private static MultiPrecision <N> BesselYLimit(MultiPrecision <N> nu, MultiPrecision <N> z)
{
Consts.BesselLimitCoef table = Consts.Bessel.LimitCoef(nu);
MultiPrecision <Plus4 <N> > z_ex = z.Convert <Plus4 <N> >();
MultiPrecision <Plus4 <N> > v = 1 / z_ex;
MultiPrecision <Plus4 <N> > w = v * v;
MultiPrecision <Plus4 <N> > x = 0, y = 0, p = 1, q = v;
Sign sign = Sign.Plus;
for (int k = 0; k <= Consts.BesselJY.LimitApproxTerms; k++, p *= w, q *= w)
{
MultiPrecision <Plus4 <N> > c = p * table.Value(k * 2);
MultiPrecision <Plus4 <N> > s = q * table.Value(k * 2 + 1);
if (sign == Sign.Plus)
{
x += c;
y += s;
sign = Sign.Minus;
}
else
{
x -= c;
y -= s;
sign = Sign.Plus;
}
if (!c.IsZero && x.Exponent - c.Exponent <= MultiPrecision <Plus2 <N> > .Bits)
{
continue;
}
if (!s.IsZero && y.Exponent - s.Exponent <= MultiPrecision <Plus2 <N> > .Bits)
{
continue;
}
break;
}
MultiPrecision <Plus4 <N> > omega = z_ex - (2 * nu.Convert <Plus4 <N> >() + 1) * MultiPrecision <Plus4 <N> > .PI / 4;
MultiPrecision <Plus4 <N> > m = x * MultiPrecision <Plus4 <N> > .Sin(omega) + y * MultiPrecision <Plus4 <N> > .Cos(omega);
MultiPrecision <Plus4 <N> > t = m * MultiPrecision <Plus4 <N> > .Sqrt(2 / (MultiPrecision <Plus4 <N> > .PI *z_ex));
return(t.Convert <N>());
}
public static MultiPrecision <N> BesselY(MultiPrecision <N> nu, MultiPrecision <N> x)
{
if (Abs(nu) > 64)
{
throw new ArgumentOutOfRangeException(
nameof(nu),
"In the calculation of the Bessel function, nu with an absolute value greater than 64 is not supported."
);
}
if (nu.IsNaN || x.IsNaN)
{
return(NaN);
}
if (x.Sign == Sign.Minus)
{
if (nu != Truncate(nu))
{
return(NaN);
}
long n = (long)nu;
return(((n & 1L) == 0) ? BesselY(nu, Abs(x)) : -BesselY(nu, Abs(x)));
}
if (!x.IsFinite)
{
return(0);
}
if (x.Exponent >= Bits)
{
return(NaN);
}
if (nu.Sign == Sign.Minus && nu == Truncate(nu))
{
long n = (long)nu;
return(((n & 1L) == 0) ? BesselY(Abs(nu), x) : -BesselY(Abs(nu), x));
}
if (nu - Point5 == Floor(nu))
{
long n = (long)Floor(nu);
if (n >= -2 && n < 2)
{
MultiPrecision <Plus1 <N> > x_ex = x.Convert <Plus1 <N> >();
MultiPrecision <Plus1 <N> > envelope = MultiPrecision <Plus1 <N> > .Sqrt(2 / (MultiPrecision <Plus1 <N> > .PI *x_ex));
if (n == -2)
{
MultiPrecision <N> y = -(envelope * (MultiPrecision <Plus1 <N> > .Sin(x_ex) / x_ex - MultiPrecision <Plus1 <N> > .Cos(x_ex))).Convert <N>();
return(y.IsNaN ? 0 : y);
}
if (n == -1)
{
MultiPrecision <N> y = (envelope * MultiPrecision <Plus1 <N> > .Sin(x_ex)).Convert <N>();
return(y.IsNaN ? 0 : y);
}
if (n == 0)
{
return(-(envelope * MultiPrecision <Plus1 <N> > .Cos(x_ex)).Convert <N>());
}
if (n == 1)
{
return(-(envelope * (MultiPrecision <Plus1 <N> > .Cos(x_ex) / x_ex + MultiPrecision <Plus1 <N> > .Sin(x_ex))).Convert <N>());
}
}
}
if (x.Exponent <= -0x1000000)
{
return(nu.IsZero ? NegativeInfinity : NaN);
}
if (x < Consts.BesselJY.ApproxThreshold)
{
return(BesselYNearZero(nu, x).Convert <N>());
}
else
{
return(BesselYLimit(nu, x));
}
}