/// <summary>
/// Only for polynomials of the degree 2!
/// Returns both zero positions of the polynomial,
/// if they are real.
/// </summary>
/// <returns>both zero positions of the polynomial</returns>
public double[] ZeroSetN2()
{
if (Degree != 2)
{
throw new Exception(
"Only for polynomials with degree 2!");
}
if (this[2] == 0)
{
throw new Exception(
"Not a quadratic function, first coefficient = 0!");
}
//normalize
Polynomial nForm;
if (this[2] != 1)
{
nForm = new Polynomial(3);
nForm[0] = this[0] / this[2];
nForm[1] = this[1] / this[2];
nForm[2] = 1;
}
else
{
nForm = this; // normal form
}
return(GeneralMath.QuadraticEquation(nForm[1], nForm[0]));
}
/// <summary>
/// Only for polynomials of the degree 3!
/// Finds all zero positions of the polynomial, if they
/// are real.
/// This method searches one zero position by using
/// the Halley method. The polynomial then will be divided
/// by polynomial division by the first found zero position.
/// The remaining two zero positions will be found by
/// solving the remaining quadratic equation.
/// </summary>
/// <param name="accuracy">
/// Number of correctly computed decimals
/// </param>
/// <returns>
/// List of all three zero positions.
/// </returns>
public double[] ZeroSetN3(int accuracy)
{
double allowedError = Math.Pow(10, -accuracy);
if (Degree != 3)
{
throw new Exception("Only for polynomials with degree 3!");
}
if (this[3] == 0)
{
throw new Exception("Not a cubic function, first coefficient = 0!");
}
// normalise term
Polynomial nForm;
if (this[3] != 1)
{
nForm = new Polynomial(3);
nForm[3] = 1;
for (int i = 0; i < 3; i++)
{
nForm[i] = this[i] / this[3];
}
}
else
{
nForm = this; // normal form
}
// check if all zeros are real
double reducedP, reducedQ;
reducedP = nForm[1] - (Math.Pow(nForm[2], 2) / 3);
reducedQ = 2 * Math.Pow(nForm[2], 3) / 27 -
nForm[2] * nForm[1] / 3 +
nForm[0];
if (Math.Pow(reducedQ / 2, 2) + Math.Pow(reducedP / 3, 3) > 0)
{
throw new Exception("not all zeroes are real!");
}
//
// Implementation of the Method of Deiters and Macias-Salinas
//
double
lowerBoundry,
upperBoundry,
initialValue,
firstZero = 0,
xInflection,
yInflection,
auxD; // "D" in Wikipedia: "Kubische Gleichungen"
xInflection = -nForm[2] / 3;
yInflection = Value(xInflection);
if (yInflection == 0)
{
firstZero = xInflection;
}
else
{
auxD = Math.Pow(nForm[2], 2) - 3 * nForm[1];
if (auxD == 0)
{
firstZero = xInflection - Math.Pow(yInflection, 1.0 / 3);
}
else
{
// original: find first zero iterative by halley method, now tangent
lowerBoundry = xInflection - (2.0 / 3) * Math.Sqrt(Math.Abs(auxD));
upperBoundry = xInflection + (2.0 / 3) * Math.Sqrt(Math.Abs(auxD));
if (auxD > 0 && yInflection > 0)
{
initialValue = lowerBoundry;
}
else if (auxD > 0 && yInflection < 0)
{
initialValue = upperBoundry;
}
else
{
initialValue = xInflection;
}
firstZero = ZeroHalley(initialValue, accuracy);
}
}
// polynomial division
double p, q;
double[]
sqret = new double[2],
ret = new double[3];
p = nForm[2] + firstZero;
q = p * firstZero + nForm[1];
ret[0] = firstZero;
// find remaining zero positions by solving quadratic equation
sqret = GeneralMath.QuadraticEquation(p, q);
ret[1] = sqret[0];
ret[2] = sqret[1];
return(ret);
}
