/// <summary>
/// Computes the definite integral within the borders a and b.
/// </summary>
/// <param name="a">Left integration border.</param>
/// <param name="b">Right integration border.</param>
/// <returns></returns>
public double Integrate(double a, double b)
{
double[] buf = new double[Degree + 2];
buf[0] = 0; // this value can be arbitrary, in fact
for (int i = 1; i < buf.Length; i++)
{
buf[i] = Coefficients[i - 1] / i;
}
Polynomial p = new Polynomial(buf);
return(p.Evaluate(b) - p.Evaluate(a));
}
/// <summary>
/// Computes the greatest value |p(z_k)|.
/// </summary>
/// <param name="p"></param>
/// <param name="z"></param>
/// <returns></returns>
public static double MaxValue(Polynomial p, double[] z)
{
double buf = 0;
for (int i = 0; i < z.Length; i++)
{
if (Math.Abs(p.Evaluate(z[i])) > buf)
{
buf = Math.Abs(p.Evaluate(z[i]));
}
}
return(buf);
}
/// <summary>
/// Computes the roots of polynomial p via Weierstrass iteration.
/// </summary>
/// <param name="p"></param>
/// <returns></returns>
public static double[] Roots(Polynomial p)
{
double tolerance = 1e-12;
int max_iterations = 30;
Polynomial q = Normalize(p);
//Polynomial q = p;
double[] z = new double[q.Degree]; // approx. for roots
double[] w = new double[q.Degree]; // Weierstraß corrections
// init z
for (int k = 0; k < q.Degree; k++)
{
//z[k] = (new Complex(.4, .9)) ^ k;
z[k] = Math.Exp(2 * Math.PI * k / q.Degree);
}
for (int iter = 0; iter < max_iterations &&
MaxValue(q, z) > tolerance; iter++)
{
for (int i = 0; i < 10; i++)
{
for (int k = 0; k < q.Degree; k++)
{
w[k] = q.Evaluate(z[k]) / WeierNull(z, k);
}
for (int k = 0; k < q.Degree; k++)
{
z[k] -= w[k];
}
}
}
// clean...
for (int k = 0; k < q.Degree; k++)
{
z[k] = Math.Round(z[k], 12);
}
return(z);
}
/// <summary>
/// Computes the definite integral within the borders a and b.
/// </summary>
/// <param name="a">Left integration border.</param>
/// <param name="b">Right integration border.</param>
/// <returns></returns>
public double Integrate(double a, double b)
{
double[] buf = new double[Degree + 2];
buf[0] = 0; // this value can be arbitrary, in fact
for (int i = 1; i < buf.Length; i++)
buf[i] = Coefficients[i - 1] / i;
Polynomial p = new Polynomial(buf);
return (p.Evaluate(b) - p.Evaluate(a));
}
/// <summary>
/// Computes the greatest value |p(z_k)|.
/// </summary>
/// <param name="p"></param>
/// <param name="z"></param>
/// <returns></returns>
public static double MaxValue(Polynomial p, double[] z)
{
double buf = 0;
for (int i = 0; i < z.Length; i++)
{
if (Math.Abs(p.Evaluate(z[i])) > buf)
buf = Math.Abs(p.Evaluate(z[i]));
}
return buf;
}