/// <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);
}
/// <summary>
/// Try to use asymptotics to generate the result
/// </summary>
/// <param name="v"></param>
/// <param name="x"></param>
/// <param name="J"></param>
/// <param name="Y"></param>
/// <param name="needJ"></param>
/// <param name="needY"></param>
/// <returns></returns>
public static bool JY_TryAsymptotics(double v, double x, out double J, out DoubleX Y, bool needJ, bool needY)
{
J = double.NaN;
Y = DoubleX.NaN;
// 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(true);
}
// 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(true);
}
}
if (UniformAsym.IsJYPrecise(v, x))
{
var(Jv, Yv) = UniformAsym.JY(v, x);
if (needJ)
{
J = Jv;
}
if (needY)
{
Y = Yv;
}
return(true);
}
// 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(true);
}
return(false);
}