private static MultiPrecision <N> BesselKIntegerNuNearZero(int n, MultiPrecision <N> z)
{
Consts.BesselIntegerFiniteTermCoef finite_table = Consts.Bessel.IntegerFiniteTermCoef(n);
Consts.BesselIntegerConvergenceTermCoef convergence_table = Consts.Bessel.IntegerConvergenceTermCoef(n);
MultiPrecision <Double <N> > z_ex = z.Convert <Double <N> >();
MultiPrecision <Double <N> > u = 1;
MultiPrecision <Double <N> > w = z_ex * z_ex;
MultiPrecision <Double <N> > r
= MultiPrecisionSandbox <Double <N> > .BesselINearZero(n, z_ex).Convert <Double <N> >() * MultiPrecision <Double <N> > .Log(z_ex / 2);
long r_exponent = r.Exponent;
MultiPrecision <Double <N> > m = MultiPrecision <Double <N> > .Pow(z_ex / 2, n);
MultiPrecision <Double <N> > x = 0, y = 0;
Sign sign = Sign.Plus;
for (int k = 0; k < int.MaxValue; k++, u *= w)
{
MultiPrecision <Double <N> > c = u * convergence_table.Value(k);
y += c;
if (k < n)
{
if (sign == Sign.Plus)
{
x += u * finite_table.Value(k);
sign = Sign.Minus;
}
else
{
x -= u * finite_table.Value(k);
sign = Sign.Plus;
}
continue;
}
if (r.Exponent < -MultiPrecision <N> .Bits * 8 && (c.IsZero || y.Exponent - c.Exponent > MultiPrecision <Plus1 <N> > .Bits))
{
break;
}
if (c.IsZero || Math.Min(y.Exponent - c.Exponent, r_exponent - c.Exponent - m.Exponent) > MultiPrecision <Double <N> > .Bits)
{
break;
}
}
MultiPrecision <Double <N> > d = (x / m + ((n & 1) == 0 ? 1 : -1) * (y * m - 2 * r)) / 2;
return(d.Convert <N>());
}
private static MultiPrecision <N> BesselKNearZero(MultiPrecision <N> nu, MultiPrecision <N> z)
{
int n = (int)MultiPrecision <N> .Round(nu);
if (nu != n)
{
MultiPrecision <N> dnu = nu - n;
if (dnu.Exponent >= -16)
{
return(MultiPrecisionSandbox <Double <Plus2 <N> > > .BesselKNonIntegerNu(nu.Convert <Double <Plus2 <N> > >(), z.Convert <Double <Plus2 <N> > >()).Convert <N>());
}
if (dnu.Exponent >= -32)
{
return(MultiPrecisionSandbox <Double <Plus4 <N> > > .BesselKNonIntegerNu(nu.Convert <Double <Plus4 <N> > >(), z.Convert <Double <Plus4 <N> > >()).Convert <N>());
}
if (dnu.Exponent >= -48)
{
return(MultiPrecisionSandbox <Double <Plus8 <N> > > .BesselKNonIntegerNu(nu.Convert <Double <Plus8 <N> > >(), z.Convert <Double <Plus8 <N> > >()).Convert <N>());
}
if (dnu.Exponent >= -80)
{
return(MultiPrecisionSandbox <Double <Plus16 <N> > > .BesselKNonIntegerNu(nu.Convert <Double <Plus16 <N> > >(), z.Convert <Double <Plus16 <N> > >()).Convert <N>());
}
if (dnu.Exponent >= -144)
{
return(MultiPrecisionSandbox <Double <Plus32 <N> > > .BesselKNonIntegerNu(nu.Convert <Double <Plus32 <N> > >(), z.Convert <Double <Plus32 <N> > >()).Convert <N>());
}
if (dnu.Exponent >= -272)
{
return(MultiPrecisionSandbox <Double <Plus64 <N> > > .BesselKNonIntegerNu(nu.Convert <Double <Plus64 <N> > >(), z.Convert <Double <Plus64 <N> > >()).Convert <N>());
}
throw new ArgumentException(
"The calculation of the BesselK function value is invalid because it loses digits" +
" when nu is extremely close to an integer. (|nu - round(nu)| < 1.32 x 10^-82 and nu != round(nu))",
nameof(nu));
}
return(MultiPrecisionSandbox <Plus4 <N> > .BesselKIntegerNuNearZero(n, z.Convert <Plus4 <N> >()).Convert <N>());
}
static int plot(StreamWriter sw, decimal nu, int min_matchbits, decimal z)
{
MultiPrecision <N> t = MultiPrecisionSandbox <N> .BesselJ(nu, z);
MultiPrecision <Plus1 <N> > s = MultiPrecisionSandbox <Plus1 <N> > .BesselJ(nu, z);
sw.WriteLine($" z: {z}");
sw.WriteLine($" f : {t}");
sw.WriteLine($" {t.ToHexcode()}");
sw.WriteLine($" {s.ToHexcode()}");
MultiPrecision <N> err = MultiPrecision <N> .Abs(t - s.Convert <N>());
sw.WriteLine($" err : {err}");
for (int keepbits = MultiPrecision <N> .Bits; keepbits >= 0; keepbits--)
{
if (keepbits == 0)
{
min_matchbits = 0;
}
else if (MultiPrecision <N> .RoundMantissa(t, MultiPrecision <N> .Bits - keepbits) == MultiPrecision <N> .RoundMantissa(s.Convert <N>(), MultiPrecision <N> .Bits - keepbits))
{
sw.WriteLine($" matchbits : {keepbits}");
if (keepbits < min_matchbits)
{
min_matchbits = keepbits;
}
break;
}
}
return(min_matchbits);
}
public static MultiPrecision <N> BesselY(MultiPrecision <N> nu, MultiPrecision <N> x)
{
if (MultiPrecision <N> .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(MultiPrecision <N> .NaN);
}
if (x.Sign == Sign.Minus)
{
if (nu != MultiPrecision <N> .Truncate(nu))
{
return(MultiPrecision <N> .NaN);
}
long n = (long)nu;
return(((n & 1L) == 0) ? BesselY(nu, MultiPrecision <N> .Abs(x)) : -BesselY(nu, MultiPrecision <N> .Abs(x)));
}
if (!x.IsFinite)
{
return(0);
}
if (x.IsZero)
{
return(MultiPrecision <N> .NegativeInfinity);
}
if (nu.Sign == Sign.Minus && nu == MultiPrecision <N> .Truncate(nu))
{
long n = (long)nu;
return(((n & 1L) == 0) ? BesselY(MultiPrecision <N> .Abs(nu), x) : -BesselY(MultiPrecision <N> .Abs(nu), x));
}
if (nu - MultiPrecision <N> .Point5 == MultiPrecision <N> .Floor(nu))
{
long n = (long)MultiPrecision <N> .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.IsNormal ? y : 0);
}
if (n == -1)
{
MultiPrecision <N> y = (envelope * MultiPrecision <Plus1 <N> > .Sin(x_ex)).Convert <N>();
return(y.IsNormal ? y : 0);
}
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 (MultiPrecision <N> .Length <= 4)
{
return(MultiPrecisionSandbox <Plus1 <N> > .BesselY(nu.Convert <Plus1 <N> >(), x.Convert <Plus1 <N> >()).Convert <N>());
}
if (x < Consts.BesselJY.ApproxThreshold)
{
return(BesselYNearZero(nu, x));
}
else
{
return(BesselYLimit(nu, x).Convert <N>());
}
}