File: [local] / OpenXM_contrib2 / asir2000 / lib / bfct (download)
Revision 1.25, Mon Apr 28 03:02:52 2003 UTC (21 years, 5 months ago) by noro
Branch: MAIN
CVS Tags: RELEASE_1_2_2 Changes since 1.24: +5 -42
lines
Improved ann0(), which now calls bfcunction() for computation of b(s).
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/*
* Copyright (c) 1994-2000 FUJITSU LABORATORIES LIMITED
* All rights reserved.
*
* FUJITSU LABORATORIES LIMITED ("FLL") hereby grants you a limited,
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* redistribute, solely for non-commercial and non-profit purposes, the
* computer program, "Risa/Asir" ("SOFTWARE"), subject to the terms and
* conditions of this Agreement. For the avoidance of doubt, you acquire
* only a limited right to use the SOFTWARE hereunder, and FLL or any
* third party developer retains all rights, including but not limited to
* copyrights, in and to the SOFTWARE.
*
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* purposes. You may use the SOFTWARE only for non-commercial and
* non-profit purposes only, such as academic, research and internal
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* international copyright treaties. If you make copies of the SOFTWARE,
* with or without modification, as permitted hereunder, you shall affix
* to all such copies of the SOFTWARE the above copyright notice.
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* for such modification or the source code of the modified part of the
* SOFTWARE.
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* MAKES ABSOLUTELY NO WARRANTIES, EXPRESSED, IMPLIED OR STATUTORY, AND
* EXPRESSLY DISCLAIMS ANY IMPLIED WARRANTY OF MERCHANTABILITY, FITNESS
* FOR A PARTICULAR PURPOSE OR NONINFRINGEMENT OF THIRD PARTIES'
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* $OpenXM: OpenXM_contrib2/asir2000/lib/bfct,v 1.25 2003/04/28 03:02:52 noro Exp $
*/
/* requires 'primdec' */
#define TMP_S ssssssss
#define TMP_DS dssssssss
#define TMP_T dtttttttt
#define TMP_DT tttttttt
#define TMP_Y1 yyyyyyyy1
#define TMP_DY1 dyyyyyyyy1
#define TMP_Y2 yyyyyyyy2
#define TMP_DY2 dyyyyyyyy2
extern LIBRARY_GR_LOADED$
extern LIBRARY_PRIMDEC_LOADED$
if(!LIBRARY_GR_LOADED) load("gr"); else ; LIBRARY_GR_LOADED = 1$
if(!LIBRARY_PRIMDEC_LOADED) load("primdec"); else ; LIBRARY_PRIMDEC_LOADED = 1$
/* toplevel */
def bfunction(F)
{
V = vars(F);
N = length(V);
D = newvect(N);
for ( I = 0; I < N; I++ )
D[I] = [deg(F,V[I]),V[I]];
qsort(D,compare_first);
for ( V = [], I = 0; I < N; I++ )
V = cons(D[I][1],V);
return bfct_via_gbfct_weight(F,V);
}
/* annihilating ideal of F^s */
def ann(F)
{
if ( member(s,vars(F)) )
error("ann : the variable 's' is reserved.");
V = vars(F);
N = length(V);
D = newvect(N);
for ( I = 0; I < N; I++ )
D[I] = [deg(F,V[I]),V[I]];
qsort(D,compare_first);
for ( V = [], I = N-1; I >= 0; I-- )
V = cons(D[I][1],V);
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
W = append([TMP_Y1,TMP_Y2,TMP_T],V);
DW = append([TMP_DY1,TMP_DY2,TMP_DT],DV);
B = [1-TMP_Y1*TMP_Y2,TMP_T-TMP_Y1*F];
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+TMP_Y1*diff(F,V[I])*TMP_DT,B);
}
/* homogenized (heuristics) */
dp_nelim(2);
G0 = dp_weyl_gr_main(B,append(W,DW),1,0,6);
G1 = [];
for ( T = G0; T != []; T = cdr(T) ) {
E = car(T); VL = vars(E);
if ( !member(TMP_Y1,VL) && !member(TMP_Y2,VL) )
G1 = cons(E,G1);
}
G2 = map(psi,G1,TMP_T,TMP_DT);
G3 = map(subst,G2,TMP_T,-1-s);
return G3;
}
/*
* compute J_f|s=r, where r = the minimal integral root of global b_f(s)
* ann0(F) returns [MinRoot,Ideal]
*/
def ann0(F)
{
F = subst(F,s,TMP_S);
Ann = ann(F);
Bf = bfunction(F);
FList = cdr(fctr(Bf));
for ( T = FList, Min = 0; T != []; T = cdr(T) ) {
LF = car(car(T));
Root = -coef(LF,0)/coef(LF,1);
if ( dn(Root) == 1 && Root < Min )
Min = Root;
}
return [Min,map(subst,Ann,s,Min,TMP_S,s,TMP_DS,ds)];
}
def psi(F,T,DT)
{
D = dp_ptod(F,[T,DT]);
Wmax = weight(D);
D1 = dp_rest(D);
for ( ; D1; D1 = dp_rest(D1) )
if ( weight(D1) > Wmax )
Wmax = weight(D1);
for ( D1 = D, Dmax = 0; D1; D1 = dp_rest(D1) )
if ( weight(D1) == Wmax )
Dmax += dp_hm(D1);
if ( Wmax >= 0 )
Dmax = dp_weyl_mul(<<Wmax,0>>,Dmax);
else
Dmax = dp_weyl_mul(<<0,-Wmax>>,Dmax);
Rmax = dp_dtop(Dmax,[T,DT]);
R = b_subst(subst(Rmax,DT,1),T);
return R;
}
def weight(D)
{
V = dp_etov(D);
return V[1]-V[0];
}
def compare_first(A,B)
{
A0 = car(A);
B0 = car(B);
if ( A0 > B0 )
return 1;
else if ( A0 < B0 )
return -1;
else
return 0;
}
/* generic b-function w.r.t. weight vector W */
def generic_bfct(F,V,DV,W)
{
N = length(V);
N2 = N*2;
/* If W is a list, convert it to a vector */
if ( type(W) == 4 )
W = newvect(length(W),W);
dp_weyl_set_weight(W);
/* create a term order M in D<x,d> (DRL) */
M = newmat(N2,N2);
for ( J = 0; J < N2; J++ )
M[0][J] = 1;
for ( I = 1; I < N2; I++ )
M[I][N2-I] = -1;
VDV = append(V,DV);
/* create a non-term order MW in D<x,d> */
MW = newmat(N2+1,N2);
for ( J = 0; J < N; J++ )
MW[0][J] = -W[J];
for ( ; J < N2; J++ )
MW[0][J] = W[J-N];
for ( I = 1; I <= N2; I++ )
for ( J = 0; J < N2; J++ )
MW[I][J] = M[I-1][J];
/* create a homogenized term order MWH in D<x,d,h> */
MWH = newmat(N2+2,N2+1);
for ( J = 0; J <= N2; J++ )
MWH[0][J] = 1;
for ( I = 1; I <= N2+1; I++ )
for ( J = 0; J < N2; J++ )
MWH[I][J] = MW[I-1][J];
/* homogenize F */
VDVH = append(VDV,[h]);
FH = map(dp_dtop,map(dp_homo,map(dp_ptod,F,VDV)),VDVH);
/* compute a groebner basis of FH w.r.t. MWH */
dp_gr_flags(["Top",1,"NoRA",1]);
GH = dp_weyl_gr_main(FH,VDVH,0,1,11);
dp_gr_flags(["Top",0,"NoRA",0]);
/* dehomigenize GH */
G = map(subst,GH,h,1);
/* G is a groebner basis w.r.t. a non term order MW */
/* take the initial part w.r.t. (-W,W) */
GIN = map(initial_part,G,VDV,MW,W);
/* GIN is a groebner basis w.r.t. a term order M */
/* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
/* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
for ( I = 0, T = 0; I < N; I++ )
T += W[I]*V[I]*DV[I];
B = weyl_minipoly(GIN,VDV,0,T); /* M represents DRL order */
return B;
}
/* all term reduction + interreduce */
def generic_bfct_1(F,V,DV,W)
{
N = length(V);
N2 = N*2;
/* If W is a list, convert it to a vector */
if ( type(W) == 4 )
W = newvect(length(W),W);
dp_weyl_set_weight(W);
/* create a term order M in D<x,d> (DRL) */
M = newmat(N2,N2);
for ( J = 0; J < N2; J++ )
M[0][J] = 1;
for ( I = 1; I < N2; I++ )
M[I][N2-I] = -1;
VDV = append(V,DV);
/* create a non-term order MW in D<x,d> */
MW = newmat(N2+1,N2);
for ( J = 0; J < N; J++ )
MW[0][J] = -W[J];
for ( ; J < N2; J++ )
MW[0][J] = W[J-N];
for ( I = 1; I <= N2; I++ )
for ( J = 0; J < N2; J++ )
MW[I][J] = M[I-1][J];
/* create a homogenized term order MWH in D<x,d,h> */
MWH = newmat(N2+2,N2+1);
for ( J = 0; J <= N2; J++ )
MWH[0][J] = 1;
for ( I = 1; I <= N2+1; I++ )
for ( J = 0; J < N2; J++ )
MWH[I][J] = MW[I-1][J];
/* homogenize F */
VDVH = append(VDV,[h]);
FH = map(dp_dtop,map(dp_homo,map(dp_ptod,F,VDV)),VDVH);
/* compute a groebner basis of FH w.r.t. MWH */
/* dp_gr_flags(["Top",1,"NoRA",1]); */
GH = dp_weyl_gr_main(FH,VDVH,0,1,11);
/* dp_gr_flags(["Top",0,"NoRA",0]); */
/* dehomigenize GH */
G = map(subst,GH,h,1);
/* G is a groebner basis w.r.t. a non term order MW */
/* take the initial part w.r.t. (-W,W) */
GIN = map(initial_part,G,VDV,MW,W);
/* GIN is a groebner basis w.r.t. a term order M */
/* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
/* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
for ( I = 0, T = 0; I < N; I++ )
T += W[I]*V[I]*DV[I];
B = weyl_minipoly(GIN,VDV,0,T); /* M represents DRL order */
return B;
}
def initial_part(F,V,MW,W)
{
N2 = length(V);
N = N2/2;
dp_ord(MW);
DF = dp_ptod(F,V);
R = dp_hm(DF);
DF = dp_rest(DF);
E = dp_etov(R);
for ( I = 0, TW = 0; I < N; I++ )
TW += W[I]*(-E[I]+E[N+I]);
RW = TW;
for ( ; DF; DF = dp_rest(DF) ) {
E = dp_etov(DF);
for ( I = 0, TW = 0; I < N; I++ )
TW += W[I]*(-E[I]+E[N+I]);
if ( TW == RW )
R += dp_hm(DF);
else if ( TW < RW )
break;
else
error("initial_part : cannot happen");
}
return dp_dtop(R,V);
}
/* b-function of F ? */
def bfct(F)
{
/* XXX */
F = replace_vars_f(F);
V = vars(F);
N = length(V);
D = newvect(N);
for ( I = 0; I < N; I++ )
D[I] = [deg(F,V[I]),V[I]];
qsort(D,compare_first);
for ( V = [], I = 0; I < N; I++ )
V = cons(D[I][1],V);
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
V1 = cons(s,V); DV1 = cons(ds,DV);
G0 = indicial1(F,reverse(V));
G1 = dp_weyl_gr_main(G0,append(V1,DV1),0,1,0);
Minipoly = weyl_minipoly(G1,append(V1,DV1),0,s);
return Minipoly;
}
/* called from bfct() only */
def indicial1(F,V)
{
W = append([y1,t],V);
N = length(V);
B = [t-y1*F];
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
DW = append([dy1,dt],DV);
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+y1*diff(F,V[I])*dt,B);
}
dp_nelim(1);
/* homogenized (heuristics) */
G0 = dp_weyl_gr_main(B,append(W,DW),1,0,6);
G1 = map(subst,G0,y1,1);
G2 = map(psi,G1,t,dt);
G3 = map(subst,G2,t,-s-1);
return G3;
}
/* b-function computation via generic_bfct() (experimental) */
def bfct_via_gbfct(F)
{
V = vars(F);
N = length(V);
D = newvect(N);
for ( I = 0; I < N; I++ )
D[I] = [deg(F,V[I]),V[I]];
qsort(D,compare_first);
for ( V = [], I = 0; I < N; I++ )
V = cons(D[I][1],V);
V = reverse(V);
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
B = [TMP_T-F];
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+diff(F,V[I])*TMP_DT,B);
}
V1 = cons(TMP_T,V); DV1 = cons(TMP_DT,DV);
W = newvect(N+1);
W[0] = 1;
R = generic_bfct(B,V1,DV1,W);
return subst(R,s,-s-1);
}
/* use an order s.t. [t,x,y,z,...,dt,dx,dy,dz,...,h] */
def bfct_via_gbfct_weight(F,V)
{
N = length(V);
D = newvect(N);
Wt = getopt(weight);
if ( type(Wt) != 4 ) {
for ( I = 0, Wt = []; I < N; I++ )
Wt = cons(1,Wt);
}
Tdeg = w_tdeg(F,V,Wt);
WtV = newvect(2*(N+1)+1);
WtV[0] = Tdeg;
WtV[N+1] = 1;
/* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
for ( I = 1; I <= N; I++ ) {
WtV[I] = Wt[I-1];
WtV[N+1+I] = Tdeg-Wt[I-1]+1;
}
WtV[2*(N+1)] = 1;
dp_set_weight(WtV);
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
B = [TMP_T-F];
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+diff(F,V[I])*TMP_DT,B);
}
V1 = cons(TMP_T,V); DV1 = cons(TMP_DT,DV);
W = newvect(N+1);
W[0] = 1;
R = generic_bfct_1(B,V1,DV1,W);
dp_set_weight(0);
return subst(R,s,-s-1);
}
/* use an order s.t. [x,y,z,...,t,dx,dy,dz,...,dt,h] */
def bfct_via_gbfct_weight_1(F,V)
{
N = length(V);
D = newvect(N);
Wt = getopt(weight);
if ( type(Wt) != 4 ) {
for ( I = 0, Wt = []; I < N; I++ )
Wt = cons(1,Wt);
}
Tdeg = w_tdeg(F,V,Wt);
WtV = newvect(2*(N+1)+1);
/* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
for ( I = 0; I < N; I++ ) {
WtV[I] = Wt[I];
WtV[N+1+I] = Tdeg-Wt[I]+1;
}
WtV[N] = Tdeg;
WtV[2*N+1] = 1;
WtV[2*(N+1)] = 1;
dp_set_weight(WtV);
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
B = [TMP_T-F];
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+diff(F,V[I])*TMP_DT,B);
}
V1 = append(V,[TMP_T]); DV1 = append(DV,[TMP_DT]);
W = newvect(N+1);
W[N] = 1;
R = generic_bfct_1(B,V1,DV1,W);
dp_set_weight(0);
return subst(R,s,-s-1);
}
def bfct_via_gbfct_weight_2(F,V)
{
N = length(V);
D = newvect(N);
Wt = getopt(weight);
if ( type(Wt) != 4 ) {
for ( I = 0, Wt = []; I < N; I++ )
Wt = cons(1,Wt);
}
Tdeg = w_tdeg(F,V,Wt);
/* a weight for the first GB computation */
/* [t,x1,...,xn,dt,dx1,...,dxn,h] */
WtV = newvect(2*(N+1)+1);
WtV[0] = Tdeg;
WtV[N+1] = 1;
WtV[2*(N+1)] = 1;
/* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
for ( I = 1; I <= N; I++ ) {
WtV[I] = Wt[I-1];
WtV[N+1+I] = Tdeg-Wt[I-1]+1;
}
dp_set_weight(WtV);
/* a weight for the second GB computation */
/* [x1,...,xn,t,dx1,...,dxn,dt,h] */
WtV2 = newvect(2*(N+1)+1);
WtV2[N] = Tdeg;
WtV2[2*N+1] = 1;
WtV2[2*(N+1)] = 1;
for ( I = 0; I < N; I++ ) {
WtV2[I] = Wt[I];
WtV2[N+1+I] = Tdeg-Wt[I]+1;
}
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
B = [TMP_T-F];
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+diff(F,V[I])*TMP_DT,B);
}
V1 = cons(TMP_T,V); DV1 = cons(TMP_DT,DV);
V2 = append(V,[TMP_T]); DV2 = append(DV,[TMP_DT]);
W = newvect(N+1,[1]);
dp_weyl_set_weight(W);
VDV = append(V1,DV1);
N1 = length(V1);
N2 = N1*2;
/* create a non-term order MW in D<x,d> */
MW = newmat(N2+1,N2);
for ( J = 0; J < N1; J++ ) {
MW[0][J] = -W[J]; MW[0][N1+J] = W[J];
}
for ( J = 0; J < N2; J++ ) MW[1][J] = 1;
for ( I = 2; I <= N2; I++ ) MW[I][N2-I+1] = -1;
/* homogenize F */
VDVH = append(VDV,[h]);
FH = map(dp_dtop,map(dp_homo,map(dp_ptod,B,VDV)),VDVH);
/* compute a groebner basis of FH w.r.t. MWH */
GH = dp_weyl_gr_main(FH,VDVH,0,1,11);
/* dehomigenize GH */
G = map(subst,GH,h,1);
/* G is a groebner basis w.r.t. a non term order MW */
/* take the initial part w.r.t. (-W,W) */
GIN = map(initial_part,G,VDV,MW,W);
/* GIN is a groebner basis w.r.t. a term order M */
/* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
/* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
for ( I = 0, T = 0; I < N1; I++ )
T += W[I]*V1[I]*DV1[I];
/* change of ordering from VDV to VDV2 */
VDV2 = append(V2,DV2);
dp_set_weight(WtV2);
for ( Pind = 0; ; Pind++ ) {
Prime = lprime(Pind);
GIN2 = dp_weyl_gr_main(GIN,VDV2,0,-Prime,0);
if ( GIN2 ) break;
}
R = weyl_minipoly(GIN2,VDV2,0,T); /* M represents DRL order */
dp_set_weight(0);
return subst(R,s,-s-1);
}
/* minimal polynomial of s; modular computation */
def weyl_minipolym(G,V,O,M,V0)
{
N = length(V);
Len = length(G);
dp_ord(O);
setmod(M);
PS = newvect(Len);
PS0 = newvect(Len);
for ( I = 0, T = G; T != []; T = cdr(T), I++ )
PS0[I] = dp_ptod(car(T),V);
for ( I = 0, T = G; T != []; T = cdr(T), I++ )
PS[I] = dp_mod(dp_ptod(car(T),V),M,[]);
for ( I = Len - 1, GI = []; I >= 0; I-- )
GI = cons(I,GI);
U = dp_mod(dp_ptod(V0,V),M,[]);
U = dp_weyl_nf_mod(GI,U,PS,1,M);
T = dp_mod(<<0>>,M,[]);
TT = dp_mod(dp_ptod(1,V),M,[]);
G = H = [[TT,T]];
for ( I = 1; ; I++ ) {
if ( dp_gr_print() )
print(".",2);
T = dp_mod(<<I>>,M,[]);
TT = dp_weyl_nf_mod(GI,dp_weyl_mul_mod(TT,U,M),PS,1,M);
H = cons([TT,T],H);
L = dp_lnf_mod([TT,T],G,M);
if ( !L[0] ) {
if ( dp_gr_print() )
print("");
return dp_dtop(L[1],[t]); /* XXX */
} else
G = insert(G,L);
}
}
/* minimal polynomial of s over Q */
def weyl_minipoly(G0,V0,O0,P)
{
HM = hmlist(G0,V0,O0);
N = length(V0);
Len = length(G0);
dp_ord(O0);
PS = newvect(Len);
for ( I = 0, T = G0, HL = []; T != []; T = cdr(T), I++ )
PS[I] = dp_ptod(car(T),V0);
for ( I = Len - 1, GI = []; I >= 0; I-- )
GI = cons(I,GI);
PSM = newvect(Len);
DP = dp_ptod(P,V0);
for ( Pind = 0; ; Pind++ ) {
Prime = lprime(Pind);
if ( !valid_modulus(HM,Prime) )
continue;
setmod(Prime);
for ( I = 0, T = G0, HL = []; T != []; T = cdr(T), I++ )
PSM[I] = dp_mod(dp_ptod(car(T),V0),Prime,[]);
NFP = weyl_nf(GI,DP,1,PS);
NFPM = dp_mod(NFP[0],Prime,[])/ptomp(NFP[1],Prime);
NF = [[dp_ptod(1,V0),1]];
LCM = 1;
TM = dp_mod(<<0>>,Prime,[]);
TTM = dp_mod(dp_ptod(1,V0),Prime,[]);
GM = NFM = [[TTM,TM]];
for ( D = 1; ; D++ ) {
if ( dp_gr_print() )
print(".",2);
NFPrev = car(NF);
NFJ = weyl_nf(GI,
dp_weyl_mul(NFP[0],NFPrev[0]),NFP[1]*NFPrev[1],PS);
NFJ = remove_cont(NFJ);
NF = cons(NFJ,NF);
LCM = ilcm(LCM,NFJ[1]);
/* modular computation */
TM = dp_mod(<<D>>,Prime,[]);
TTM = dp_mod(NFJ[0],Prime,[])/ptomp(NFJ[1],Prime);
NFM = cons([TTM,TM],NFM);
LM = dp_lnf_mod([TTM,TM],GM,Prime);
if ( !LM[0] )
break;
else
GM = insert(GM,LM);
}
if ( dp_gr_print() )
print("");
U = NF[0][0]*idiv(LCM,NF[0][1]);
Coef = [];
for ( J = D-1; J >= 0; J-- ) {
Coef = cons(strtov("u"+rtostr(J)),Coef);
U += car(Coef)*NF[D-J][0]*idiv(LCM,NF[D-J][1]);
}
for ( UU = U, Eq = []; UU; UU = dp_rest(UU) )
Eq = cons(dp_hc(UU),Eq);
M = etom([Eq,Coef]);
B = henleq(M,Prime);
if ( dp_gr_print() )
print("");
if ( B ) {
R = 0;
for ( I = 0; I < D; I++ )
R += B[0][I]*s^I;
R += B[1]*s^D;
return R;
}
}
}
def weyl_nf(B,G,M,PS)
{
for ( D = 0; G; ) {
for ( U = 0, L = B; L != []; L = cdr(L) ) {
if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
GCD = igcd(dp_hc(G),dp_hc(R));
CG = idiv(dp_hc(R),GCD); CR = idiv(dp_hc(G),GCD);
U = CG*G-dp_weyl_mul(CR*dp_subd(G,R),R);
if ( !U )
return [D,M];
D *= CG; M *= CG;
break;
}
}
if ( U )
G = U;
else {
D += dp_hm(G); G = dp_rest(G);
}
}
return [D,M];
}
def weyl_nf_mod(B,G,PS,Mod)
{
for ( D = 0; G; ) {
for ( U = 0, L = B; L != []; L = cdr(L) ) {
if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
CR = dp_hc(G)/dp_hc(R);
U = G-dp_weyl_mul_mod(CR*dp_mod(dp_subd(G,R),Mod,[]),R,Mod);
if ( !U )
return D;
break;
}
}
if ( U )
G = U;
else {
D += dp_hm(G); G = dp_rest(G);
}
}
return D;
}
def remove_zero(L)
{
for ( R = []; L != []; L = cdr(L) )
if ( car(L) )
R = cons(car(L),R);
return R;
}
def z_subst(F,V)
{
for ( ; V != []; V = cdr(V) )
F = subst(F,car(V),0);
return F;
}
def flatmf(L) {
for ( S = []; L != []; L = cdr(L) )
if ( type(F=car(car(L))) != NUM )
S = append(S,[F]);
return S;
}
def intersection(A,B)
{
for ( L = []; A != []; A = cdr(A) )
if ( member(car(A),B) )
L = cons(car(A),L);
return L;
}
def b_subst(F,V)
{
D = deg(F,V);
C = newvect(D+1);
for ( I = D; I >= 0; I-- )
C[I] = coef(F,I,V);
for ( I = 0, R = 0; I <= D; I++ )
if ( C[I] )
R += C[I]*v_factorial(V,I);
return R;
}
def v_factorial(V,N)
{
for ( J = N-1, R = 1; J >= 0; J-- )
R *= V-J;
return R;
}
def w_tdeg(F,V,W)
{
dp_set_weight(newvect(length(W),W));
T = dp_ptod(F,V);
for ( R = 0; T; T = cdr(T) ) {
D = dp_td(T);
if ( D > R ) R = D;
}
return R;
}
def replace_vars_f(F)
{
return subst(F,s,TMP_S,t,TMP_T,y1,TMP_Y1,y2,TMP_Y2);
}
def replace_vars_v(V)
{
V = replace_var(V,s,TMP_S);
V = replace_var(V,t,TMP_T);
V = replace_var(V,y1,TMP_Y1);
V = replace_var(V,y2,TMP_Y2);
return V;
}
def replace_var(V,X,Y)
{
V = reverse(V);
for ( R = []; V != []; V = cdr(V) ) {
Z = car(V);
if ( Z == X )
R = cons(Y,R);
else
R = cons(Z,R);
}
return R;
}
end$