/// <summary>
/// Simultaneously compute I{v}(x) and K{v}(x)
/// </summary>
/// <param name="v"></param>
/// <param name="x"></param>
/// <param name="needI"></param>
/// <param name="needK"></param>
/// <returns>A tuple of (I, K). If needI == false or needK == false, I = NaN or K = NaN respectively</returns>
public static (double I, double K) IK(double v, double x, bool needI, bool needK)
{
Debug.Assert(needI || needK, "NeedI and NeedK cannot both be false");
// set initial values for parameters
double I = double.NaN;
double K = double.NaN;
// check v first so that there are no integer throws later
if (Math.Abs(v) > int.MaxValue)
{
Policies.ReportNotImplementedError("BesselIK(v: {0}, x: {1}): Requires |v| <= {2}", v, x, int.MaxValue);
return(I, K);
}
if (v < 0)
{
v = -v;
// for integer orders only, we can use the following identities:
// I{-n}(x) = I{n}(v)
// K{-n}(x) = K{n}(v)
// for any v, use reflection rule:
// I{-v}(x) = I{v}(x) + (2/PI) * sin(pi*v)*K{v}(x)
// K{-v}(x) = K{v}(x)
if (needI && !Math2.IsInteger(v))
{
var(IPos, KPos) = IK(v, x, true, true);
I = IPos + (2 / Math.PI) * Math2.SinPI(v) * KPos;
if (needK)
{
K = KPos;
}
return(I, K);
}
}
if (x < 0)
{
Policies.ReportDomainError("BesselIK(v: {0}, x: {1}): Complex result not supported. Requires x >= 0", v, x);
return(I, K);
}
// both x and v are non-negative from here
Debug.Assert(x >= 0 && v >= 0);
if (x == 0)
{
if (needI)
{
I = (v == 0) ? 1.0 : 0.0;
}
if (needK)
{
K = double.PositiveInfinity;
}
return(I, K);
}
if (needI && (x < 2 || 3 * (v + 1) > x * x))
{
I = I_SmallArg(v, x);
needI = false;
if (!needK)
{
return(I, K);
}
}
// Hankel is fast, and reasonably accurate.
// at x == 32, it will converge in about 15 iterations
if (x >= HankelAsym.IKMinX(v))
{
if (needK)
{
K = HankelAsym.K(v, x);
}
if (needI)
{
I = HankelAsym.I(v, x);
}
return(I, K);
}
// the uniform expansion is here as a last resort
// to limit the number of recurrences, but it is less accurate.
if (UniformAsym.IsIKAvailable(v, x))
{
if (needK)
{
K = UniformAsym.K(v, x);
}
if (needI)
{
I = UniformAsym.I(v, x);
}
return(I, K);
}
// K{v}(x) and K{v+1}(x), binary scaled
var(Kv, Kv1, binaryScale) = K_CF(v, x);
if (needK)
{
K = Math2.Ldexp(Kv, binaryScale);
}
if (needI)
{
// use the Wronskian relationship
// Since CF1 is O(x) for x > v, try to weed out obvious overflows
// Note: I{v+1}(x)/I{v}(x) is in [0, 1].
// I{0}(713. ...) == MaxDouble
const double I0MaxX = 713;
if (x > I0MaxX && x > v)
{
I = Math2.Ldexp(1.0 / (x * (Kv + Kv1)), -binaryScale);
if (double.IsInfinity(I))
{
return(I, K);
}
}
double W = 1 / x;
double fv = I_CF1(v, x);
I = Math2.Ldexp(W / (Kv * fv + Kv1), -binaryScale);
}
return(I, K);
}
/// <summary>
/// Returns K{n}(x) for integer order
/// </summary>
/// <param name="n"></param>
/// <param name="x"></param>
/// <returns></returns>
public static double KN(int n, double x)
{
if (x < 0)
{
Policies.ReportDomainError("BesselK(v: {0}, x: {1}): Requires x >= 0 for real result", n, x);
return(double.NaN);
}
if (x == 0)
{
return(double.PositiveInfinity);
}
// even function
// K{-n}(z) = K{n}(z)
if (n < 0)
{
n = -n;
}
if (n == 0)
{
return(K0(x));
}
if (n == 1)
{
return(K1(x));
}
double v = n;
// Hankel is fast, and reasonably accurate, saving us from many recurrences.
if (x >= HankelAsym.IKMinX(v))
{
return(HankelAsym.K(v, x));
}
// the uniform expansion is here as a last resort
// to limit the number of recurrences, but it is less accurate.
if (UniformAsym.IsIKAvailable(v, x))
{
return(UniformAsym.K(v, x));
}
// Since K{v}(x) has a (e^-x)/sqrt(x) multiplier
// using recurrence can underflow too quickly for large x,
// so, use a scaled version
double result;
if (x > 1)
{
double prev = K0(x, true);
double current = K1(x, true);
// for large v and x this number can get very large
// maximum observed K(1000,10) = 2^6211
var(Kv, _, binaryScale) = Recurrence.ForwardK_B(1, x, n - 1, current, prev);
// Compute: value * 2^(binaryScale) * e^-x
if (x < -DoubleX.MinLogValue)
{
DoubleX exs = DoubleX.Ldexp(DoubleX.Exp(-x), binaryScale);
result = Math2.Ldexp(Kv * exs.Mantissa, exs.Exponent);
}
else
{
result = Math.Exp(-x + Math.Log(Kv) + binaryScale * Constants.Ln2);
}
}
else
{
double prev = K0(x);
double current = K1(x);
result = Recurrence.ForwardK(1, x, n - 1, current, prev).Kvpn;
}
return(result);
}