polynomial.h

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#ifndef __POLYNOMIAL_H
#define __POLYNOMIAL_H
#include "math.h"
#include "complex.h"
double cbrt(double x){//Kubikwurzel
        if (x<0) return -pow(-x,1.0/3.0);
        return pow(x,1.0/3.0);
}

//      double epsilon=0.0000001;

//x1/2=(-b+-sqrt(b^2-4ac))/(2a)
int solve_quadratic(double a,double b,double c,double* x1,double* x2)
{
        if (a==0)//Lineares Gleichungssystem
        {
                if (b==0) return 0;//c ist konstant
                *x1=-c/b;
                return 1;
        }
        double d=b*b-4*a*c;
        if (d<0) return 0;
        *x1=(-b+sqrt(d))/2*a;
        *x2=(-b-sqrt(d))/2*a;
        return 2;


}

//Solve cubic nach http://www.montgelas-gymnasium.de/mathe/kubfa/leitkubgleich.html
int solve_cubic(double a,double b,double c,double d,double* x1,double* x2,double* x3)
{       if (a==0) return solve_quadratic(b,c,d,x1,x2);
        double pi=3.141592653589793;
        double p=(3*a*c-b*b)/(3*a*a);
        double q=(2*b*b*b)/(27*a*a*a)-(b*c)/(3*a*a)+d/a;
        double D=(q*q/4)+(p*p*p/27);
        if (D>epsilon) 
        { //Nur eine reelle L�ung
                *x1=cbrt(-q/2+sqrt(D))+cbrt(-q/2-sqrt(D))-b/(3*a);
                *x2=HUGE_VAL;
                *x3=HUGE_VAL;
                return 1;
        }
        else if (D<-epsilon)
        {
                double alpha=acos(-q/(2*sqrt(-p*p*p/27)));
                *x1=2*sqrt(-p/3)*cos(alpha/3+0.0/3.0*pi)-b/(3*a);
                *x2=2*sqrt(-p/3)*cos(alpha/3+2.0/3.0*pi)-b/(3*a);
                *x3=2*sqrt(-p/3)*cos(alpha/3+4.0/3.0*pi)-b/(3*a);
                return 3;
        }
        else //D is nearby zero
        {
                if (abs(p)<epsilon)
                {
                        *x1=*x2=*x3=-b/(3*a);
                        return 3;
                } else
                {       *x1=cbrt(-4*q)-b/(3*a);
                        *x2=*x3=cbrt(q/2)-b/(3*a);
                        return 3;
                }
        }
}

double sqrt2(double x)
{
        return sqrt(abs(x));
}


void setpos(int p,complex a,double* x1,double* x2,double* x3,double* x4)
{
switch (p)
{
case 1:*x1=a.r;break;
case 2:*x2=a.r;break;
case 3:*x3=a.r;break;
case 4:*x4=a.r;
}
}

//Nach http://de.wikipedia.org/wiki/Biquadratische_Gleichung
int solve_biquadratic(double a4,double a3,double a2,double a1,double a0,double* x1,double* x2,double* x3,double* x4)
{
        if (a4==0) return solve_cubic(a3,a2,a1,a0,x1,x2,x3);
        if ((a3==0) && (a1==0))
        {
                if (solve_quadratic(a4,a2,a0,x1,x3)==0) return 0;
                int s=0;
                if (*x1>=0)
                {
                        *x1=sqrt(*x1);
                        *x2=-*x1;
                        s+=2;
                }
                if (*x3>=0)
                {
                        if (s==0)
                        {
                        *x1=sqrt(*x3);
                        *x2=-*x1;
                        s+=2;
                        } else
                        {
                        *x3=sqrt(*x3);
                        *x4=-*x3;
                        s+=2;
                        }
                }
                return s;
        }
        complex A=r2c(a4);
        complex B=r2c(a3);
        complex C=r2c(a2);
        complex D=r2c(a1);
        complex E=r2c(a0);
        complex a = cadd(cdiv (cmul(cmul(r2c(-3),B),B) , cmul(cmul(r2c(8),A),A)) , cdiv(C,A));

        complex b = cadd(csub(cdiv(cpow(B,3),cmul(r2c(8),cpow(A,3))),
                                  cdiv(cmul(B,C),cmul(cmul(r2c(2),A),A))),
                                  cdiv(D,A));

        complex c = cadd(csub(cadd(cdiv(cmul(r2c(-3),cpow(B,4)),cmul(r2c(256),cpow(A,4))),
                        cdiv(cmul(C,cmul(B,B)),cmul(r2c(16),cpow(A,3)))),
                                cdiv(cmul(B,D),cmul(cmul(r2c(4),A),A))),
                                cdiv(E,A));

        if (isEqualE(b,czero))
        {   if (solve_quadratic(1,a.r,c.r,x1,x3)==0) return 0;
                int s=0;
                if (*x1>=0)
                {
                        *x1=sqrt(*x1);
                        *x2=-*x1-a3/(4*a4);
                        *x1-=a3/(4*a4);
                        s+=2;
                }
                if (*x3>=0)
                {
                        if (s==0)
                        {
                        *x1=sqrt(*x3);
                        *x2=-*x1-a3/(4*a4);
                        *x1-=a3/(4*a4);
                        s+=2;
                        } else
                        {
                        *x3=sqrt(*x3);
                        *x4=-*x3-a3/(4*a4);
                        *x3-=a3/(4*a4);
                        s+=2;
                        }
                }
                return s;
        }
        complex P = csub(cdiv(cmul(a,a),r2c(-12)),c);
        complex Q = csub(cadd(cdiv(cpow(a,3),r2c(-108)),cdiv(cmul(a,c),r2c(3))),cdiv(cmul(b,b),r2c(8)));
        complex R = cadd(cdiv(Q,r2c(2)),csqrt( cadd(cdiv(cmul(Q,Q),r2c(4)),cdiv(cpow(P,3),r2c(27))) ));
        complex U = czero;
        if (isEqualE(R,czero))
        {
                U=czero;
        } else
        {
                 U = cexp(cdiv(cln(R),r2c(3)));
        }
        complex y=csub(cmul(r2c(-5.0/6.0),a),U);
        if (!isEqualE(U,czero))
        {
                y=cadd(y,cdiv(P,cmul(r2c(3),U)));
        }
        //          -B/(4*A) + (sqrt(a+2*y) + sqrt(-(a+2*y) - 2*(a + b/sqrt(a+2*y) )))/2;
        //            L1            L2       L5/L7   L3      |        L4/L6
        complex a2y = cadd(a,cmul(r2c(2),y));
        complex L1 = cneg(cdiv(B,cmul(r2c(4),A)));
        complex L2 = csqrt(a2y);
        complex L3 = cneg(a2y);
        complex L4 = cmul(r2c(2),cadd(a,cdiv(b,L2)));
        complex L5 = csqrt(csub(L3,L4));
        complex L6 = cmul(r2c(2),csub(a,cdiv(b,L2)));
        complex L7 = csqrt(csub(L3,L6));
        complex X1 = cadd(L1,cdiv(cadd(     L2 ,     L5),r2c(2)));
        complex X2 = cadd(L1,cdiv(cadd(cneg(L2),     L7),r2c(2)));
        complex X3 = cadd(L1,cdiv(cadd(     L2 ,cneg(L5)),r2c(2)));
        complex X4 = cadd(L1,cdiv(cadd(cneg(L2),cneg(L7)),r2c(2)));
        int s=0;
        if (isReal(X1)) {s++;setpos(s,X1,x1,x2,x3,x4);}
        if (isReal(X2)) {s++;setpos(s,X2,x1,x2,x3,x4);}
        if (isReal(X3)) {s++;setpos(s,X3,x1,x2,x3,x4);}
        if (isReal(X4)) {s++;setpos(s,X4,x1,x2,x3,x4);}
        return s;
}
#endif