/// <summary>
/// Returns the sum of a hypergeometric series 3F1.
/// <para>Sum3F1 = addend + Σ(( (a1)_n * (a2)_n * (a3)_n)/(b1)_n * (z^n/n!)) n={0,∞}</para>
/// </summary>
/// <param name="a1">First numerator</param>
/// <param name="a2">Second numerator</param>
/// <param name="a3">Third numerator</param>
/// <param name="b1">First denominator. Requires b1 > 0</param>
/// <param name="z">Argument</param>
/// <param name="addend">The addend to the series</param>
/// <returns>
/// The sum of the series, or NaN if the series does not converge
/// within the policy maximum number of iterations
/// </returns>
public static double Sum3F1(double a1, double a2, double a3, double b1, double z, double addend = 0)
{
if ((double.IsNaN(a1) || double.IsInfinity(a1))
|| (double.IsNaN(a2) || double.IsInfinity(a2))
|| (double.IsNaN(a3) || double.IsInfinity(a3))
|| (double.IsNaN(z) || double.IsInfinity(z))
|| (double.IsNaN(b1) || double.IsInfinity(b1))
|| double.IsNaN(addend)) {
Policies.ReportDomainError("Sum3F1(a1: {0}, a2: {1}, a3: {2}, b1: {3}, z: {4}, addend: {5}): Requires finite arguments", a1, a2, a3, b1, z, addend);
return double.NaN;
}
if (b1 <= 0 && Math2.IsInteger(b1)) {
Policies.ReportDomainError("Sum3F1(a1: {0}, a2: {1}, a3: {2}, b1: {3}, z: {4}, addend: {5}): Requires b1 is not zero or a negative integer", a1, a2, a3, b1, z, addend);
return double.NaN;
}
double sum = 1.0 + addend;
if (z == 0)
return sum;
if (a1 == b1)
return Sum2F0(a2, a3, z, addend);
if (a2 == b1)
return Sum2F0(a1, a3, z, addend);
if (a3 == b1)
return Sum2F0(a1, a2, z, addend);
#if false
// TODO:
if (a1 == 1)
return Sum3F1_A1(a2, a3, b1, z, addend);
if (a2 == 1)
return Sum3F1_A1(a1, a3, b1, z, addend);
if (a3 == 1)
return Sum3F1_A1(a1, a2, b1, z, addend);
#endif
double term = 1.0;
int n = 0;
for (; n < Policies.MaxSeriesIterations; n++) {
double prevSum = sum;
term *= (z / (n + 1)) * ((a1 + n) / (b1 + n)) * (a2 + n) * (a3 + n);
sum += term;
#if CMP_FLOAT
if (Math.Abs(term) <= Math.Abs(prevSum) * Policies.SeriesTolerance)
return sum;
#else
if (sum == prevSum)
return sum;
#endif
}
Policies.ReportConvergenceError("Sum3F1(a1: {0}, a2: {1}, a3: {2}, b1: {3}, z: {4}, addend: {5}): No convergence after {6} iterations", a1, a2, a3, b1, z, addend, n);
return double.NaN;
}
/// <summary>
/// Returns the sum of a hypergeometric series 1F2.
/// <para>Sum2F1 = addend + Σ( ((a1)_n)/((b1)_n * (b2)_n) * (z^n/n!)) n={0,∞}</para>
/// </summary>
/// <param name="a1">First numerator</param>
/// <param name="b1">First denominator. Requires b1 > 0</param>
/// <param name="b2">Second denominator. Requires b2 > 0</param>
/// <param name="z">Argument</param>
/// <param name="addend">The addend to the series</param>
/// <returns>
/// The sum of the series, or NaN if the series does not converge
/// within the policy maximum number of iterations
/// </returns>
public static double Sum1F2(double a1, double b1, double b2, double z, double addend = 0)
{
if ((double.IsNaN(a1) || double.IsInfinity(a1))
|| (double.IsNaN(z) || double.IsInfinity(z))
|| (double.IsNaN(b1) || double.IsInfinity(b1))
|| (double.IsNaN(b2) || double.IsInfinity(b2))
|| double.IsNaN(addend)) {
Policies.ReportDomainError("Sum1F2(a1: {0}, b1: {1}, b2: {2}, z: {3}, addend: {4}): Requires finite arguments", a1, b1, b2, z, addend);
return double.NaN;
}
if ((b1 <= 0 && Math2.IsInteger(b1))
|| (b2 <= 0 && Math2.IsInteger(b2))) {
Policies.ReportDomainError("Sum1F2(a1: {0}, b1: {1}, b2: {2}, z: {3}, addend: {4}): Requires b1, b2 are not zero or negative integers", a1, b1, b2, z, addend);
return double.NaN;
}
double sum = 1.0 + addend;
if (z == 0)
return sum;
if (a1 == b1)
return Sum0F1(b2, z, addend);
if (a1 == b2)
return Sum0F1(b1, z, addend);
if (a1 == 1)
return Sum1F2_A1(b1, b2, z, addend);
double term = 1.0;
int n = 0;
for (; n < Policies.MaxSeriesIterations; n++) {
double prevSum = sum;
term *= (z / (n + 1)) * (((a1 + n) / (b1 + n)) / (b2 + n));
sum += term;
#if CMP_FLOAT
if (Math.Abs(term) <= Math.Abs(prevSum) * Policies.SeriesTolerance)
return sum;
#else
if (sum == prevSum)
return sum;
#endif
}
Policies.ReportConvergenceError("Sum1F2(a1: {0}, b1: {1}, b2: {2}, z: {3}, addend: {4}): No convergence after {0} iterations", a1, b1, b2, z, addend, n);
return double.NaN;
}
//
// Series form for BesselY as z -> 0,
// see: http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/06/01/04/01/01/0003/
// This series is only useful when the second term is small compared to the first
// otherwise we get catestrophic cancellation errors.
//
// Approximating tgamma(v) by v^v, and assuming |tgamma(-z)| < eps we end up requiring:
// eps/2 * v^v(x/2)^-v > (x/2)^v or Math.Log(eps/2) > v Math.Log((x/2)^2/v)
//
/// <summary>
/// Series form for Y{v}(x) as z -> 0 and v is not an integer
/// <para>Y{v}(z) = (-((z/2)^v Cos(πv) Γ(-v))/π)) * 0F1(; 1+v; -z^2/4) - ((z/2)^-v Γ(v))/π) * 0F1(; 1-v; -z^2/4)</para>
/// </summary>
/// <param name="v"></param>
/// <param name="x"></param>
/// <returns></returns>
/// <seealso href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/26/01/02/0001/"/>
public static DoubleX Y_SmallArgSeries(double v, double x)
{
Debug.Assert(v >= 0 && x >= 0);
Debug.Assert( !Math2.IsInteger(v), "v cannot be an integer");
DoubleX powDivGamma;
double lnp = v * Math.Log(x / 2);
if (v > DoubleLimits.MaxGamma || lnp < DoubleLimits.MinLogValue || lnp > DoubleLimits.MaxLogValue) {
powDivGamma = DoubleX.Exp(lnp - Math2.Lgamma(v));
} else {
DoubleX p = Math.Pow(x / 2, v);
DoubleX gam = Math2.Tgamma(v);
powDivGamma = p / gam;
}
// Series 1: -((z/2)^-v Γ(v))/π) * 0F1(; 1-v; -z^2/4)
double s1 = HypergeometricSeries.Sum0F1(1 - v, -x * x / 4);
DoubleX result1 = -(s1/Math.PI)/ powDivGamma;
// Series2: -((z/2)^v Cos(πv) Γ(-v))/π)) * 0F1(; 1+v; -z^2/4)
// using the reflection formula: Γ(-v) = -π/(Γ(v) * v * Sin(π * v))
// prefix = (z/2)^v * Cos(π * v)/(Γ(v) * v * Sin(π * v))
// prefix = (z/2)^v * Cot(π * v) / (Γ(v) * v)
DoubleX result2 = DoubleX.Zero;
double cot = Math2.CotPI(v);
if (cot != 0) {
double s2 = HypergeometricSeries.Sum0F1(v + 1, -x * x / 4);
result2 = powDivGamma * cot * s2 / v; // Do this all in DoubleX
}
return (result1 - result2);
}
/// <summary>
/// Returns the sum of a hypergeometric series PFQ, which has a variable number of numerators and denominators.
/// <para> PFQ = addend + Σ(( Prod(a[j]_n)/Prod(b[j])_n * (z^n/n!)) n={0,∞}</para>
/// </summary>
/// <param name="a">The array of numerators</param>
/// <param name="b">The array of denominators. (b != 0 or b != negative integer)</param>
/// <param name="z">Argument</param>
/// <param name="addend">The addend to the series</param>
/// <returns>
/// The sum of the series, or NaN if the series does not converge
/// within the policy maximum number of iterations
/// </returns>
public static double SumPFQ(double[] a, double[] b, double z, double addend = 0)
{
if ( (a == null || b == null)
|| (double.IsNaN(z) || double.IsInfinity(z))
|| double.IsNaN(addend)) {
string aStr = (a == null) ? "null" : a.ToString();
string bStr = (b == null) ? "null" : b.ToString();
Policies.ReportDomainError("SumPFQ(a: {0}; b: {1}, z: {2}, addend: {3}): Requires finite non-null arguments", aStr, bStr, z, addend);
return double.NaN;
}
if (a.Length == 0 && b.Length == 0)
return Sum0F0(z, addend);
// check for infinities or non-positive integer b
bool hasBadParameter = false;
for (int i = 0; i < a.Length; i++) {
if (double.IsInfinity(a[i])) {
hasBadParameter = true;
break;
}
}
if (!hasBadParameter) {
for (int i = 0; i < b.Length; i++) {
if (double.IsInfinity(b[i]) || (b[i] <= 0 && Math2.IsInteger(b[i]))) {
hasBadParameter = true;
break;
}
}
}
if (hasBadParameter) {
Policies.ReportDomainError("SumPFQ(a: {0}; b: {1}, z: {2}, addend: {3}): Requires finite arguments and b not zero or a negative integer", a, b, z, addend);
return double.NaN;
}
double sum = 1.0 + addend;
if (z == 0)
return sum;
double term = 1.0;
int n = 0;
for (; n < Policies.MaxSeriesIterations; n++) {
double term_part = (z / (n + 1));
// use a/b where we can for max(a.Length, b.Length)
int j = 0;
for (; j < a.Length && j < b.Length; j++) {
term_part *= (a[j] + n) / (b[j] + n);
}
// otherwise multiply or divide the rest of the factors
// either a or b
for (; j < a.Length; j++) {
term_part *= (a[j] + n);
}
for (; j < b.Length; j++) {
term_part /= (b[j] + n);
}
term *= term_part;
double prevSum = sum;
sum += term;
#if CMP_FLOAT
if (Math.Abs(term) <= Math.Abs(prevSum) * Policies.SeriesTolerance)
return sum;
#else
if (sum == prevSum)
return sum;
#endif
}
Policies.ReportConvergenceError("No convergence after {0} iterations", n);
return double.NaN;
}
/// <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 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>
/// 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);
}