CAD-QE on Risa/Asir（ソースコード）
Risa/Asir 上に CAD ベースの RCF-QE を実装しましたので，ソースコードと使用例を公開します．
環境は，Core i5-3210M 2.50GHz/8GB/Ubuntu 14.04，Risa/Asir Version 20170917 (Kobe Distribution)，OpenXM/Risa/Asir-Contrib $Revision: 1.143 $ (20170210) です．
/*%% 18.03.24.Sat. 20:33:27*/
load("sp")$load("gr")$load("sturm")$load("os_muldif.rr")$
def deld(L) " delete dupl. " {if(L==[]){return [];}
else {M=[C=car(L=qsort(L))];L=cdr(L);
while(L!=[]) {
if(C!=(C=car(L))) {L=cdr(L);M=cons(C,M);} else L=cdr(L);}
return M;}
}
def delc(L) " without os_md " {A=deld(L);B=[];
while(A!=[]) {if(type(F=car(A))==2) B=cons(F,B);A=cdr(A);}
return B;}
def factors(L) { return L==[] ? [] : delc(taka_base_flatten(map(fctr,delc(L)))); }
def proj(VV,L) " without os_md "
{W=reverse(cdr(VV));A=L;PP=[];
while(W!=[]) {V=car(W);W=cdr(W);C=factors(A);A=B=[]; /*print(V);*/
while(C!=[]) {F=car(C);C=cdr(C);
if((DG=deg(F,V))>0)
{if(vars(F)!=[V] || count_real_roots(F)>0) {A=cons(F,A);}
B=cons(coef(F,DG,V),B);
B=cons(res(V,F,diff(F,V)),B);
CL=[];for(K=0;K<=DG;K++) {if((CF=coef(F,K,V))!=0) CL=cons(CF,CL);}
if((GR=nd_f4(CL,vars(CL),0,0))!=[1] && GR!=[-1])
B=cons(coef(F,0,V),B);}
else B=cons(F,B);}
PP=cons(A,PP);
for(I=0;I<(LA=length(A))-1;I++) for(J=I+1;J<LA;J++)
B=cons(res(V,A[I],A[J]),B);
A=B;}
A=factors(A);B=[];V=car(VV);while(A!=[]) {F=car(A);A=cdr(A);
if(count_real_roots(F)>0) B=cons(F,B);}
return cons(B,PP);
}
def prod(L) {P=1;while(L!=[]){P*=car(L);L=cdr(L);} return P;}
def adp(L)
{R=car(L)[1];/*R=rdp(L); maybe long */L=cdr(L);
while(L!=[]){R=res((C=car(L))[0],C[1],R);L=cdr(L);}
return polsqf1(R);}
def polrootsreal(F,P) {R=pari(roots,F,P);
if(type(R)!=5) return [R];
else
{R=vtol(R);RR=[];
while(R!=[]){ if(imag(C=car(R))==0) {RR=cons(C,RR);} R=cdr(R); }
return reverse(RR);}
}
def setG(N) { pari(allocatemem,10^9*N); }$
setG(1)$
/* 1G */
ctrl("bigfloat",1)$
extern REALPRECISION$
REALPRECISION=50$setprec(REALPRECISION)$
def polrootsreal4(P) {A=polrootsreal(P,LP=REALPRECISION);
if((NA=length(A))!=0)
{if(NA==1) R=+oo;
else while((R=(os_md.lmin(ltov(cdr(A))-ltov(reverse(cdr(reverse(A)))))*6/25))*10^LP<=os_md.lmax(map(number_abs,A))) {A=polrootsreal(P,LP=2*LP);/*print(A);*/}
return([A,R]);}
else return [];
}
def toint(L) {R=L[1];return base_makelist([s-R,s+R],s,L[0],0);}
def toint2(VF,L) {V=VF[0];F=VF[1];R=L[1];
return base_makelist([subst(F,V,s-R),subst(F,V,s+R)],s,L[0],0);}
def toint4(VF,L) {V=VF[0];F=VF[1];M=L[0];R=(L[1]==oo)?1:L[1];Z=[];
while(M!=[]){S=car(M);M=cdr(M);
if(subst(F,V,S-R)*subst(F,V,S+R)<0) Z=cons(S,Z);}
return Z;
}
extern CAD,CAD0;
def ans() { if(CAD==[]){ print("false");}
else {for(K=0;K<length(CAD);K++){printf(rtostr(1+K));print(CAD[K]);}}
}
def lambda1(X) { return car(X); }
extern PP0,PPj,Vj,FF,AFS,US,UN,UA,QQj,EXS,ALLS,TC,FC,TCS,TFUNC;
def id(F) {F;}
def not(F){ 1-F; }
def or(F,G){ 1-(1-F)*(1-G); } /* F || G */
def and(F,G){ F * G; } /* F && G */
def imp(F,G){ 1+F*(G-1); }
def eq(F,G){ (1+F*(G-1))*(1+G*(F-1)); }
def tfunc(F,G){ F * G; }
def tuf(N,S,A) {L=AFS;M=[];SUS=cons([Vj,S],US);NUN=cons(Vj,cons(N,UN));
while(L!=[]){F=car(car(L));G=car(L)[1];L=cdr(L); /*print([F,SUS]);*/
F=call(subst,cons(F2=simpalg_noalg(F,SUS),NUN)); /*print("->"+rtostr(F2));print("->"+rtostr(F));*/
if(type(F)==2) M=cons(1/2,M);
else M=cons(eval_str(rtostr(F)+G),M);
}
FC=TC=0;
V=call(TFUNC,reverse(M));
if(V!=0 && V!=1) {return [NUN,SUS,cons(A,UA)];}
else {if(V==0) {FC=1;return 0;} else {TC=1;TCS++;
return [cons(A,UA)];}}
}
def addinf(U) {
return (length(U)==1)?[cons([-oo,oo],U[0])]
:[cons(Vj,cons(0,U[0])),cons([Vj,Vj],U[1]),cons([-oo,oo],U[2])];
}
def lambda3(L) { return reverse(car(L)); }
def makeANS1(ANS0) {
if(ANS0==[[]]){
TUF=tuf(0,Vj,[-oo,oo]);
if(TUF) {ANS1=[TUF];}}
else {
ANS1=[];LAST=[car(car(ANS0))-2,0,-oo];
while(ANS0!=[])
{Last=car(LAST);TOP=car(ANS0);Top=car(TOP);DDD=2;
do {DDD*=2;Mid=floor((Last+Top)/2*DDD)/DDD; /*print(DDD);*/ }
while(Mid<=Last || Top<=Mid);
TUF=tuf(Mid,ptozp(Vj-Mid),[LAST[2],TOP[2]]);
if(EXS*TC) {return [TUF];}
if(ALLS*FC) {return [];}
if(TUF) {ANS1=cons(TUF,ANS1);}
TUF=tuf(Top,TOP[1],map(vect,[TOP[2],TOP[2]]));
if(EXS*TC) {return [TUF];}
if(ALLS*FC) {return [];}
if(TUF) {ANS1=cons(TUF,ANS1);}
LAST=TOP;ANS0=cdr(ANS0);}
TUF=tuf(Etr=ceil(Top+1),ptozp(Vj-Etr),[LAST[2],oo]);
if(EXS*TC) {return [TUF];}
if(ALLS*FC) {return [];}
if(TUF) {ANS1=cons(TUF,ANS1);}}
return ANS1; }
def deltopn(N,L) { for(K=1;K<=N;K++){L=cdr(L);} return L; }
def consinfn(N,L) { for(K=1;K<=N;K++){L=cons([-oo,oo],L);} return L; }
def delnth(L,N)
{ M=[];
for(K=0;K<N;K++){M=cons(car(L),M);L=cdr(L);}
L=cdr(L);
for(K=0;K<N;K++){L=cons(car(M),L);M=cdr(M);}
return L; }
def repnth(L,N,A)
{ M=[];
for(K=0;K<N;K++){M=cons(car(L),M);L=cdr(L);}
L=cdr(L);L=cons(A,L);
for(K=0;K<N;K++){L=cons(car(M),L);M=cdr(M);}
return L; }
def cnn(LenVV,ANS2)
{
LAST0=base_makelist([[0,0],[0,0]],k,1,LenVV);
for(K=0;K<LenVV;K++){
ANS3=[];LAST=LAST0;
while(ANS2!=[]){TOP=car(ANS2);ANS2=cdr(ANS2);
if(delnth(LAST,K)!=delnth(TOP,K)){ANS3=cons(TOP,ANS3);LAST=TOP;}
else
{LASTK=LAST[K];TOPK=TOP[K];
if(ltov(LASTK[1])==TOPK[0] || LASTK[1]==ltov(TOPK[0]))
{LAST=repnth(LAST,K,[LASTK[0],TOPK[1]]);
ANS3=cdr(ANS3);ANS3=cons(LAST,ANS3);}
else{ANS3=cons(TOP,ANS3);LAST=TOP;}
}}
ANS2=reverse(ANS3);
}
return ANS2;
}
def lambda4(L) { C=car(L);
if(C==8){C="==0";}else{
if(C==14){C="!=0";}else{
if(C==9){C="<=0";}else{
if(C==11){C="<0";}else{
if(C==13){C=">=0";}else{
if(C==15){C=">0";}}}}}}
return [L[1],C];
}
def at2afs(AT) { return map(lambda4,map(fopargs,AT)); }
extern RVV,TOP;
def lambda5(F) { TpF=type(F);
if(TpF==2){ return F; }
if(TpF==4){ return [prod(flatmf(fctr(gr([F[0],TOP],RVV,2)[1]))),F[1]]; }
if(TpF==5){ return ltov([prod(flatmf(fctr(gr([F[0],TOP],RVV,2)[1]))),F[1]]); }
}
extern LenQQ;
def lambda6(F) { for(K=1;K<=LenQQ;K++){F=cdr(F);} return F; }
def ppzeros(V,PP/*must be !=[]*/,L1,L2)
{
M=[];while(PP!=[]){M=append(M,zeros(V,car(PP),L1,L2));PP=cdr(PP);}
M=qsort(M,sort2);
N=[LAST=car(M)];M=cdr(M);
while(M!=[]){
if(LAST[0]!=((TOP=car(M))[0])) {N=cons(TOP,N);LAST=TOP;}
/* else {if(lambda2(LAST[1])!=lambda2(TOP[1])) {*/
/* else {if(subst(LAST[1],U0)!=subst(TOP[1],U0)) { */
else {if(ptozp(simpalg_noalg(ptozp( LAST[1] ),L1))
!=ptozp(simpalg_noalg(ptozp( TOP[1] ),L1))) {
/* else {if(LAST[1]!=TOP[1]) {*/
print(" *** common-roots fault in ppzeros:");print([LAST,TOP,US,UN]);
print(" *** Increase the value of REALPRECISION.");}}
M=cdr(M);
}
/* print(N); */
return N;
}
def cadqe(QQ,VV,Tfunc,AF)
"
cadqe([ex],[a,b,c,x],id,[a*x^2+b*x+c@==0])$
cadqe([all],[a,b,x],imp,[x^2-1@<=0,x^3+a*x+b@<=0])$
cadqe([all,all],[a,b,c,y,x],eq,[a*x^2+b*y^2@<=c,x^2+y^2@<=1])$
def _(A,B,C){and(A,and(B,C));}$cadqe([ex,ex,ex],[a,b,c,x,y,z],_,[x+y+z@==a,x*y+y*z+z*x@==b,x*y*z@==c]|grob=1)$
"
{
JJ0=(LenVV=length(VV))-(LenQQ=length(QQ));EXS=ALLS=0;
if(type(car(AF))==14){AF=at2afs(AF);}
TFUNC=Tfunc;
/*if(length(AF)==1){TFUNC=id;}else{TFUNC=tfunc;}*/
FF=map(lambda1,AF);
if(type(getopt(grob))!=-1){FF=gr(FF,reverse(VV),2);}
PP=proj(VV,FF);ANS2=[[[],[],[]]];PP0=PP;AFS=AF;
for(J=0;J<LenVV;J++){Vj=VV[J];PPj=PP[J];
if(J==LenVV-1){PPZ=ppzeros;}else{PPZ=ppzeros;}
if((JJ=J-JJ0)>=0){QQj=QQ[JJ];
if(QQj==ex){EXS++;ALLS=0;}else{EXS=0;ALLS++;}; /*print([QQj,EXS,ALLS]);*/ }
if(PPj==[]){ANS3=[];while(ANS2!=[])
{/*print(ANS2);*/U=car(ANS2);ANS2=cdr(ANS2);
if(length(U)==1){TUF=[cons([-oo,oo],UA=car(U))]; /*print(TUF);*/}
else{US=U[1];UN=U[0];UA=U[2];TUF=tuf(0,Vj,[-oo,oo]);}
if(TUF) {ANS3=cons(TUF,ANS3);}}
ANS2=reverse(ANS3);}
else{ANS3=[];while(ANS2!=[])
{U=car(ANS2);ANS2=cdr(ANS2);ETC=TCA=TCS=0;
if(length(U)==1){ANS1=[[cons([-oo,oo],UA=car(U))]]; /*print(ANS1);*/}
else {ANS0=( *PPZ)(Vj,PPj,US=U[1],UN=U[0]);UA=U[2]; /*if(Vj==x){print(ANS0);};*/
ANS1FULL=2*length(ANS0)+1;
ANS1=makeANS1(ANS0); /*print([CNT,ANS1]);*/
/*print([EXS,TC,TCS,ANS1FULL,ALLS,FC,J]);*/
if((ETC=EXS*TC) || ALLS*FC) {
UA=deltopn(EAS=(EXS+ALLS-1),UA); /*print(UA);*/
/*print([ETC,TCA,TCS,UA,EAS]);print(ANS2);*/
while(deltopn(EAS,car(reverse(car(ANS2))))==UA){ANS2=cdr(ANS2);}
/*print(ANS2);print(" ");*/
while(deltopn(EAS+1,car(reverse(car(car(ANS3)))))==UA){ANS3=cdr(ANS3);}}
if(ETC /*|| TCS==ANS1FULL*/){ANS1=[[consinfn(EAS+1,UA)]];}
else{if(TCS==ANS1FULL){ANS1=[[cons([-oo,oo],UA)]];}} }
if(ANS1){ANS3=cons(reverse(ANS1),ANS3);}; /*print(["ANS3",ANS3]);print(["ANS1",ANS1]);*/ }
ANS2=os_md.m2l(reverse(ANS3)|flat=1); /*print([J,ANS2]);*/
}
}
if(ANS2==[]){ return CAD=[];}
CAD0=ANS2=map(lambda1,ANS2);
do {ANS3=ANS2;ANS2=cnn(LenVV,ANS2);} while (ANS3!=ANS2);
for(K=JJ;0<=K;K--){
if(QQ[K]==all){ANS3=[];
while(ANS2!=[]){TOP=car(ANS2);ANS2=cdr(ANS2);
if(TOP[JJ-K]==[-oo,oo]){ANS3=cons(TOP,ANS3);}}
ANS2=reverse(ANS3);
do {ANS3=ANS2;ANS2=cnn(LenVV,ANS2);} while (ANS3!=ANS2);
}
if(QQ[K]==ex){ANS3=[LAST=repnth(car(ANS2),JJ-K,[-oo,oo])];
while(ANS2!=[]){TOP=repnth(car(ANS2),JJ-K,[-oo,oo]);ANS2=cdr(ANS2);
if(LAST!=TOP){ANS3=cons(TOP,ANS3);LAST=TOP;}}
ANS2=reverse(ANS3);
do {ANS3=ANS2;ANS2=cnn(LenVV,ANS2);} while (ANS3!=ANS2);
}
}
LenVV=JJ0;
ANS2=map(reverse,map(lambda6,ANS2));
ANS2=call(mat,ANS2);RVV=reverse(VV);
for(I=0;I<size(ANS2)[0];I++){
for(J=0;J<LenVV-1;J++){Aij=ANS2[I][J];
if((Aij0=Aij[0])==Aij[1]){TOP=Aij0[0];
for(K=J+1;K<LenVV;K++){ANS2[I][K]=map(lambda5,ANS2[I][K]);}
}
}
}
CAD=ANS2=os_md.m2ll(ANS2);
ans();
return CAD;
}
def ansm() { print(call(mat,CAD)); }
def af_noalg(F,DL)
{
DL = reverse(DL);
N = length(DL);
Tab = newvect(N);
/* Tab = [[a1,r1],...]; ri is a root of di(t,r(i-1),...,r1). */
AL = [];
for ( I = 0; I < N; I++ ) {
T = DL[I];
for ( J = 0, DP = T[1]; J < I; J++ )
DP = subst(DP,Tab[J][0],Tab[J][1]);
B = newalg(DP);
Tab[I] = [T[0],B];
F = subst(F,T[0],B);
AL = cons(B,AL);
}
FL = af(F,deld(AL)); /* modified */
for ( T = FL, R = []; T != []; T = cdr(T) )
R = cons([conv_noalg(T[0][0],Tab),T[0][1]],R);
return reverse(R);
}
def sort2(X,Y) { return (X[0]<Y[0])?1:0; }
def lambda2(F) { return ptozp(simpalg_noalg(ptozp(F),US)); }
def sort1(X,Y) { return (X[0]<Y[0])?1:0; }
def zeros(V,F,L1,L2)
{
G=simpalg_noalg(F,L1); /*print(G);*/
if(G==0) F=G=subst3(F,L1); else 0; /*print([V,F,L1,L2,G]);*/
L=flatmf(af_noalg(G,L1));
if(L==[1]) return [];
W=[];
while(L!=[]){G=car(L);L=cdr(L);
if((S=polrootsreal4(adp(cons([V,G],L1))))!=[])
{ if((S=toint4([V,call(subst,cons(G,L2))],S))!=[])
W=append(W,base_makelist([s,G],s,S,0));}
}
W=qsort(W,sort1);U=[];C=length(W);
while(W!=[]){U=cons(append(car(W),[[F,C]]),U);W=cdr(W);C--;}
return U;
}
def zeros2(V,F,L1,L2)
{
G=simpalg_noalg(F,L1); /*print(G);*/
if(G==0) G=subst3(F,L1); else 0; /*print([V,F,L1,L2]);*/
G=prod(factors([prod(flatmf(asq_noalg(cons([V,G],L1))))]));
if(G==1) return [];
W=[];L=factors([adp(cons([V,G],L1))]);
while(L!=[]){A=car(L);L=cdr(L);
if((S=polrootsreal4(A))!=[])
{ if((S=toint4([V,call(subst,cons(G,L2))],S))!=[])
W=append(W,base_makelist([s,A],s,S,0));}
}
W=qsort(W,sort1);U=[];C=length(W);
while(W!=[]){U=cons(append(car(W),[[G,C],[F,G]]),U);W=cdr(W);C--;}
return U;
}
def asq_noalg(DL)
{ F = car(DL)[1]; V = car(DL)[0]; DL = cdr(DL);
DL = reverse(DL);
N = length(DL);
Tab = newvect(N);
/* Tab = [[a1,r1],...]; ri is a root of di(t,r(i-1),...,r1). */
AL = [];
for ( I = 0; I < N; I++ ) {
T = DL[I];
for ( J = 0, DP = T[1]; J < I; J++ )
DP = subst(DP,Tab[J][0],Tab[J][1]);
B = newalg(DP);
Tab[I] = [T[0],B];
F = subst(F,T[0],B);
AL = cons(B,AL);
}
FL = asq(red(monica(simpalg(F),V)));
for ( T = FL, R = []; T != []; T = cdr(T) )
R = cons([conv_noalg(T[0][0],Tab),T[0][1]],R);
return reverse(R);
}
def simpalg_noalg(F,DL)
{
DL = reverse(DL);
N = length(DL);
Tab = newvect(N);
/* Tab = [[a1,r1],...]; ri is a root of di(t,r(i-1),...,r1). */
AL = [];
for ( I = 0; I < N; I++ ) {
T = DL[I];
for ( J = 0, DP = T[1]; J < I; J++ )
DP = subst(DP,Tab[J][0],Tab[J][1]);
B = newalg(DP);
Tab[I] = [T[0],B];
F = subst(F,T[0],B);
AL = cons(B,AL);
}
return conv_noalg(simpalg(F),Tab);
}
def subst3(F/* must be banish */,L)
/*
subst3((a+1)*x^2+(a+b)*x+b^2-1,[[b,b+a],[a,a+1]]);
*/
{L=reverse(L);M=[];
while(L!=[]) {M=cons(car(L),M);L=cdr(L);
while((G=simpalg_noalg(F,M))==0) {F=diff(F,M[0][0]);}
F=G;
}
return F;
}
def rdp0(L)
/* relative def. poly. using af_noalg.
rdp0([[x,a*x^2+b*x+1],[b,b^2-4*a],[a,a^2-2]]); */
{ return prod(flatmf(af_noalg(car(L)[1],cdr(L)))); }
def rdp(L)
/* relative def. poly. using asq_noalg. faster than by af_noalg.
rdp([[x,a*x^2+b*x+1],[b,b^2-4*a],[a,a^2-2]]);
*/
{ return prod(flatmf(asq_noalg(L))); }
def monica(F,V) {return F==0?0:F/coef(F,deg(F,V),V);}
def polsqf(F,V) {return F==0?0:sdiv(F,gcd(F,diff(F,V)),V);}
def polsqf1(F) {return prod(flatmf(sqfr(F)));}
def polsqfa(A,P,B,X) " polsqfa(p^2-2,p,(x-p)^4*(p*x^2+4*x+2*p)^5,x); "
{
if(B==0) return 0;
else {C=simpcoef(simpalg(monica(subst(B,P,P1=newalg(A)),X)));
return subst(algptorat(simpalg(
sdiv(C,cr_gcda(C,diff(C,X)),X))),algptorat(P1),P);}
}
cputime(1)$
end$
