Factor.cpp

/*
Copyright (c) 2006, Michael Kazhdan and Matthew Bolitho
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*/

//////////////////////
// Polynomial Roots //
//////////////////////
#include <math.h>
#include "Factor.h"
int Factor(double a1,double a0,double roots[1][2],const double& EPS){
	if(fabs(a1)<=EPS){return 0;}
	roots[0][0]=-a0/a1;
	roots[0][1]=0;
	return 1;
}
int Factor(double a2,double a1,double a0,double roots[2][2],const double& EPS){
	double d;
	if(fabs(a2)<=EPS){return Factor(a1,a0,roots,EPS);}

	d=a1*a1-4*a0*a2;
	a1/=(2*a2);
	if(d<0){
		d=sqrt(-d)/(2*a2);
		roots[0][0]=roots[1][0]=-a1;
		roots[0][1]=-d;
		roots[1][1]= d;
	}
	else{
		d=sqrt(d)/(2*a2);
		roots[0][1]=roots[1][1]=0;
		roots[0][0]=-a1-d;
		roots[1][0]=-a1+d;
	}
	return 2;
}
// Solution taken from: http://mathworld.wolfram.com/CubicFormula.html
// and http://www.csit.fsu.edu/~burkardt/f_src/subpak/subpak.f90
int Factor(double a3,double a2,double a1,double a0,double roots[3][2],const double& EPS){
	double q,r,r2,q3;

	if(fabs(a3)<=EPS){return Factor(a2,a1,a0,roots,EPS);}
	a2/=a3;
	a1/=a3;
	a0/=a3;

	q=-(3*a1-a2*a2)/9;
	r=-(9*a2*a1-27*a0-2*a2*a2*a2)/54;
	r2=r*r;
	q3=q*q*q;

	if(r2<q3){
		double sqrQ=sqrt(q);
		double theta = acos ( r / (sqrQ*q) );
		double cTheta=cos(theta/3)*sqrQ;
		double sTheta=sin(theta/3)*sqrQ*SQRT_3/2;
		roots[0][1]=roots[1][1]=roots[2][1]=0;
		roots[0][0]=-2*cTheta;
		roots[1][0]=-2*(-cTheta*0.5-sTheta);
		roots[2][0]=-2*(-cTheta*0.5+sTheta);
	}
	else{
		double s1,s2,sqr=sqrt(r2-q3);
		double t;
		t=-r+sqr;
		if(t<0){s1=-pow(-t,1.0/3);}
		else{s1=pow(t,1.0/3);}
		t=-r-sqr;
		if(t<0){s2=-pow(-t,1.0/3);}
		else{s2=pow(t,1.0/3);}
		roots[0][1]=0;
		roots[0][0]=s1+s2;
		s1/=2;
		s2/=2;
		roots[1][0]= roots[2][0]=-s1-s2;
		roots[1][1]= SQRT_3*(s1-s2);
		roots[2][1]=-roots[1][1];
	}
	roots[0][0]-=a2/3;
	roots[1][0]-=a2/3;
	roots[2][0]-=a2/3;
	return 3;
}
double ArcTan2(const double& y,const double& x){
	/* This first case should never happen */
	if(y==0 && x==0){return 0;}
	if(x==0){
		if(y>0){return PI/2.0;}
		else{return -PI/2.0;}
	}
	if(x>=0){return atan(y/x);}
	else{
		if(y>=0){return atan(y/x)+PI;}
		else{return atan(y/x)-PI;}
	}
}
double Angle(const double in[2]){
	if((in[0]*in[0]+in[1]*in[1])==0.0){return 0;}
	else{return ArcTan2(in[1],in[0]);}
}
void Sqrt(const double in[2],double out[2]){
	double r=sqrt(sqrt(in[0]*in[0]+in[1]*in[1]));
	double a=Angle(in)*0.5;
	out[0]=r*cos(a);
	out[1]=r*sin(a);
}
void Add(const double in1[2],const double in2[2],double out[2]){
	out[0]=in1[0]+in2[0];
	out[1]=in1[1]+in2[1];
}
void Subtract(const double in1[2],const double in2[2],double out[2]){
	out[0]=in1[0]-in2[0];
	out[1]=in1[1]-in2[1];
}
void Multiply(const double in1[2],const double in2[2],double out[2]){
	out[0]=in1[0]*in2[0]-in1[1]*in2[1];
	out[1]=in1[0]*in2[1]+in1[1]*in2[0];
}
void Divide(const double in1[2],const double in2[2],double out[2]){
	double temp[2];
	double l=in2[0]*in2[0]+in2[1]*in2[1];
	temp[0]= in2[0]/l;
	temp[1]=-in2[1]/l;
	Multiply(in1,temp,out);
}
// Solution taken from: http://mathworld.wolfram.com/QuarticEquation.html
// and http://www.csit.fsu.edu/~burkardt/f_src/subpak/subpak.f90
int Factor(double a4,double a3,double a2,double a1,double a0,double roots[4][2],const double& EPS){
	double R[2],D[2],E[2],R2[2];

	if(fabs(a4)<EPS){return Factor(a3,a2,a1,a0,roots,EPS);}
	a3/=a4;
	a2/=a4;
	a1/=a4;
	a0/=a4;

	Factor(1.0,-a2,a3*a1-4.0*a0,-a3*a3*a0+4.0*a2*a0-a1*a1,roots,EPS);

	R2[0]=a3*a3/4.0-a2+roots[0][0];
	R2[1]=0;
	Sqrt(R2,R);
	if(fabs(R[0])>10e-8){
		double temp1[2],temp2[2];
		double p1[2],p2[2];

		p1[0]=a3*a3*0.75-2.0*a2-R2[0];
		p1[1]=0;

		temp2[0]=((4.0*a3*a2-8.0*a1-a3*a3*a3)/4.0);
		temp2[1]=0;
		Divide(temp2,R,p2);

		Add     (p1,p2,temp1);
		Subtract(p1,p2,temp2);

		Sqrt(temp1,D);
		Sqrt(temp2,E);
	}
	else{
		R[0]=R[1]=0;
		double temp1[2],temp2[2];
		temp1[0]=roots[0][0]*roots[0][0]-4.0*a0;
		temp1[1]=0;
		Sqrt(temp1,temp2);
		temp1[0]=a3*a3*0.75-2.0*a2+2.0*temp2[0];
		temp1[1]=                  2.0*temp2[1];
		Sqrt(temp1,D);
		temp1[0]=a3*a3*0.75-2.0*a2-2.0*temp2[0];
		temp1[1]=                 -2.0*temp2[1];
		Sqrt(temp1,E);
	}

	roots[0][0]=-a3/4.0+R[0]/2.0+D[0]/2.0;
	roots[0][1]=        R[1]/2.0+D[1]/2.0;

	roots[1][0]=-a3/4.0+R[0]/2.0-D[0]/2.0;
	roots[1][1]=        R[1]/2.0-D[1]/2.0;

	roots[2][0]=-a3/4.0-R[0]/2.0+E[0]/2.0;
	roots[2][1]=       -R[1]/2.0+E[1]/2.0;

	roots[3][0]=-a3/4.0-R[0]/2.0-E[0]/2.0;
	roots[3][1]=       -R[1]/2.0-E[1]/2.0;
	return 4;
}

int Solve(const double* eqns,const double* values,double* solutions,const int& dim){
	int i,j,eIndex;
	double v,m;
	int *index=new int[dim];
	int *set=new int[dim];
	double* myEqns=new double[dim*dim];
	double* myValues=new double[dim];

	for(i=0;i<dim*dim;i++){myEqns[i]=eqns[i];}
	for(i=0;i<dim;i++){
		myValues[i]=values[i];
		set[i]=0;
	}
	for(i=0;i<dim;i++){
		// Find the largest equation that has a non-zero entry in the i-th index
		m=-1;
		eIndex=-1;
		for(j=0;j<dim;j++){
			if(set[j]){continue;}
			if(myEqns[j*dim+i]!=0 && fabs(myEqns[j*dim+i])>m){
				m=fabs(myEqns[j*dim+i]);
				eIndex=j;
			}
		}
		if(eIndex==-1){
			delete[] index;
			delete[] myValues;
			delete[] myEqns;
			delete[] set;
			return 0;
		}
		// The position in which the solution for the i-th variable can be found
		index[i]=eIndex;
		set[eIndex]=1;

		// Normalize the equation
		v=myEqns[eIndex*dim+i];
		for(j=0;j<dim;j++){myEqns[eIndex*dim+j]/=v;}
		myValues[eIndex]/=v;

		// Subtract it off from everything else
		for(j=0;j<dim;j++){
			if(j==eIndex){continue;}
			double vv=myEqns[j*dim+i];
			for(int k=0;k<dim;k++){myEqns[j*dim+k]-=myEqns[eIndex*dim+k]*vv;}
			myValues[j]-=myValues[eIndex]*vv;
		}
	}
	for(i=0;i<dim;i++){solutions[i]=myValues[index[i]];}
	delete[] index;
	delete[] myValues;
	delete[] myEqns;
	delete[] set;
	return 1;
}