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// mequiv.cc: implementation of quartic equivalence functions
//////////////////////////////////////////////////////////////////////////
//
// Copyright 1990-2012 John Cremona
//
// This file is part of the eclib package.
//
// eclib is free software; you can redistribute it and/or modify it
// under the terms of the GNU General Public License as published by the
// Free Software Foundation; either version 2 of the License, or (at your
// option) any later version.
//
// eclib is distributed in the hope that it will be useful, but WITHOUT
// ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
// FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
// for more details.
//
// You should have received a copy of the GNU General Public License
// along with eclib; if not, write to the Free Software Foundation,
// Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA
//
//////////////////////////////////////////////////////////////////////////
#include <eclib/mequiv.h>
//#ifdef NEW_EQUIV
int new_equiv(quartic* q1, quartic* q2, int info)
{
if(info)
{
cout<<"Checking equivalence of " << *q1 << " and " << *q2 << "\n";
}
bigint& ii=q1->ii;
bigint& jj=q1->jj;
if (!(ii==q2->ii && jj==q2->jj && q1->disc==q2->disc && q1->type== q2->type))
{
if (info)
{
cout << "equiv failed on first test!\n";
cout << "First has I="<<q1->ii<<", J="<<q1->jj<<",";
cout << " disc="<<q1->disc<<", type="<<q1->type<<endl;
cout << "Second has I="<<q2->ii<<", J="<<q2->jj<<",";
cout << " disc="<<q2->disc<<", type="<<q2->type<<endl;
}
return 0;
}
if(q1->equiv_code!=q2->equiv_code)
{
if(info) cout << "--equiv_codes not equal\n";
return 0;
}
q1->make_zpol();
q2->make_zpol();
bigint& p1=q1->p; bigint& p2=q2->p;
bigint& r1=q1->r; bigint& r2=q2->r;
bigint& p1sq=q1->psq; bigint& p2sq=q2->psq;
bigint& a1=q1->a; bigint& a2=q2->a;
bigint& a1sq=q1->asq; bigint& a2sq=q2->asq;
const bigint& a1a2=a1*a2;
const bigint& p1p2=p1*p2;
const bigint& p = (32*a1a2*ii + p1p2)/3;
const bigint& s = (-256*jj*a1a2*(a1*p2+a2*p1)
+ 64*ii*(p1sq*a2sq+p2sq*a1sq+p1p2*a1a2)
- p1sq*p2sq) / 27;
const bigint& r = r1*r2;
if(info) cout<<"u-poly = [1,0, " << -2*p<< ", "<<-8*r<<", "<<s<<"]\n";
// Compute the roots of "u-poly", see if any are integral, and check:
// The roots are sqrt(z1*w1)+sqrt(z2*w2)+sqrt(z3*w3) with some choice of signs,
// where the wi are the z-values of the second quartic. We compute the zi
// from the roots of the first, then to ensure that conjugates are matched
// we compute the wi from the zi (via, invisibly, the phi_i).
bigcomplex* roots1 = q1->getroots();
bigfloat xa1=I2bigfloat(a1), xa2=3*I2bigfloat(a2);
bigcomplex rz1=xa1*(roots1[0]+roots1[1]-roots1[2]-roots1[3]);
bigcomplex rz2=xa1*(roots1[0]-roots1[1]+roots1[2]-roots1[3]);
bigcomplex rz3=xa1*(roots1[0]-roots1[1]-roots1[2]+roots1[3]);
bigcomplex t = I2bigfloat(a1*p2-a2*p1), tt=1/I2bigfloat(3*a1);
bigcomplex rzw1 = rz1*sqrt((xa2*rz1*rz1+t)*tt);
bigcomplex rzw2 = rz2*sqrt((xa2*rz2*rz2+t)*tt);
bigcomplex rzw3 = rz3*sqrt((xa2*rz3*rz3+t)*tt);
bigcomplex u;
for(long k=0; k<4; k++)
{
switch(k) {
case 0: u=rzw1+rzw2+rzw3; break;
case 1: u=rzw1+rzw2-rzw3; break;
case 2: u=rzw1-rzw2+rzw3; break;
case 3: u=rzw1-rzw2-rzw3; break;
}
if(is_approx_zero(imag(u)))
{
bigint uu = Iround(real(u));
bigint uu2=uu*uu;
bigint fuu1 = (uu2-2*p)*uu2+s;
bigint fuu2 = 8*r*uu;
if(fuu1==fuu2)
{
if(info) cout<<"Root u = "<<uu<<endl;
return 1;
}
if(fuu1==-fuu2)
{
if(info) cout<<"Root u = "<<-uu<<endl;
return 1;
}
}
}
if(info) cout<<"No integral roots"<<endl;
return 0;
}
//#else // not using new_equiv so no need to compile this stuff
int testd(const bigint& a, const bigint& b, const bigint& c,
const bigint& d, const bigint& e, const bigint& as,
const bigint& bs, const bigint& cs, const bigint& ds,
const bigint& es, const bigint& dd, const bigint& al,
const bigint& be, const bigint& ga, const bigint& de,
int info);
bigcomplex crossratio(const bigcomplex& x1,const bigcomplex& x2,const bigcomplex& x3,const bigcomplex& x4);
int rootsequiv(const quartic* q1, const quartic* q2, int i, const vector<bigint>& dlist, int info);
int allperms[24][4] =
{{0,1,2,3},{1,0,2,3},{0,1,3,2},{1,0,3,2}, // Up to here for Type III
{2,3,0,1},{2,3,1,0},{3,2,0,1},{3,2,1,0}, // Up to here for Type I
{0,2,1,3},{0,2,3,1},{0,3,1,2},{0,3,2,1},
{1,2,0,3},{1,2,3,0},{1,3,0,2},{1,3,2,0},
{2,0,1,3},{2,0,3,1},{2,1,0,3},{2,1,3,0},
{3,0,1,2},{3,0,2,1},{3,1,0,2},{3,1,2,0},}; // All for Type II
int testd(const bigint& a, const bigint& b, const bigint& c,
const bigint& d, const bigint& e, const bigint& as,
const bigint& bs, const bigint& cs, const bigint& ds,
const bigint& es, const bigint& dd, const bigint& al,
const bigint& be, const bigint& ga, const bigint& de,
int info)
{
bigint d2 = dd*dd;
bigint al2=al*al, al3=al2*al, al4=al3*al;
bigint ga2=ga*ga, ga3=ga2*ga, ga4=ga3*ga;
bigint temp = ga4*es + al*ga3*ds + al2*ga2*cs + al3*ga*bs + al4*as - d2*a;
if (!is_zero(temp)) return 0;
bigint de2=de*de, de3=de2*de, de4=de3*de;
bigint be2=be*be, be3=be2*be, be4=be3*be;
temp = de4*es + be*de3*ds + be2*de2*cs + be3*de*bs + be4*as - d2*e;
if (!is_zero(temp)) return 0;
temp = 4*ga3*de*es + (3*al*ga2*de+be*ga3)*ds
+ 2*(al2*ga*de+al*be*ga2) * cs
+ (3*al2*be*ga+al3*de)*bs + 4*al3*be*as - d2*b;
if (!is_zero(temp)) return 0;
temp = 4*ga*de3*es + (3*be*ga*de2+al*de3)*ds
+ 2*(be2*ga*de+al*be*de2)*cs
+ (be3*ga+ 3*al*be2*de)*bs + 4*al*be3*as - d2*d;
if (!is_zero(temp)) return 0;
temp = 6*ga2*de2*es + 3*(be*ga2*de+al*ga*de2) * ds
+ (be2*ga2+ 4*al*be*ga*de+al2*de2) * cs
+ 3*(al*be2*ga+al2*be*de) * bs + 6*al2*be2*as - d2*c;
if (!is_zero(temp)) return 0;
return 1;
} /* end of testd() */
bigcomplex crossratio(const bigcomplex& x1,const bigcomplex& x2,const bigcomplex& x3,const bigcomplex& x4)
{
return ((x1-x3)*(x2-x4))/((x1-x4)*(x2-x3));
}
int rootsequiv(const quartic* q1, const quartic* q2, int i, const vector<bigint>& dlist, int info)
{ int ans=0;
bigcomplex *x = q1->getroots();
bigcomplex *y = q2->getroots();
bigcomplex x1=x[0], x2=x[1], x3=x[2], x4=x[3];
bigcomplex y1=y[allperms[i][0]],y2=y[allperms[i][1]],
y3=y[allperms[i][2]],y4=y[allperms[i][3]];
//cout<<"X-roots ("<<x<<"): "<<x1<<x2<<x3<<x4<<endl;
//cout<<"Y-roots ("<<y<<"): "<<y1<<y2<<y3<<y4<<endl;
bigcomplex xtheta = crossratio(x1,x2,x3,x4);
bigcomplex ytheta = crossratio(y1,y2,y3,y4);
if (abs(xtheta-ytheta)>0.1)
{if (info)
{cout << i+1 << ": Cross-ratios unequal: "<<xtheta<<" and "<<ytheta<<".\n";
}
return 0;
}
if(info) cout << i+1 << ": Cross-ratios equal: "<<xtheta<<" and "<<ytheta<<".\n";
bigcomplex xy1=x1*y1, xy2=x2*y2,
xy3=x3*y3, // xy4=x4*y4,
xy23=(x2*y3)-(x3*y2),
xy13=(y1*x3)-(x1*y3),
xy12=(x1*y2)-(y1*x2);
bigcomplex calpha = (xy1*(y2-y3)) + (xy2*(y3-y1)) + (xy3*(y1-y2)),
cbeta = (xy1*xy23) + (xy2*xy13) + (xy3*xy12),
cgamma = xy13 + xy12 + xy23,
cdelta = (xy1*(x2-x3)) + (xy2*(x3-x1)) + (xy3*(x1-x2));
bigcomplex scale = calpha;
if (abs(scale)<abs(cbeta)) scale=cbeta;
if (abs(scale)<abs(cgamma)) scale=cgamma;
if (abs(scale)<abs(cdelta)) scale=cdelta;
if(is_small(scale))
{
cout << "Warning from rootsequiv(): scale = " << scale << endl;
cout << "alpha, beta, gamma, delta = " << calpha << cbeta << cgamma << cdelta << endl;
}
calpha /= scale;
cbeta /= scale;
cgamma /= scale;
cdelta /= scale;
if (!(is_real(calpha) && is_real(cbeta)
&& is_real(cgamma)
&& is_real(cdelta)))
{if (info)
{cout << "Transformation not real.\n";
cout << "alpha, beta, gamma, delta = " << calpha << cbeta << cgamma << cdelta << endl;
}
return 0;
}
bigfloat alpha = real(calpha),
beta = real(cbeta),
gamma = real(cgamma),
delta = real(cdelta);
bigfloat det = alpha*delta-beta*gamma;
if (info)
{
cout << "Real transformation has alpha, beta, gamam, delta = ";
cout <<alpha<<" "<<beta<<" "<<gamma<<" "<<delta<<endl;
cout << "Testing divisors of "<<q1->getdisc()<<":\n";
// cout << dlist << endl;
}
bigint d;
vector<bigint>::const_iterator dvar;
for (dvar=dlist.begin(); dvar!=dlist.end() && (!ans); dvar++)
{
d = *dvar;
bigfloat rscale = sqrt(abs(I2bigfloat(d)/det));
bigfloat rscaler = floor(rscale+0.5);
if(abs(rscale-rscaler)<0.001)
{
bigint al = Iround(rscale*alpha),
be = Iround(rscale*beta ),
ga = Iround(rscale*gamma),
de = Iround(rscale*delta);
bigint ddet = abs(al*de-be*ga);
if(d==ddet)
{
if (info)
{cout << "d = " << d << endl;
cout<<"rscale = "<<rscale<<endl;
cout << "al,be,ga,de = "<<al<<" "<<be<<" "<<ga<<" "<<de<<endl;
}
ans=testd(q1->geta(),q1->getb(),q1->getcc(),q1->getd(),q1->gete(),
q2->geta(),q2->getb(),q2->getcc(),q2->getd(),q2->gete(),
d,al,be,ga,de,info);
}
}
}
return ans ;
} // of rootsequiv()
int equiv(const quartic* q1, const quartic* q2, const vector<bigint>& dlist, int info)
{
bigint iiq1 = q1->getI(), jjq1 = q1->getJ(), discq1 = q1->getdisc();
bigint iiq2 = q2->getI(), jjq2 = q2->getJ(), discq2 = q2->getdisc();
int typeq1 = q1->gettype(), typeq2 = q2->gettype();
if(info)
{
cout<<"Checking equivalence of \n"; q1->dump(cout);
cout<<"and\n"; q2->dump(cout);
}
if (iiq1==iiq2 && jjq1==jjq2 && discq1==discq2 && typeq1== typeq2)
{
int nperms = (typeq1==1)? 8 : (typeq1==2)? 24 : 4;
if (info) cout << "Params agree; calling rootsequiv "<<nperms<<" times.\n";
int i,ans = 0;
for (i=0; i<nperms && (!ans); i++)
{
ans=rootsequiv(q1,q2,i,dlist,info);
}
if(info) {if(!ans) cout << "Not "; cout<<"equiv\n";}
return ans;
}
else
{
if (info) {cout << "equiv failed on first test!\n";
cout << "First has I="<<iiq1<<", J="<<jjq1<<",";
cout << " disc="<<discq1<<", type="<<typeq1<<endl;
cout << "Second has I="<<iiq2<<", J="<<jjq2<<",";
cout << " disc="<<discq2<<", type="<<typeq2<<endl;
}
return 0;
}
} // of equiv()
//#endif
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