, [groebner basis, backward transformation]]
+で答えを戻す. (sm1 の getAttribute を参照)
+@item ring_var オプションの既定値は ring_var_for_asir である. sm1 はこの大域変数名で計算に用いた ring 構造体を保存する.  reduction を参照.
 @end itemize
 */
 /*&C
@@ -710,6 +716,12 @@ $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
 ,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]]
 @end example
 */
+/*&C
+@example
+[1834] sm1.gb([[dx^2-x,dx],[x]] | needBack=1);
+[[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]]]]
+@end example
+*/
 
 /*&en
 @table @t
@@ -1151,6 +1163,9 @@ the inputs @var{f} and @var{g} are left ideals of D.
 @findex sm1.reduction
 @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}]|proc=@var{p})
+@item sm1.reduction([@var{f},@var{g}]|proc=@var{p},ring_var=@var{r})
+@item sm1.reduction_verbose([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
 ::  
 @end table
 
@@ -1180,6 +1195,10 @@ in lower order terms.
 @item  The functions 
 sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G)
 are for distributed polynomials.
+@item When the arguments are two, the function mod_reduction is called. It uses the ring
+structure saved in the global variable ring_var in the ox_sm1 server.
+@item The return value of reduction_verbose is of the form
+[r,c0,[c1,...,cm],[g1,...gm],init,order] where init is the initial of r and order is the order structure used.
 @end itemize
 */
 /*&ja
@@ -1188,6 +1207,9 @@ are for distributed polynomials.
 @findex sm1.reduction
 @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}]|proc=@var{p})
+@item sm1.reduction([@var{f},@var{g}]|proc=@var{p},ring_var=@var{r})
+@item sm1.reduction_verbose([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
 ::  
 @end table
 
@@ -1219,6 +1241,11 @@ r/c0 が normal form である.
 @item  函数
 sm1.reduction_d(P,F,G) および sm1.reduction_noH_d(P,F,G)
 は, 分散多項式用である.
+@item 引数が２つの時は mod_reduction 関数が呼ばれる. これは ox_sm1 の大域変数 
+ring_var 変数に保存された ring において簡約を行う. auto_reduce(1) が自動でセットされる.
+gb を参照.
+@item reduction_verbose の戻り値は
+[r,c0,[c1,...,cm],[g1,...gm],init,order] ここで init は 順序 order による r の initial.
 @end itemize
 */
 /*&C
@@ -1227,8 +1254,37 @@ sm1.reduction_d(P,F,G) および sm1.reduction_noH_
 [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]]]);
 [0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]]
+
+[1837] XM_debug=0$ S=sm1.syz([ [x^2-1,x^3-1,x^4-1],[x]])$                        
+[1838] sm1.auto_reduce(1);
+1
+[1839] S0=sm1.gb([S[0],[x]]);
+[[[-x^2-x-1,x+1,0],[x^2+1,0,-1]],[[0,x,0],[0,0,-1]]]
+[1840] sm1.reduction([ [-x^4-x^3-x^2-x,x^3+x^2+x+1,-1], S0[0]]);
+[[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
 */
+
+/*&C
+@example
+XM_debug=0$
+sm1.auto_reduce(1)$
+F=[x*y-1,x^2+y^2-4]$
+Weight_vec=[[x,10,y,1]]$  
+printf("\n\nsyz----\n")$
+S=sm1.syz([F,[x,y],Weight_vec]);  // When Weight_vec is given, the TOP order is used.
+// If the Weight_vec is not given, the POT order (e.g., (1,0,0)<(0,1,0)<(0,0,1)) with grlex is used.
+Sgb=sm1.gb([S[0],[x,y],Weight_vec]);
+R0=[x+y,x^2*y+x];
+P=R0[0]*F[0]+R0[1]*F[1];
+R=sm1.reduction_verbose([R0,Sgb[0],[x,y],Weight_vec]);
+
+printf("\nMinimal representation=%a\n",R[0])$
+printf("The initial of minimal rep=%a\n",R[4])$
+printf("Order=%a\n",R[5][1][1])$
+@end example
+*/
+
 /*&en
 @table @t
 @item Reference