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Diff for /OpenXM/src/asir-doc/parts/builtin/poly.texi between version 1.4 and 1.6

version 1.4, 2003/04/19 15:44:59 version 1.6, 2003/11/27 15:56:08
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 @comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/poly.texi,v 1.3 2002/09/03 01:50:59 noro Exp $  @comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/poly.texi,v 1.5 2003/04/20 08:01:29 noro Exp $
 \BJP  \BJP
 @node $BB?9`<0$*$h$SM-M}<0$N1i;;(B,,, $BAH$_9~$_H!?t(B  @node $BB?9`<0$*$h$SM-M}<0$N1i;;(B,,, $BAH$_9~$_H!?t(B
 @section $BB?9`<0(B, $BM-M}<0$N1i;;(B  @section $BB?9`<0(B, $BM-M}<0$N1i;;(B
Line 451  objects which are created under different variable ord
Line 451  objects which are created under different variable ord
   
 @example  @example
 [0] ord();  [0] ord();
 [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,  [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,
 _q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,exp(_x),(_x)^(_y),  _w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,
 log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),(-_x^2+1)^(-1/2),cosh(_x),sinh(_x),  exp(_x),(_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),
 tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]  (-_x^2+1)^(-1/2),cosh(_x),sinh(_x),tanh(_x),
   (_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]
 [1] ord([dx,dy,dz,a,b,c]);  [1] ord([dx,dy,dz,a,b,c]);
 [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,  [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,
 _w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,exp(_x),  _z,_u,_v,_w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,
 (_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),(-_x^2+1)^(-1/2),  _o,exp(_x),(_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),
 cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]  (-_x^2+1)^(-1/2),cosh(_x),sinh(_x),tanh(_x),
   (_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]
 @end example  @end example
   
 \JP @node sdiv sdivm srem sremm sqr sqrm,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B  \JP @node sdiv sdivm srem sremm sqr sqrm,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
Line 645  to the polynomials repeatedly yields the multiplicity.
Line 647  to the polynomials repeatedly yields the multiplicity.
   
 @example  @example
 [11] Y=(x+y+z)^5*(x-y-z)^3;  [11] Y=(x+y+z)^5*(x-y-z)^3;
 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  x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6
 +(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  +(-6*y^3-18*z*y^2-18*z^2*y-6*z^3)*x^5
 +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  +(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
 -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  +(2*y^6+12*z*y^5+30*z^2*y^4+40*z^3*y^3+30*z^4*y^2+12*z^5*y+2*z^6)*x^2
 -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  +(-2*y^7-14*z*y^6-42*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^7-28*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
 [12] for(I=0,F=x+y+z,T=Y; T=tdiv(T,F); I++);  [12] for(I=0,F=x+y+z,T=Y; T=tdiv(T,F); I++);
 [13] I;  [13] I;
 5  5
Line 783  if arguments are repeated.)
Line 787  if arguments are repeated.)
 $B$J$k$Y$/J,Jl(B, $BJ,;R$,Bg$-$/$J$i$J$$$h$&$KG[N8$9$k$3$H$b$7$P$7$PI,MW$H$J$k(B.  $B$J$k$Y$/J,Jl(B, $BJ,;R$,Bg$-$/$J$i$J$$$h$&$KG[N8$9$k$3$H$b$7$P$7$PI,MW$H$J$k(B.
 @item  @item
 $BJ,?t$rBeF~$9$k>l9g$bF1MM$G$"$k(B.  $BJ,?t$rBeF~$9$k>l9g$bF1MM$G$"$k(B.
   @item
   @code{subst}$B$N0z?t(B@var{rat}$B$,%j%9%H(B,$BG[Ns(B,$B9TNs(B,$B$"$k$$$OJ,;6I=8=B?9`<0$G(B
   $B$"$C$?>l9g$K$O(B, $B$=$l$>$l$NMWAG$^$?$O78?t$KBP$7$F:F5"E*$K(B@code{subst}$B$r(B
   $B9T$&(B.
 \E  \E
 \BEG  \BEG
 @item  @item
 Substitutes rational expressions for specified kernels in a rational  Substitutes rational expressions for specified kernels in a rational
 expression.  expression.
 @item  @item
 @t{subst}(@var{rat},@var{var1},@var{rat1},@var{var2},@var{rat2},@dots{})  @t{subst}(@var{r},@var{v1},@var{r1},@var{v2},@var{r2},@dots{})
 has the same effect as  has the same effect as
 @t{subst}(@t{subst}(@var{rat},@var{var1},@var{rat1}),@var{var2},@var{rat2},@dots{}).  @t{subst}(@t{subst}(@var{r},@var{v1},@var{r1}),@var{v2},@var{r2},@dots{}).
 @item  @item
 Note that repeated substitution is done from left to right successively.  Note that repeated substitution is done from left to right successively.
 You may get different result by changing the specification order.  You may get different result by changing the specification order.
Line 1085  has a degree that is a multiple of @var{d}.
Line 1093  has a degree that is a multiple of @var{d}.
 t^9-15*t^6-87*t^3-125  t^9-15*t^6-87*t^3-125
 0msec  0msec
 [11] N=res(t,subst(A,t,x-2*t),A);  [11] N=res(t,subst(A,t,x-2*t),A);
 -x^81+1215*x^78-567405*x^75+139519665*x^72-19360343142*x^69+1720634125410*x^66  -x^81+1215*x^78-567405*x^75+139519665*x^72-19360343142*x^69
 -88249977024390*x^63-4856095669551930*x^60+1999385245240571421*x^57  +1720634125410*x^66-88249977024390*x^63-4856095669551930*x^60
 -15579689952590251515*x^54+15956967531741971462865*x^51  +1999385245240571421*x^57-15579689952590251515*x^54
   +15956967531741971462865*x^51
 ...  ...
 +140395588720353973535526123612661444550659875*x^6  +140395588720353973535526123612661444550659875*x^6
 +10122324287343155430042768923500799484375*x^3  +10122324287343155430042768923500799484375*x^3
Line 1105  t^9-15*t^6-87*t^3-125
Line 1114  t^9-15*t^6-87*t^3-125
 [[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],  [[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],
 [x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3  [x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3
 +296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],  +296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],
 [x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3+31524548679,1],  [x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3
   +31524548679,1],
 [x^18+10773*x^12+2784051*x^6+307546875,1]]  [x^18+10773*x^12+2784051*x^6+307546875,1]]
 167.050sec + gc : 1.890sec  167.050sec + gc : 1.890sec
 [14] ufctrhint(N,9);  [14] ufctrhint(N,9);
 [[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],  [[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],
 [x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3  [x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3
 +296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],  +296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],
 [x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3+31524548679,1],  [x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3
   +31524548679,1],
 [x^18+10773*x^12+2784051*x^6+307546875,1]]  [x^18+10773*x^12+2784051*x^6+307546875,1]]
 119.340sec + gc : 1.300sec  119.340sec + gc : 1.300sec
 @end example  @end example
Line 1130  t^9-15*t^6-87*t^3-125
Line 1141  t^9-15*t^6-87*t^3-125
   
 @table @t  @table @t
 @item modfctr(@var{poly},@var{mod})  @item modfctr(@var{poly},@var{mod})
 \JP :: $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B  \JP :: $BM-8BBN>e$G$NB?9`<0$N0x?tJ,2r(B
 \EG :: Univariate factorizer over small finite fields  \EG :: Factorizer over small finite fields
 @end table  @end table
   
 @table @var  @table @var
Line 1139  t^9-15*t^6-87*t^3-125
Line 1150  t^9-15*t^6-87*t^3-125
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item poly  @item poly
 \JP $B@0?t78?t$N(B 1 $BJQ?tB?9`<0(B  \JP $B@0?t78?t$NB?9`<0(B
 \EG univariate polynomial with integer coefficients  \EG Polynomial with integer coefficients
 @item mod  @item mod
 \JP $B<+A3?t(B  \JP $B<+A3?t(B
 \EG non-negative integer  \EG non-negative integer
Line 1149  t^9-15*t^6-87*t^3-125
Line 1160  t^9-15*t^6-87*t^3-125
 @itemize @bullet  @itemize @bullet
 \BJP  \BJP
 @item  @item
 2^31 $BL$K~$N<+A3?t(B @var{mod} $B$rI8?t$H$9$kAGBN>e$G0lJQ?tB?9`<0(B  2^29 $BL$K~$N<+A3?t(B @var{mod} $B$rI8?t$H$9$kAGBN>e$GB?9`<0(B
 @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B.  @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B.
 @item  @item
 $B7k2L$O(B [[@b{$B?t78?t(B},1],[@b{$B0x;R(B},@b{$B=EJ#EY(B}],...] $B$J$k%j%9%H(B.  $B7k2L$O(B [[@b{$B?t78?t(B},1],[@b{$B0x;R(B},@b{$B=EJ#EY(B}],...] $B$J$k%j%9%H(B.
Line 1161  t^9-15*t^6-87*t^3-125
Line 1172  t^9-15*t^6-87*t^3-125
 \E  \E
 \BEG  \BEG
 @item  @item
 This function factorizes a univarate polynomial @var{poly} over  This function factorizes a polynomial @var{poly} over
 the finite prime field of characteristic @var{mod}, where  the finite prime field of characteristic @var{mod}, where
 @var{mod} must be smaller than 2^31.  @var{mod} must be smaller than 2^29.
 @item  @item
 The result is represented by a list, whose elements are a pair  The result is represented by a list, whose elements are a pair
 represented as  represented as
Line 1183  To factorize polynomials over large finite fields, use
Line 1194  To factorize polynomials over large finite fields, use
 [[1,1],[x+1513477736,1],[x+2055628767,1],[x+91854880,1],  [[1,1],[x+1513477736,1],[x+2055628767,1],[x+91854880,1],
 [x+634005911,1],[x+1513477735,1],[x+634005912,1],  [x+634005911,1],[x+1513477735,1],[x+634005912,1],
 [x^4+1759639395*x^2+2045307031,1]]  [x^4+1759639395*x^2+2045307031,1]]
   [1] modfctr(2*x^6+(y^2+z*y)*x^4+2*z*y^3*x^2+(2*z^2*y^2+z^3*y)*x+z^4,3);
   [[2,1],[2*x^3+z*y*x+z^2,1],[2*x^3+y^2*x+2*z^2,1]]
 @end example  @end example
   
 @table @t  @table @t
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