/// <summary>
/// チェビシェフ有理式(elliptic rational)を計算する。
/// </summary>
/// <param name="n">次数</param>
/// <param name="l">x > 1 における極小値</param>
/// <returns>次数 n のチェビシェフ有理式</returns>
public static Rational Elliptic(int n, double l)
{
double m1 = 1 / (l * l);
double m1p = 1 - m1;
double Kk1 = Ellip.K(m1);
double Kk1p = Ellip.K(m1p);
double m = Ellip.InverseQ(Math.Exp(-Math.PI * Kk1p / (n * Kk1)));
double Kk = Ellip.K(m);
Polynomial num = Polynomial.X(0, 1);
Polynomial denom = Polynomial.X(0, 1);
Rational r = new Rational(Polynomial.X(0, 1), Polynomial.X(0, 1));
double g = 1;
for(int i=n-1; i>0; i-=2)
{
double u = Kk * (double)i / n;
double sn = Ellip.Sn(u, m);
double w = sn * sn;
g *= (m*w - 1) / (1/w - 1);
num *= Polynomial.X(2, 1/w) - 1;
denom *= Polynomial.X(2, m*w) - 1;
}
if((n & 1) == 1)
{
num *= Polynomial.X(1);
}
num *= g;
return new Rational(num, denom);
}
/// <summary>
/// 分子/分母多項式を指定して初期化。
/// </summary>
/// <param name="num">分子多項式</param>
/// <param name="denom">分母多項式</param>
public Rational(Polynomial num, Polynomial denom)
{
this.num = num;
this.denom = denom;
}
/// <summary>
/// x の n 乗を返す。
/// </summary>
/// <param name="n">指数</param>
/// <returns>x の n 乗</returns>
public static Polynomial X(int n)
{
return(Polynomial.X(n, 1));
}
