internal static double Imp(int n)
{
// For n >= 0 this routine uses a table lookup for speed
//
// For n < 0, the following routines are used:
//
// http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/03/01/0001/
// ζ(-n) = (-1)^n/(n+1) * B{n+1}, where B is the Bernoulli number
//
// http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/17/01/01/0001/
// ζ(n) = Γ(1-n) * (2^n) * π^(n-1) * Sin(nπ/2) * ζ(1-n)
if (n >= 0)
{
if (n < Math2.ZetaTable.Length)
{
return(Math2.ZetaTable[n]);
}
return(1.0);
}
// ζ(-2n) = 0
if (!Math2.IsOdd(n))
{
return(0.0);
}
// underflow
if (n < -DoubleLimits.MaxLogValue)
{
return(0);
}
int sc = 1 - n;
if (sc < 2 * Math2.Bernoulli2nTable.Length)
{
return(-Math2.Bernoulli2nTable[sc / 2] / sc);
}
double sign = Math2.SinPI(0.5 * n);
// This won't be called because of the current length of the bernoulli table
// if (sc <= Math2.MaxFactorialIndex)
// return sign * 2 * Math.Pow(2 * Math.PI, -sc) * Math2.FactorialTable[sc-1] * Imp(sc);
// use duplication
// Γ(2n) = 2^(2n-1)/Sqrt(Pi) * Γ(n) * Γ(n+1/2)
if (sc <= 2 * (Math2.MaxFactorialIndex - 1))
{
return(sign * Math.Pow(Math.PI, -sc - 0.5) * Math2.Tgamma(sc / 2.0) * Math2.Tgamma(sc / 2.0 + 0.5) * Imp(sc));
}
// We're going to overflow
return(sign * Math.Exp(Math2.Lgamma(sc) - sc * Constants.Ln2PI + Constants.Ln2 + Math.Log(Imp(sc))));
}
/// <summary>
/// Returns J{v}(x) for integer v
/// </summary>
/// <param name="v"></param>
/// <param name="x"></param>
/// <returns></returns>
public static double JN(double v, double x)
{
Debug.Assert(Math2.IsInteger(v));
// For integer orders only, we can use two identities:
// #1: J{-n}(x) = (-1)^n * J{n}(x)
// #2: J{n}(-x) = (-1)^n * J{n}(x)
double sign = 1;
if (v < 0)
{
v = -v;
if (Math2.IsOdd(v))
{
sign = -sign;
}
}
if (x < 0)
{
x = -x;
if (Math2.IsOdd(v))
{
sign = -sign;
}
}
Debug.Assert(v >= 0 && x >= 0);
if (v > int.MaxValue)
{
Policies.ReportNotImplementedError("BesselJ(v: {0}): Large integer values not yet implemented", v);
return(double.NaN);
}
int n = (int)v;
//
// Special cases:
//
if (n == 0)
{
return(sign * J0(x));
}
if (n == 1)
{
return(sign * J1(x));
}
// n >= 2
Debug.Assert(n >= 2);
if (x == 0)
{
return(0);
}
if (x < 5 || (v > x * x / 4))
{
return(sign * J_SmallArg(v, x));
}
// if v > MinAsymptoticV, use the asymptotics
const int MinAsymptoticV = 7; // arbitrary constant to reduce iterations
if (v > MinAsymptoticV)
{
double Jv;
DoubleX Yv;
if (JY_TryAsymptotics(v, x, out Jv, out Yv, true, false))
{
return(sign * Jv);
}
}
// v < abs(x), forward recurrence stable and usable
// v >= abs(x), forward recurrence unstable, use Miller's algorithm
if (v < x)
{
// use forward recurrence
double prev = J0(x);
double current = J1(x);
return(sign * Recurrence.ForwardJY(1.0, x, n - 1, current, prev).JYvpn);
}
else
{
// Steed is somewhat more accurate when n gets large
if (v >= 200)
{
var(J, Y) = JY_Steed(n, x, true, false);
return(J);
}
// J{n+1}(x) / J{n}(x)
var(fv, s) = J_CF1(n, x);
var(Jvmn, Jvmnp1, scale) = Recurrence.BackwardJY_B(n, x, n, s, fv * s);
// scale the result
double Jv = (J0(x) / Jvmn);
Jv = Math2.Ldexp(Jv, -scale);
return((s * sign) * Jv); // normalization
}
}
/// <summary>
/// Computes Y{v}(x), where v is an integer
/// </summary>
/// <param name="v">Integer order</param>
/// <param name="x">Argument. Requires x > 0</param>
/// <returns></returns>
public static DoubleX YN(double v, double x)
{
if ((x == 0) && (v == 0))
{
return(double.NegativeInfinity);
}
if (x <= 0)
{
Policies.ReportDomainError("BesselY(v: {0}, x: {1}): Complex number result not supported. Requires x >= 0", v, x);
return(double.NaN);
}
//
// Reflection comes first:
//
double sign = 1;
if (v < 0)
{
// Y_{-n}(z) = (-1)^n Y_n(z)
if (Math2.IsOdd(v))
{
sign = -1;
}
v = -v;
}
Debug.Assert(v >= 0 && x >= 0);
if (v > int.MaxValue)
{
Policies.ReportNotImplementedError("BesselY(v: {0}, x: {1}): Large integer values not yet implemented", v, x);
return(double.NaN);
}
int n = (int)v;
if (n == 0)
{
return(Y0(x));
}
if (n == 1)
{
return(sign * Y1(x));
}
if (x < DoubleLimits.RootMachineEpsilon._2)
{
DoubleX smallArgValue = YN_SmallArg(n, x);
return(smallArgValue * sign);
}
// if v > MinAsymptoticV, use the asymptotics
const int MinAsymptoticV = 7; // arbitrary constant to reduce iterations
if (v > MinAsymptoticV)
{
double Jv;
DoubleX Yv;
if (JY_TryAsymptotics(v, x, out Jv, out Yv, false, true))
{
return(sign * Yv);
}
}
// forward recurrence always OK (though unstable)
double prev = Y0(x);
double current = Y1(x);
var(Yvpn, Yvpnm1, YScale) = Recurrence.ForwardJY_B(1.0, x, n - 1, current, prev);
return(DoubleX.Ldexp(sign * Yvpn, YScale));
}
/// <summary>
/// Compute J{v}(x) and Y{v}(x)
/// </summary>
/// <param name="v"></param>
/// <param name="x"></param>
/// <param name="needJ"></param>
/// <param name="needY"></param>
/// <returns></returns>
public static (double J, DoubleX Y) JY(double v, double x, bool needJ, bool needY)
{
Debug.Assert(needJ || needY);
// uses Steed's method
// see Barnett et al, Computer Physics Communications, vol 8, 377 (1974)
// set [out] parameters
double J = double.NaN;
DoubleX Y = DoubleX.NaN;
// check v first so that there are no integer throws later
if (Math.Abs(v) > int.MaxValue)
{
Policies.ReportNotImplementedError("BesselJY(v: {0}): Large |v| > int.MaxValue not yet implemented", v);
return(J, Y);
}
if (v < 0)
{
v = -v;
if (Math2.IsInteger(v))
{
// for integer orders only, we can use the following identities:
// J{-n}(x) = (-1)^n * J{n}(x)
// Y{-n}(x) = (-1)^n * Y{n}(x)
if (Math2.IsOdd(v))
{
var(JPos, YPos) = JY(v, x, needJ, needY);
if (needJ)
{
J = -JPos;
}
if (needY)
{
Y = -YPos;
}
return(J, Y);
}
}
if (v - Math.Floor(v) == 0.5)
{
Debug.Assert(v >= 0);
// use reflection rule:
// for integer m >= 0
// J{-(m+1/2)}(x) = (-1)^(m+1) * Y{m+1/2}(x)
// Y{-(m+1/2)}(x) = (-1)^m * J{m+1/2}(x)
// call the general bessel functions with needJ and needY reversed
var(JPos, YPos) = JY(v, x, needY, needJ);
double m = v - 0.5;
bool isOdd = Math2.IsOdd(m);
if (needJ)
{
double y = (double)YPos;
J = isOdd ? y : -y;
}
if (needY)
{
Y = isOdd ? -JPos : JPos;
}
return(J, Y);
}
else
{
// use reflection rule:
// J{-v}(x) = cos(pi*v)*J{v}(x) - sin(pi*v)*Y{v}(x)
// Y{-v}(x) = sin(pi*v)*J{v}(x) + cos(pi*v)*Y{v}(x)
var(JPos, YPos) = JY(v, x, true, true);
double cp = Math2.CosPI(v);
double sp = Math2.SinPI(v);
J = cp * JPos - (double)(sp * YPos);
Y = sp * JPos + cp * YPos;
return(J, Y);
}
}
// both x and v are positive from here
Debug.Assert(x >= 0 && v >= 0);
if (x == 0)
{
// For v > 0
if (needJ)
{
J = 0;
}
if (needY)
{
Y = DoubleX.NegativeInfinity;
}
return(J, Y);
}
int n = (int)Math.Floor(v + 0.5);
double u = v - n; // -1/2 <= u < 1/2
// is it an integer?
if (u == 0)
{
if (v == 0)
{
if (needJ)
{
J = J0(x);
}
if (needY)
{
Y = Y0(x);
}
return(J, Y);
}
if (v == 1)
{
if (needJ)
{
J = J1(x);
}
if (needY)
{
Y = Y1(x);
}
return(J, Y);
}
// for integer order only
if (needY && x < DoubleLimits.RootMachineEpsilon._2)
{
Y = YN_SmallArg(n, x);
if (!needJ)
{
return(J, Y);
}
needY = !needY;
}
}
if (needJ && ((x < 5) || (v > x * x / 4)))
{
// always use the J series if we can
J = J_SmallArg(v, x);
if (!needY)
{
return(J, Y);
}
needJ = !needJ;
}
if (needY && x <= 2)
{
// J should have already been solved above
Debug.Assert(!needJ);
// Evaluate using series representations.
// Much quicker than Y_Temme below.
// This is particularly important for x << v as in this
// area Y_Temme may be slow to converge, if it converges at all.
// for non-integer order only
if (u != 0)
{
if ((x < 1) && (Math.Log(DoubleLimits.MachineEpsilon / 2) > v * Math.Log((x / 2) * (x / 2) / v)))
{
Y = Y_SmallArgSeries(v, x);
return(J, Y);
}
}
// Use Temme to find Yu where |u| <= 1/2, then use forward recurrence for Yv
var(Yu, Yu1) = Y_Temme(u, x);
var(Yvpn, Yvpnm1, YScale) = Recurrence.ForwardJY_B(u + 1, x, n, Yu1, Yu);
Y = DoubleX.Ldexp(Yvpnm1, YScale);
return(J, Y);
}
Debug.Assert(x > 2 && v >= 0);
// Try asymptotics directly:
if (x > v)
{
// x > v*v
if (x >= HankelAsym.JYMinX(v))
{
var result = HankelAsym.JY(v, x);
if (needJ)
{
J = result.J;
}
if (needY)
{
Y = result.Y;
}
return(J, Y);
}
// Try Asymptotic Phase for x > 47v
if (x >= MagnitudePhase.MinX(v))
{
var result = MagnitudePhase.BesselJY(v, x);
if (needJ)
{
J = result.J;
}
if (needY)
{
Y = result.Y;
}
return(J, Y);
}
}
// fast and accurate within a limited range of v ~= x
if (UniformAsym.IsJYPrecise(v, x))
{
var(Jv, Yv) = UniformAsym.JY(v, x);
if (needJ)
{
J = Jv;
}
if (needY)
{
Y = Yv;
}
return(J, Y);
}
// Try asymptotics with recurrence:
if (x > v && x >= HankelAsym.JYMinX(v - Math.Floor(v) + 1))
{
var(Jv, Yv) = JY_AsymRecurrence(v, x, needJ, needY);
if (needJ)
{
J = Jv;
}
if (needY)
{
Y = Yv;
}
return(J, Y);
}
// Use Steed's Method
var(SteedJv, SteedYv) = JY_Steed(v, x, needJ, needY);
if (needJ)
{
J = SteedJv;
}
if (needY)
{
Y = SteedYv;
}
return(J, Y);
}
