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// compproc.cc: declarations of functions using complex numbers
//////////////////////////////////////////////////////////////////////////
//
// 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/compproc.h>
int is_real(const bigcomplex& z) {return(is_approx_zero(imag(z)));}
int is_small(bigfloat x) {return is_approx_zero(x);}
int is_small(const bigcomplex& z){return is_approx_zero(z);}
void orderreal(bigfloat& e1, bigfloat& e2, bigfloat& e3) // puts in decreasing order
{
bigfloat t;
if (e1 < e3) {t=e1; e1=e3; e3=t;} //swap(e1,e3);
if (e1 < e2) {t=e1; e1=e2; e2=t;} //swap(e1,e2);
else if (e2 < e3) {t=e2; e2=e3; e3=t;} //swap(e2,e3);
}
//#define DEBUG_CAGM
bigcomplex cagm(const bigcomplex& a, const bigcomplex& b)
{
bigcomplex x=a, y=b, oldx;
#ifdef DEBUG_CAGM
cout<<"cagm("<<x<<","<<y<<"):"<<endl;
#endif
static bigfloat two=to_bigfloat(2);
bigfloat theta, piby2=Pi()/two;
while (1)
{
oldx=x;
x=(x+y)/two;
y= sqrt(oldx*y);
theta = arg(y/x);
if ((theta>piby2) || (theta<=-piby2)) y=-y;
#ifdef DEBUG_CAGM
cout<<"Relative error = "<<abs((x-y)/x)<<endl;
#endif
if(is_approx_zero(abs((x-y)/x))) return x;
}
return x;
}
bigcomplex normalize(bigcomplex& w1, bigcomplex& w2)
{
bigcomplex tau = w1/w2, w3;
if (tau.imag() < 0) { w1=-w1 ; tau=-tau ; }
w1=w1-w2*round(tau.real());
tau=w1/w2;
for (int i=1;i<50 && (abs(tau)<1);i++)
// {Just to stop infinite loop due to rounding}
{ w3=-w1; w1=w2; w2=w3; tau=w1/w2;
w1=w1-w2*round(tau.real());
tau=w1/w2;
}
return tau;
}
void getc4c6(const bigcomplex& w1, const bigcomplex& w2,
bigcomplex& c4, bigcomplex &c6)
{
bigcomplex tau= w1/w2;
static bigfloat zero(to_bigfloat(0)), one(to_bigfloat(1)), two(to_bigfloat(2));
bigfloat pi(Pi());
bigfloat x = two*pi*tau.real();
bigfloat y = two*pi*tau.imag();
bigfloat nx = x, ny=y;
bigcomplex q = exp(-y) * bigcomplex(cos(x),sin(x));
bigcomplex f = two*pi/w2;
bigcomplex f2 = f*f;
bigcomplex f4=f2*f2;
bigcomplex term = bigcomplex(one);
bigcomplex qpower = bigcomplex(one);
bigcomplex sum4 = bigcomplex(zero);
bigcomplex sum6 = bigcomplex(zero);
bigfloat n, n2;
for (n=1;
#ifdef MPFP
(n==1)|| (!is_approx_zero(term/sum6));
#else
(n==1)|| (!is_zero(term/sum6));
#endif
n+=1)
{ n2 = n*n;
qpower = exp(-ny) * bigcomplex(cos(nx),sin(nx));
nx += x; ny += y;
term = n*n2*qpower/(one-qpower);
sum4 += term;
term *= n2;
sum6 += term;
}
c4= (one + to_bigfloat(240)*sum4)*f4;
c6= (one - to_bigfloat(504)*sum6)*f4*f2;
}
bigcomplex discriminant(const bigcomplex& b, const bigcomplex& c, const bigcomplex& d)
{
bigcomplex bb = b*b, cc = c*c, bc = b*c;
return to_bigfloat(27)*d*d - bc*bc + to_bigfloat(4)*bb*b*d
- to_bigfloat(18)*bc*d + to_bigfloat(4)*c*cc;
}
bigcomplex cube_root(const bigcomplex& z)
{
#ifdef DEBUG_CUBIC
cout << "Taking complex cube root of z = "<<z<<endl;
#endif
if(is_zero(z)) return z;
return exp(log(z)/to_bigfloat(3));
}
vector<bigcomplex> solvecubic(const bigcomplex& c1, const bigcomplex& c2, const bigcomplex& c3)
{
#ifdef DEBUG_CUBIC
cout << "In solvecubic with c1 = "<<c1<<", c2 = "<<c2<<", c3 = "<<c3<<"\n";
#endif
long i, iter, niter=2; // number of iterations in Newton refinement of roots
bigfloat three(to_bigfloat(3)), two(to_bigfloat(2)), one(to_bigfloat(1));
bigfloat third = one/three;
bigcomplex w = bigcomplex(to_bigfloat(-1), sqrt(three))/two;
bigcomplex disc = discriminant(c1,c2,c3);
bigcomplex p3= three*c2 - c1*c1;
bigcomplex mc1over3 = -c1*third;
#ifdef DEBUG_CUBIC
cout << "p3 = "<<p3<<", -c1/3 = "<<mc1over3<<"\n";
#endif
vector<bigcomplex> roots(3);
if (is_zero(abs(disc)))
{if (is_zero(abs(p3))) // triple root
{roots[0]=roots[1]=roots[2]= mc1over3;
}
else // double root
{roots[0]=roots[1]= (c1*c2 - to_bigfloat(9)*c3)/( p3+p3);
roots[2]=-( roots[0] + roots[0]+c1);
}
}
else // distinct roots
{
bigcomplex q = (((mc1over3+c1)*mc1over3 +c2)*mc1over3 +c3);
// = F(mc1over3);
if (is_approx_zero(abs(p3))) // pure cubic
{
roots[0]=-cube_root(q);
roots[1]=w*roots[0];
roots[2]=w*roots[1];
roots[0]+=mc1over3;
roots[1]+=mc1over3;
roots[2]+=mc1over3;
}
else
{bigcomplex d = to_bigfloat(729)*q*q+ to_bigfloat(4)*p3*p3*p3;
#ifdef DEBUG_CUBIC
cout << "q = " << q << ", p3 = " << p3 << ", d = " << d << "\n";
#endif
bigcomplex t1cubed = to_bigfloat(0.5)*(sqrt(d)- to_bigfloat(27)*q);
#ifdef DEBUG_CUBIC
cout << "t1cubed = " << t1cubed << endl;
#endif
if (is_approx_zero(abs(t1cubed))) // approximately pure cubic
{
#ifdef DEBUG_CUBIC
cout << "t1cubed approx 0 so treating as pure cubic " << endl;
#endif
roots[0]=-cube_root(q);
roots[1]=w*roots[0];
roots[2]=w*roots[1];
roots[0]+=mc1over3;
roots[1]+=mc1over3;
roots[2]+=mc1over3;
for(i=0; i<3; i++)
{
bigcomplex z = roots[i], fz, fdashz;
for(iter=0; iter<niter; iter++)
{
fz = ((z+c1)*z+c2)*z+c3;
fdashz = (three*z+two*c1)*z+c2;
if(!is_zero(fdashz)) z -= fz/fdashz;
}
roots[i] = z;
}
}
else
{
bigcomplex t1 = cube_root(t1cubed);
bigcomplex t2 = t1*w;
bigcomplex t3 = t2*w;
#ifdef DEBUG_CUBIC
cout<<"t1cubed = "<<t1cubed<<"\n";
cout<<"t1 = "<<t1<<", t2 = "<<t2<<", t3 = "<<t3<<"\n";
#endif
roots[0] = (-c1+t1-p3/t1)* third;
roots[1] = (-c1+t2-p3/t2)* third;
roots[2] = (-c1+t3-p3/t3)* third;
}
}
}
#ifdef DEBUG_CUBIC
cout << "refining roots using Newton with " << niter << "iterations\n";
cout << "unrefined roots: ";
for(i=0; i<3;i++) cout << roots[i] << "\n";
#endif
for(i=0; i<3; i++)
{
bigcomplex z = roots[i], fz, fdashz;
for(iter=0; iter<niter; iter++)
{
fz = ((z+c1)*z+c2)*z+c3;
fdashz = (three*z+two*c1)*z+c2;
if(!is_zero(fdashz)) z -= fz/fdashz;
}
roots[i] = z;
}
#ifdef DEBUG_CUBIC
cout << "refined roots: ";
for(i=0; i<3;i++) cout << roots[i] << "\n";
#endif
return roots;
}
vector<bigcomplex> solverealquartic(const bigfloat& a, const bigfloat& b, const bigfloat& c, const bigfloat& d, const bigfloat& e)
{
#ifdef DEBUG
cout<<"In solverealquartic with (a,b,c,d,e)=("<<a<<","<<b<<","<<c<<","<<d<<","<<e<<")\n";
#endif
bigfloat three = to_bigfloat(3);
bigfloat ii = 12*a*e - 3*b*d + c*c;
bigfloat jj = (72*a*e + 9*b*d - 2*c*c) * c - 27*(a*d*d + b*b*e);
#ifdef DEBUG
cout<<"ii="<<ii<<"\njj="<<jj<<"\n";
#endif
bigfloat disc = 4*ii*ii*ii-jj*jj;
bigfloat H = 8*a*c - 3*b*b, R = b*b*b + 8*d*a*a - 4*a*b*c;
bigfloat Q = H*H-16*a*a*ii; // = 3*Q really
int type;
if(disc<0)
{type=3;} // 2 real roots
else
{
if((H<0)&&(Q>0))
{type=2;} // 4 real roots
else
{type=1;} // 0 real roots
}
bigcomplex c1(to_bigfloat(0)), c2(-3*ii), c3(jj);
#ifdef DEBUG
int nrr = (type=1? 0 : (type=2? 4 : 2));
cout<<"Type = " << type << " ("<<nrr<<" real roots)\n";
cout<<"Coeffs of resolvent cubic are:\n"<<c1<<"\n"<<c2<<"\n"<<c3<<endl;
#endif
vector<bigcomplex> cphi = solvecubic( c1, c2, c3);
vector<bigcomplex> roots(4);
bigfloat a4=4*a;
bigfloat oneover4a = to_bigfloat(1)/a4;
#ifdef DEBUG
cout<<"Roots of cubic are:\n"<<cphi<<endl;
#endif
if(type<3)
{
#ifdef DEBUG
cout<<"Positive discriminant\n";
#endif
// all the phi are real; order them so that a*phi[i] decreases
bigfloat phi1 = real(cphi[0]);
bigfloat phi2 = real(cphi[1]);
bigfloat phi3 = real(cphi[2]);
if(a>0) orderreal(phi1,phi2,phi3);
else orderreal(phi3,phi2,phi1);
#ifdef DEBUG
cout<<"phi = "<<phi1<<", "<<phi2<<", "<<phi3<<"\n";
#endif
if(type==2) // all roots are real
{
#ifdef DEBUG
cout<<"Type 2\n";
#endif
bigfloat r1 = sqrt((a4*phi1-H)/three);
bigfloat r2 = sqrt((a4*phi2-H)/three);
bigfloat r3 = sqrt((a4*phi3-H)/three);
if(R<0) r3 = -r3;
#ifdef DEBUG
cout<<"r_i = "<<r1<<", "<<r2<<", "<<r3<<"\n";
cout<<"product = "<<r1*r2*r3<<", R = "<<R<<endl;
#endif
roots[0] = bigcomplex(( r1 + r2 - r3 - b) * oneover4a);
roots[1] = bigcomplex(( r1 - r2 + r3 - b) * oneover4a);
roots[2] = bigcomplex((-r1 + r2 + r3 - b) * oneover4a);
roots[3] = bigcomplex((-r1 - r2 - r3 - b) * oneover4a);
// Those are all real and in descending order of size
}
else // no roots are real
{
#ifdef DEBUG
cout<<"Type 1\n";
#endif
bigfloat r1 = sqrt((a4*phi1-H)/3);
bigfloat ir2 = sqrt(-((a4*phi2-H)/3));
bigfloat ir3 = sqrt(-((a4*phi3-H)/3));
if(R>0) r1 = -r1;
#ifdef DEBUG
cout<<"r_i = "<<r1<<", "<<ir2<<"i, "<<ir3<<"i\n";
cout<<"product = "<<-r1*ir2*ir3<<", R = "<<R<<endl;
#endif
roots[0] = bigcomplex( r1-b, ir2 - ir3) * oneover4a;
roots[1] = conj(roots[0]); // bigcomplex( r1-b, ir2 - ir2) * oneover4a;
roots[2] = bigcomplex(-r1-b, ir2 + ir3 ) * oneover4a;
roots[3] = conj(roots[2]); // bigcomplex(-r1-b, -ir2 - ir3 ) * oneover4a;
}
}
else // disc < 0
{
#ifdef DEBUG
cout<<"Negative discriminant\nType 3\n";
#endif
bigfloat realphi; // will hold the real root, which will be cphi[2]
if (is_real(cphi[1]))
{
realphi=real(cphi[1]);
cphi[1]=cphi[2];
cphi[2]=realphi;
}
else
if (is_real(cphi[2]))
{
realphi=real(cphi[2]);
}
else
{
realphi=real(cphi[0]);
cphi[0]=cphi[2];
cphi[2]=realphi;
}
#ifdef DEBUG
cout<<"Sorted roots of cubic (real one last) are \n";
cout<<cphi[0]<<"\n"<<cphi[1]<<"\n"<<cphi[2]<<endl;
#endif
bigcomplex r1 = sqrt((a4*cphi[0]-H)/three);
bigfloat r3 = sqrt((a4*realphi-H)/three);
if(R<0) r3 = -r3;
#ifdef DEBUG
cout<<"r_i = "<<r1<<", "<<conj(r1)<<", "<<r3<<"\n";
cout<<"product = "<<r1*conj(r1)*r3<<", R = "<<R<<endl;
#endif
roots[0] = bigcomplex( r3 - b, 2*imag(r1) ) * oneover4a;
roots[1] = conj(roots[0]);
roots[2] = bigcomplex(( 2*real(r1) - r3 - b)) * oneover4a;
roots[3] = bigcomplex((-2*real(r1) - r3 - b)) * oneover4a;
// roots[2] and roots[3] are real
}
return roots;
}
vector<bigcomplex> solvequartic(const bigcomplex& a, const bigcomplex& b, const bigcomplex& c, const bigcomplex& d)
{ bigcomplex p,q,r,aa,e,f1,f2;
static bigfloat zero = to_bigfloat(0);
static bigfloat two = to_bigfloat(2);
static bigfloat three = to_bigfloat(3);
static bigfloat four = to_bigfloat(4);
static bigfloat eight = to_bigfloat(8);
static bigfloat x16 = to_bigfloat(16);
static bigfloat x64 = to_bigfloat(64);
static bigfloat x256 = to_bigfloat(256);
bigcomplex a4=a/four;
vector<bigcomplex> roots(4);
if(is_zero(d))
{
roots[0]= zero;
vector<bigcomplex> cuberoots=solvecubic(a,b,c);
roots[1] = cuberoots[0];
roots[2] = cuberoots[1];
roots[3] = cuberoots[2];
}
else
{
p = b - three*a*a / eight;
q = ((a / two) * (a*a4 - b)) + c;
r = ((x256*d) - (x64*a*c) + (x16*a*a*b) - (three*a*a*a*a)) / x256;
if( is_approx_zero(q) )
{
bigcomplex s = sqrt(p*p - four*r);
roots[0] = sqrt((-p + s) / two) - a4;
roots[1] = -sqrt((-p + s) / two) - a4;
roots[2] = sqrt((-p - s) / two) - a4;
roots[3] = -sqrt((-p - s) / two) - a4;
}
else
{
vector<bigcomplex> aaroots=
solvecubic(-p / two,-r,((p * r) / two - (q * q) / eight));
aa = aaroots[0];
if( is_approx_zero(aa) ) aa=zero;
e = sqrt(-p + two*aa);
f1 = (aa + q / (two*e));
f2 = (aa - q / (two*e));
bigcomplex s1 = sqrt(e*e - four*f1);
bigcomplex s2 = sqrt(e*e - four*f2);
roots[0] = (( e + s1) / two) - a4;
roots[1] = (( e - s1) / two) - a4;
roots[2] = ((-e + s2) / two) - a4;
roots[3] = ((-e - s2) / two) - a4;
}
}
long i, iter, niter=2; // number of iterations in Newton refinement of roots
#ifdef DEBUG_QUARTIC
cout << "refining roots using Newton with " << niter << "iterations\n";
cout << "unrefined roots: ";
for(i=0; i<4;i++) cout << roots[i] << "\n";
#endif
for(i=0; i<4; i++)
{
bigcomplex z = roots[i], fz, fdashz;
for(iter=0; iter<niter; iter++)
{
fz = (((z+a)*z+b)*z+c)*z+d;
fdashz = ((four*z+three*a)*z+two*b)*z+c;
if(!is_zero(fdashz)) z -= fz/fdashz;
}
roots[i] = z;
}
#ifdef DEBUG_QUARTIC
cout << "refined roots: ";
for(i=0; i<4;i++) cout << roots[i] << "\n";
#endif
return roots;
}
void quadsolve(const bigfloat& p, const bigfloat& q,
bigcomplex& root1, bigcomplex& root2)
{
static bigfloat two = to_bigfloat(2);
static bigfloat four = to_bigfloat(4);
bigcomplex disc(p*p- four*q);
bigcomplex rootdisc = sqrt(disc);
root1 = ( rootdisc-p)/ two;
root2 = (-rootdisc-p)/ two;
}
vector<long> introotscubic(long a, long b, long c, int& nr)
{ bigcomplex za(to_bigfloat(a)), zb(to_bigfloat(b)), zc(to_bigfloat(c));
vector<bigcomplex> croots = solvecubic(za,zb,zc);
vector<long> iroots;
int i; long x,cx;
for (i=0; i<3; i++)
{
cout << "Complex root = " << croots[i] << endl;
bigfloat xx = croots[i].real();
Iasb(x,xx);
cout << "Rounds to " << x << endl;
if (x==0) {if (c==0) iroots.push_back(x);}
else
{
cx = c/x;
if (x*cx==c)
if (((x+a)*x+b+cx) == 0)
iroots.push_back(x);
}
}
return iroots;
}
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