/*
* COMPLEX ROOTS
*/
// deg0
public static void RootsAll(double a,
out int numRoot, out int numRootReal, out CNum[] root)
{
numRoot=(Math.Abs(a)>MConsts.EPS_DEC)? 0: Equation.NumRootInfinite;
numRootReal=numRoot;
root=null;
}
public CNum Pow(double power)
{
if (this.IsZero)
return new CNum(0,0);
CNum res=new CNum();
res.FromRadiusAngle(Math.Pow(this.Radius,power),this.Angle*power);
return res;
}
public CNum Pow(double power)
{
if (this.IsZero)
{
return(new CNum(0, 0));
}
CNum res = new CNum();
res.FromRadiusAngle(Math.Pow(this.Radius, power), this.Angle * power);
return(res);
}
// deg1
public static void RootsAll(double a, double b,
out int numRoot, out int numRootReal, out CNum[] root)
{
if (Math.Abs(a) < MConsts.EPS_DEC)
{
Equation.RootsAll(b, out numRoot, out numRootReal, out root);
}
else
{
numRoot = 1;
numRootReal = numRoot;
root = new CNum[1];
root[0] = new CNum(-b / a, 0);
}
}
// deg1
public static void RootsAll(double a, double b,
out int numRoot, out int numRootReal, out CNum[] root)
{
if (Math.Abs(a)<MConsts.EPS_DEC)
{
Equation.RootsAll(b,out numRoot,out numRootReal,out root);
}
else
{
numRoot=1;
numRootReal=numRoot;
root=new CNum[1];
root[0]=new CNum(-b/a,0);
}
}
// deg2
public static void RootsAll(double a, double b, double c,
out int numRoot, out int numRootReal, out CNum[] root)
{
// multiply root is referred as different roots;
// in case of multiply roots output:
// numRoot=2; root[0]=root[1];
// root[0].Multiplicity=root[1].Multiplicity=2
// TODO: check LOGIC above !!!
if (Math.Abs(a) < MConsts.EPS_DEC)
{
Equation.RootsAll(b, c, out numRoot, out numRootReal, out root);
}
else
{
numRoot = 2;
root = new CNum[2];
double D = b * b - 4.0 * a * c;
if (D < 0) // TODO: check D<=-MConsts.EPS_DEC
{
numRootReal = 0;
double sqrtD = Math.Sqrt(Math.Abs(D));
root[0] = new CNum(-b / (2.0 * a), -sqrtD / (2.0 * a));
root[1] = new CNum(-b / (2.0 * a), sqrtD / (2.0 * a));
}
else
{
numRootReal = 2;
if (D == 0)
{
root[0] = new CNum(-b / (2.0 * a), 0.0, 2); // TODO: CHECK MULTIPLICITY !!!
root[1] = new CNum(-b / (2.0 * a), 0.0, 2);
}
else
{
double sqrtD = Math.Sqrt(D);
root[0] = new CNum((-b - sqrtD) / (2.0 * a), 0);
root[1] = new CNum((-b + sqrtD) / (2.0 * a), 0);
}
}
}
}
// deg2
public static void RootsAll(double a, double b, double c,
out int numRoot, out int numRootReal, out CNum[] root)
{
// multiply root is referred as different roots;
// in case of multiply roots output:
// numRoot=2; root[0]=root[1];
// root[0].Multiplicity=root[1].Multiplicity=2
// TODO: check LOGIC above !!!
if (Math.Abs(a)<MConsts.EPS_DEC)
{
Equation.RootsAll(b,c,out numRoot, out numRootReal, out root);
}
else
{
numRoot = 2;
root=new CNum[2];
double D = b*b - 4.0*a*c;
if (D < 0) // TODO: check D<=-MConsts.EPS_DEC
{
numRootReal = 0;
double sqrtD = Math.Sqrt(Math.Abs(D));
root[0]=new CNum(-b/(2.0*a),-sqrtD/(2.0*a));
root[1]=new CNum(-b/(2.0*a), sqrtD/(2.0*a));
}
else
{
numRootReal = 2;
if (D==0)
{
root[0]=new CNum(-b/(2.0*a), 0.0, 2); // TODO: CHECK MULTIPLICITY !!!
root[1]=new CNum(-b/(2.0*a), 0.0, 2);
}
else
{
double sqrtD = Math.Sqrt(D);
root[0]=new CNum((-b-sqrtD)/(2.0*a), 0);
root[1]=new CNum((-b+sqrtD)/(2.0*a), 0);
}
}
}
}
/*
* INITIALIZERS
*/
public void From(CNum z)
{
this.re = z.Re;
this.im = z.Im;
this.multiplicity = z.Multiplicity;
}
public CNum(CNum z) : this(z.Re, z.Im, z.Multiplicity)
{
}
// deg3
public static void RootsAll(double a, double b, double c, double d,
out int numRoot, out int numRootReal, out CNum[] root)
{
// output:
// case of 3 real roots =>
// the roots are sorted in increasing order
// case of 1 real root => real, complex-, complex+
//
// multiply real roots referred as different roots
/*
* Catch EXACT TANGENTS when they are not obvious
* from the control points specification
*/
double rootToAdd = -1.0; // asign an invalid parametric value
int numRootRed = -1, numRootRealRed = -1; // roots after reduction
CNum[] rootRed = null;
if (Equation.Evaluate(0.0, a, b, c, d) == 0.0)
{
rootToAdd = 0.0;
Equation.RootsAll(a, b, c,
out numRootRed, out numRootRealRed, out rootRed);
}
if (Equation.Evaluate(1.0, a, b, c, d) == 0.0)
{
rootToAdd = 1.0;
Equation.RootsAll(a, a + b, a + b + c,
out numRootRed, out numRootRealRed, out rootRed);
}
if (rootToAdd != -1.0)
{
root = new CNum[numRootRed + 1];
numRoot = numRootRed + 1;
numRootReal = numRootRealRed + 1;
//
// two complex roots of the reduced (quadratic) equation
//
if ((numRootRed - numRootRealRed) == 2)
{
root[0] = new CNum(rootToAdd, 0.0);
root[1] = new CNum(rootRed[0]);
root[2] = new CNum(rootRed[1]);
return;
}
//
// all roots of the reduced equation are real
//
CNum rootNew = new CNum(rootToAdd, 0.0);
// adjust multiplicities
for (int iRootRed = 0; iRootRed < numRootRed; iRootRed++)
{
if (rootToAdd == rootRed[iRootRed].Re)
{
rootNew.Multiplicity += 1;
rootRed[iRootRed].Multiplicity += 1;
}
}
if (numRootRed == 0)
{
root[0] = new CNum(rootNew);
return;
}
if (numRootRed == 1)
{
if (rootToAdd <= rootRed[0].Re)
{
root[0] = new CNum(rootNew);
root[1] = new CNum(rootRed[0]);
}
else
{
root[0] = new CNum(rootRed[0]);
root[1] = new CNum(rootNew);
}
}
if (numRootRed == 2)
{
if (rootToAdd <= rootRed[0].Re)
{
root[0] = new CNum(rootNew);
root[1] = new CNum(rootRed[0]);
root[2] = new CNum(rootRed[1]);
}
else if (rootToAdd <= rootRed[1].Re)
{
root[0] = new CNum(rootRed[0]);
root[1] = new CNum(rootNew);
root[2] = new CNum(rootRed[1]);
}
else
{
root[0] = new CNum(rootRed[0]);
root[1] = new CNum(rootRed[1]);
root[2] = new CNum(rootNew);
}
}
return; // end - catch EXACT TANGENTS
}
if (Math.Abs(a) < MConsts.EPS_DEC)
{
Equation.RootsAll(b, c, d, out numRoot, out numRootReal, out root);
}
else
{
numRoot = 3;
root = new CNum[3];
double bb, cc, dd, pp, qq, D;
bb = b / a;
cc = c / a;
dd = d / a;
pp = -(bb * bb / 3.0) + cc;
qq = (2.0 * bb * bb * bb / 27.0) - (bb * cc / 3.0) + dd;
// solve quadratic equation (1,qq,-pp^3/27)
D = qq * qq + (4.0 * pp * pp * pp / 27.0);
D = (Math.Abs(D) < MConsts.EPS_DEC * MConsts.EPS_DEC) ? 0.0 : D; //tolerancing for double roots
if (D > 0) //D>0, => 1 real & 2 complex roots
{
numRootReal = 1;
double zp = 0.5 * (-qq + Math.Sqrt(D)); // two real roots of the quadratic equation
double zm = 0.5 * (-qq - Math.Sqrt(D));
double zpRoot3 = Math.Sign(zp) * Math.Pow(Math.Abs(zp), (double)(1.0 / 3.0));
double zmRoot3 = Math.Sign(zm) * Math.Pow(Math.Abs(zm), (double)(1.0 / 3.0));
double sumRoots = zpRoot3 + zmRoot3;
double difRoots = zpRoot3 - zmRoot3;
root[0] = new CNum(-bb / 3.0 + sumRoots, 0);
root[1] = new CNum(-bb / 3.0 - 0.5 * sumRoots, -0.5 * Math.Sqrt(3.0) * difRoots);
root[2] = new CNum(-bb / 3.0 - 0.5 * sumRoots, 0.5 * Math.Sqrt(3.0) * difRoots);
}
else //D<=0, => 3 real roots
{
numRootReal = 3;
ArrayList arr = new ArrayList(); // roots of the depressed equation, root = y-bb/3
int i;
if (D < 0)
{
CNum z = new CNum(-0.5 * qq, 0.5 * Math.Sqrt(-D)); // root of the quartatic equation
if (z.IsZero)
{
throw new ExceptionGMath("Equation", "RootsAll", "Degree 3");
}
double zRadius = z.Radius;
double zAngle = z.Angle;
double zRadiusRoot3 = Math.Pow(zRadius, 1.0 / 3.0);
for (i = 0; i <= 2; i++)
{
arr.Add(2.0 * zRadiusRoot3 * Math.Cos((zAngle + 2.0 * i * Math.PI) / 3.0));
}
arr.Sort(); // TODO: check
for (i = 0; i <= 2; i++)
{
root[i] = new CNum(-bb / 3.0 + (double)arr[i], 0);
}
}
if (D == 0) // case of the multiply roots
{
double z = -0.5 * qq; // double real root of the quadratic equation
double zRoot3 = Math.Sign(z) * Math.Pow(Math.Abs(z), 1.0 / 3.0);
arr.Add(2.0 * zRoot3);
arr.Add(-zRoot3);
arr.Add(-zRoot3);
if (qq > 0)
{
root[0] = new CNum(-bb / 3.0 + (double)arr[0], 0, 1);
root[1] = new CNum(-bb / 3.0 + (double)arr[1], 0, 2);
root[2] = new CNum(root[1]);
}
else if (qq == 0)
{
root[0] = new CNum(-bb / 3.0, 0, 3);
root[1] = new CNum(root[0]);
root[2] = new CNum(root[0]);
}
else
{
root[0] = new CNum(-bb / 3.0 + (double)arr[1], 0, 2);
root[1] = new CNum(root[0]);
root[2] = new CNum(-bb / 3.0 + (double)arr[0], 0, 2);
}
}
}
}
}
/*
* REAL ROOTS
*/
private static void RootsAllToReal(int numRootReal, CNum[] croot, out double[] root)
{
root=null;
if (numRootReal>0)
{
root=new double[numRootReal];
for (int i=0; i<numRootReal; i++)
{
root[i]=croot[i].Re;
}
}
}
// deg3
public static void RootsAll(double a, double b, double c, double d,
out int numRoot, out int numRootReal, out CNum[] root)
{
// output:
// case of 3 real roots =>
// the roots are sorted in increasing order
// case of 1 real root => real, complex-, complex+
//
// multiply real roots referred as different roots
/*
* Catch EXACT TANGENTS when they are not obvious
* from the control points specification
*/
double rootToAdd=-1.0; // asign an invalid parametric value
int numRootRed=-1, numRootRealRed=-1; // roots after reduction
CNum[] rootRed=null;
if (Equation.Evaluate(0.0,a,b,c,d)==0.0)
{
rootToAdd=0.0;
Equation.RootsAll(a,b,c,
out numRootRed, out numRootRealRed, out rootRed);
}
if (Equation.Evaluate(1.0,a,b,c,d)==0.0)
{
rootToAdd=1.0;
Equation.RootsAll(a,a+b,a+b+c,
out numRootRed, out numRootRealRed, out rootRed);
}
if (rootToAdd!=-1.0)
{
root=new CNum[numRootRed+1];
numRoot=numRootRed+1;
numRootReal=numRootRealRed+1;
//
// two complex roots of the reduced (quadratic) equation
//
if ((numRootRed-numRootRealRed)==2)
{
root[0]=new CNum(rootToAdd,0.0);
root[1]=new CNum(rootRed[0]);
root[2]=new CNum(rootRed[1]);
return;
}
//
// all roots of the reduced equation are real
//
CNum rootNew=new CNum(rootToAdd,0.0);
// adjust multiplicities
for (int iRootRed=0; iRootRed<numRootRed; iRootRed++)
{
if (rootToAdd==rootRed[iRootRed].Re)
{
rootNew.Multiplicity+=1;
rootRed[iRootRed].Multiplicity+=1;
}
}
if (numRootRed==0)
{
root[0]=new CNum(rootNew);
return;
}
if (numRootRed==1)
{
if (rootToAdd<=rootRed[0].Re)
{
root[0]=new CNum(rootNew);
root[1]=new CNum(rootRed[0]);
}
else
{
root[0]=new CNum(rootRed[0]);
root[1]=new CNum(rootNew);
}
}
if (numRootRed==2)
{
if (rootToAdd<=rootRed[0].Re)
{
root[0]=new CNum(rootNew);
root[1]=new CNum(rootRed[0]);
root[2]=new CNum(rootRed[1]);
}
else if (rootToAdd<=rootRed[1].Re)
{
root[0]=new CNum(rootRed[0]);
root[1]=new CNum(rootNew);
root[2]=new CNum(rootRed[1]);
}
else
{
root[0]=new CNum(rootRed[0]);
root[1]=new CNum(rootRed[1]);
root[2]=new CNum(rootNew);
}
}
return; // end - catch EXACT TANGENTS
}
if (Math.Abs(a)<MConsts.EPS_DEC)
{
Equation.RootsAll(b,c,d,out numRoot, out numRootReal, out root);
}
else
{
numRoot = 3;
root=new CNum[3];
double bb, cc, dd, pp, qq, D;
bb = b/a;
cc = c/a;
dd = d/a;
pp = -(bb*bb/3.0)+cc;
qq = (2.0*bb*bb*bb/27.0)-(bb*cc/3.0)+dd;
// solve quadratic equation (1,qq,-pp^3/27)
D = qq*qq+(4.0*pp*pp*pp/27.0);
D = (Math.Abs(D)<MConsts.EPS_DEC*MConsts.EPS_DEC) ? 0.0 : D; //tolerancing for double roots
if (D>0) //D>0, => 1 real & 2 complex roots
{
numRootReal = 1;
double zp = 0.5*(-qq+Math.Sqrt(D)); // two real roots of the quadratic equation
double zm = 0.5*(-qq-Math.Sqrt(D));
double zpRoot3 = Math.Sign(zp)*Math.Pow(Math.Abs(zp),(double)(1.0/3.0));
double zmRoot3 = Math.Sign(zm)*Math.Pow(Math.Abs(zm),(double)(1.0/3.0));
double sumRoots = zpRoot3+zmRoot3;
double difRoots = zpRoot3-zmRoot3;
root[0]=new CNum(-bb/3.0+sumRoots,0);
root[1]=new CNum(-bb/3.0-0.5*sumRoots, -0.5*Math.Sqrt(3.0)*difRoots);
root[2]=new CNum(-bb/3.0-0.5*sumRoots, 0.5*Math.Sqrt(3.0)*difRoots);
}
else //D<=0, => 3 real roots
{
numRootReal = 3;
ArrayList arr=new ArrayList(); // roots of the depressed equation, root = y-bb/3
int i;
if (D<0)
{
CNum z=new CNum(-0.5*qq, 0.5*Math.Sqrt(-D)); // root of the quartatic equation
if (z.IsZero)
{
throw new ExceptionGMath("Equation","RootsAll","Degree 3");
}
double zRadius=z.Radius;
double zAngle=z.Angle;
double zRadiusRoot3 = Math.Pow(zRadius,1.0/3.0);
for (i=0; i<=2; i++)
{
arr.Add(2.0*zRadiusRoot3*Math.Cos((zAngle+2.0*i*Math.PI)/3.0));
}
arr.Sort(); // TODO: check
for (i=0; i<=2; i++)
{
root[i]=new CNum(-bb/3.0+(double)arr[i],0);
}
}
if (D==0) // case of the multiply roots
{
double z = -0.5*qq; // double real root of the quadratic equation
double zRoot3 = Math.Sign(z)*Math.Pow(Math.Abs(z),1.0/3.0);
arr.Add(2.0*zRoot3);
arr.Add(-zRoot3);
arr.Add(-zRoot3);
if (qq>0)
{
root[0]=new CNum(-bb/3.0+(double)arr[0],0,1);
root[1]=new CNum(-bb/3.0+(double)arr[1],0,2);
root[2]=new CNum(root[1]);
}
else if (qq==0)
{
root[0]=new CNum(-bb/3.0,0,3);
root[1]=new CNum(root[0]);
root[2]=new CNum(root[0]);
}
else
{
root[0]=new CNum(-bb/3.0+(double)arr[1],0,2);
root[1]=new CNum(root[0]);
root[2]=new CNum(-bb/3.0+(double)arr[0],0,2);
}
}
}
}
}
public CNum(CNum z): this(z.Re,z.Im,z.Multiplicity)
{
}
/*
* INITIALIZERS
*/
public void From(CNum z)
{
this.re=z.Re;
this.im=z.Im;
this.multiplicity=z.Multiplicity;
}