int main(int argc, char * argv[])
{
unsigned int counts;
double ce[4]= {1.87832,-0.0950376,-1.87832,-0.882024};
unsigned int j=5;
if(argc < j ) j=argc;
for(unsigned int i=1; i< j; i++) {
ce[i-1]=strtod(argv[i],NULL);
}
std::cout<<"x^4";
if(argc == 1) j=4;
else j--;
for(unsigned int i=0; i< j; i++) {
std::cout<<" + ("<<ce[i]<<")";
if ( i != 3 ) {
std::cout<<"*x^"<<3-i;
} else {
std::cout<<" =0\n";
}
}
double roots[4];
switch (j) {
case 4:
std::cout<<"quadratic:\n";
counts=quarticSolver(ce,roots);
break;
case 3:
std::cout<<"cubic:\n";
counts=cubicSolver(ce,roots);
break;
default:
std::cout<<"quartic:\n";
counts=quadraticSolver(ce,roots);
}
std::cout<<"Number of real roots="<<counts<<std::endl;
for(unsigned int i=0; i<counts; i++) {
std::cout<<"x= "<<roots[i]<<'\t';
}
std::cout<<std::endl;
return 0;
}
unsigned int RS_Math::quarticSolver(double * ce, double *roots)
//quartic solver
// x^4 + ce[0] x^3 + ce[1] x^2 + ce[2] x + ce[3] = 0
{
// x^4 + a x^3 + b x^2 +c x + d = 0
// depressed quartic, x= t - a/4
// t^4 + ( b - 3/8 a^2 ) t^2 + (c - a b/2 + a^3/8) t + d - a c /4 + a^2 b/16 - 3 a^4/256 =0
// t^4 + p t^2 + q t + r =0
// p= b - (3./8)*a*a;
// q= c - 0.5*a*b+(1./8)*a*a*a;
// r= d - 0.25*a*c+(1./16)*a*a*b-(3./256)*a^4
unsigned int ret=0;
double shift=0.25*ce[0];
double shift2=shift*shift;
double a2=ce[0]*ce[0];
double p= ce[1] - (3./8)*a2;
double q= ce[2] + ce[0]*((1./8)*a2 - 0.5*ce[1]);
double r= ce[3] - shift*ce[2] + (ce[1] - 3.*shift2)*shift2;
// std::cout<<"quartic_solver:: p="<<p<<"\tq="<<q<<"\tr="<<r<<std::endl;
if (fabs(q) <= RS_TOLERANCE) {// Biquadratic equations
double discriminant= 0.25*p*p -r;
if (discriminant < 0.) {
return 0;
}
double t2[2];
t2[0]=-0.5*p-sqrt(discriminant);
t2[1]= -p - t2[0];
// std::cout<<"t2[0]="<<t2[0]<<std::endl;
// std::cout<<"t2[1]="<<t2[1]<<std::endl;
if ( t2[0] >= 0. ) {// four real roots
roots[0]=sqrt(t2[0])-shift;
roots[1]= -sqrt(t2[0])-shift;
roots[2]=sqrt(t2[1])-shift;
roots[3]= -sqrt(t2[1])-shift;
return 4;
}
if ( t2[1] >= 0.) { // two real roots
roots[0]=sqrt(t2[1])-shift;
roots[1]= -sqrt(t2[1])-shift;
return 2;
}
return 0;
}
if ( fabs(r)< 1.0e-75 ) {
double cubic[3]= {0.,p,q};
roots[0]=0.;
ret=1+cubicSolver(cubic,roots+1);
for(unsigned int i=0; i<ret; i++) roots[i] -= shift;
return ret;
}
// depressed quartic to two quadratic equations
// t^4 + p t^2 + q t + r = ( t^2 + u t + v) ( t^2 - u t + w)
// so,
// p + u^2= w+v
// q/u= w-v
// r= wv
// so,
// (p+u^2)^2 - (q/u)^2 = 4 r
// y=u^2,
// y^3 + 2 p y^2 + ( p^2 - 4 r) y - q^2 =0
//
double cubic[3]= {2.*p,p*p-4.*r,-q*q},croots[3];
ret = cubicSolver(cubic,croots);
//std::cout<<"quartic_solver:: real roots from cubic: "<<ret<<std::endl;
//for(unsigned int i=0; i<ret; i++)
// std::cout<<"cubic["<<i<<"]="<<cubic[i]<<" x= "<<croots[i]<<std::endl;
if (ret==1) { //one real root from cubic
if (croots[0]< 0.) {//this should not happen
std::cerr<<"Quartic Error:: Found one real root for cubic, but negative\n";
return 0;
}
double sqrtz0=sqrt(croots[0]);
double ce2[2];
ce2[0]= -sqrtz0;
ce2[1]=0.5*(p+croots[0])+0.5*q/sqrtz0;
ret=quadraticSolver(ce2,roots);
if (! ret ) {
ce2[0]= sqrtz0;
ce2[1]=0.5*(p+croots[0])-0.5*q/sqrtz0;
ret=quadraticSolver(ce2,roots);
}
ret=2;
for(unsigned int i=0; i<ret; i++) roots[i] -= shift;
return ret;
}
if ( croots[0]> 0. && croots[1] > 0. ) {
double sqrtz0=sqrt(croots[0]);
double ce2[2];
ce2[0]= -sqrtz0;
ce2[1]=0.5*(p+croots[0])+0.5*q/sqrtz0;
ret=quadraticSolver(ce2,roots);
ce2[0]= sqrtz0;
ce2[1]=0.5*(p+croots[0])-0.5*q/sqrtz0;
ret=quadraticSolver(ce2,roots+2);
ret=4;
for(unsigned int i=0; i<ret; i++) roots[i] -= shift;
return ret;
}
return 0;
}
unsigned int RS_Math::cubicSolver(double * ce, double *roots)
//cubic equation solver
// x^3 + ce[0] x^2 + ce[1] x + ce[2] = 0
{
// depressed cubic, Tschirnhaus transformation, x= t - b/(3a)
// t^3 + p t +q =0
unsigned int ret=0;
double shift=(1./3)*ce[0];
double p=ce[1] -shift*ce[0];
double q=ce[0]*( (2./27)*ce[0]*ce[0]-(1./3)*ce[1])+ce[2];
//Cardano's method,
// t=u+v
// u^3 + v^3 + ( 3 uv + p ) (u+v) + q =0
// select 3uv + p =0, then,
// u^3 + v^3 = -q
// u^3 v^3 = - p^3/27
// so, u^3 and v^3 are roots of equation,
// z^2 + q z - p^3/27 = 0
// and u^3,v^3 are,
// -q/2 \pm sqrt(q^2/4 + p^3/27)
// discriminant= q^2/4 + p^3/27
//std::cout<<"p="<<p<<"\tq="<<q<<std::endl;
double discriminant= (1./27)*p*p*p+(1./4)*q*q;
if ( fabs(p)< 1.0e-75) {
ret=1;
*roots=(q>0)?-pow(q,(1./3)):pow(-q,(1./3));
*roots -= shift;
return ret;
}
//std::cout<<"discriminant="<<discriminant<<std::endl;
if(discriminant>0) {
double ce2[2]= {q, -1./27*p*p*p},u3[2];
ret=quadraticSolver(ce2,u3);
if (! ret ) { //should not happen
std::cerr<<"cubicSolver()::Error cubicSolver("<<ce[0]<<' '<<ce[1]<<' '<<ce[2]<<")\n";
}
ret=1;
double u,v;
u= (q<=0) ? pow(u3[0], 1./3): -pow(-u3[1],1./3);
//u=(q<=0)?pow(-0.5*q+sqrt(discriminant),1./3):-pow(0.5*q+sqrt(discriminant),1./3);
v=(-1./3)*p/u;
//std::cout<<"u="<<u<<"\tv="<<v<<std::endl;
//std::cout<<"u^3="<<u*u*u<<"\tv^3="<<v*v*v<<std::endl;
*roots=u+v - shift;
return ret;
}
ret=3;
std::complex<double> u(q,0),rt[3];
u=std::pow(-0.5*u-sqrt(0.25*u*u+p*p*p/27),1./3);
rt[0]=u-p/(3.*u)-shift;
std::complex<double> w(-0.5,sqrt(3.)/2);
rt[1]=u*w-p/(3.*u*w)-shift;
rt[2]=u/w-p*w/(3.*u)-shift;
// std::cout<<"Roots:\n";
// std::cout<<rt[0]<<std::endl;
// std::cout<<rt[1]<<std::endl;
// std::cout<<rt[2]<<std::endl;
roots[0]=rt[0].real();
roots[1]=rt[1].real();
roots[2]=rt[2].real();
return ret;
}