/// <summary>
/// Returns (J{v+n}(x), J{v+n-1}(x)) OR (Y{v+n}(x), Y{v+n-1}(x)) after n forward recurrences
/// </summary>
/// <param name="v">Order of the current parameter</param>
/// <param name="x">Argument</param>
/// <param name="n">Number of forward recurrence steps. 0 returns (current,prev)</param>
/// <param name="current">J{v}(x) OR Y{v}(x)</param>
/// <param name="prev">J{v-1}(x) OR Y{v-1}(x)</param>
/// <returns></returns>
public static (double JYvpn, double JYvpnm1) ForwardJY(double v, double x, int n, double current, double prev)
{
int scale = 0;
for (int k = 0; k < n; k++)
{
double fact = 2 * (v + k) / x;
double value = fact * current - prev;
if (double.IsInfinity(value))
{
int exp = 0;
current = Math2.Frexp(current, out exp);
prev = Math2.Ldexp(prev, -exp);
scale += exp;
value = fact * current - prev;
}
prev = current;
current = value;
}
if (scale != 0)
{
current = Math2.Ldexp(current, scale);
prev = Math2.Ldexp(prev, scale);
}
return(current, prev);
}
/// <summary>
/// Returns (K{v+n}, K{v+n-1}) after n forward recurrences
/// </summary>
/// <param name="v">Order of the current parameter</param>
/// <param name="x">Argument</param>
/// <param name="n">Number of forward recurrence steps. 0 returns (current,prev)</param>
/// <param name="current">K{v}(x)</param>
/// <param name="prev">K{v-1}(x)</param>
/// <returns></returns>
public static (double Kvpn, double Kvpnm1) ForwardK(double v, double x, int n, double current, double prev)
{
Debug.Assert(n >= 0);
// binary scale
int binaryScale = 0;
for (int k = 0; k < n; k++)
{
double factor = 2 * (v + k) / x;
double value = factor * current + prev;
if (double.IsInfinity(value))
{
int exp = 0;
current = Math2.Frexp(current, out exp);
prev = Math2.Ldexp(prev, -exp);
binaryScale += exp;
value = factor * current + prev;
}
prev = current;
current = value;
}
if (binaryScale != 0)
{
current = Math2.Ldexp(current, binaryScale);
prev = Math2.Ldexp(prev, binaryScale);
}
return(current, prev);
}
/// <summary>
/// Returns a tuple (J{v+n}(x), J{v+n-1}(x), BScale) OR (Y{v+n}(x), Y{v+n-1}(x), BScale) after n forward recurrences.
/// Note: to get actual values multiply by 2^BScale.
/// </summary>
/// <param name="v">Order of the current parameter</param>
/// <param name="x">Argument</param>
/// <param name="n">Number of forward recurrence steps. 0 returns (current,prev)</param>
/// <param name="current">J{v}(x) OR Y{v}(x)</param>
/// <param name="prev">J{v-1}(x) OR Y{v-1}(x)</param>
/// <returns></returns>
public static (double JYvpn, double JYvpnm1, int BScale) ForwardJY_B(double v, double x, int n, double current, double prev)
{
Debug.Assert(n >= 0);
Debug.Assert(!double.IsInfinity(2 * (v + n) / x));
int binaryScale = 0;
for (int k = 0; k < n; k++)
{
double fact = 2 * (v + k) / x;
double value = fact * current - prev;
if (double.IsInfinity(value))
{
int exp = 0;
current = Math2.Frexp(current, out exp);
prev = Math2.Ldexp(prev, -exp);
binaryScale += exp;
value = fact * current - prev;
}
prev = current;
current = value;
}
{
int exp = 0;
current = Math2.Frexp(current, out exp);
prev = Math2.Ldexp(prev, -exp);
binaryScale += exp;
}
return(current, prev, binaryScale);
}
/// <summary>
/// Returns (J{v-n}(x), J{v-n+1}(x)) OR (Y{v-n}(x), Y{v-n+1}(x)) after n backward recurrences
/// </summary>
/// <param name="v">Order of the current parameter</param>
/// <param name="x">Argument</param>
/// <param name="n">Number of backward recurrence steps. 0 returns (current,prev)</param>
/// <param name="current">J{v}(x) OR Y{v}(x)</param>
/// <param name="prev">J{v+1}(x) OR Y{v+1}(x)</param>
/// <returns></returns>
public static (double JYvmn, double JYvmnp1) BackwardJY(double v, double x, int n, double current, double prev)
{
Debug.Assert(n >= 0);
Debug.Assert(!double.IsInfinity(2 * v / x));
int scale = 0;
for (int k = 0; k < n; k++)
{
double fact = 2 * (v - k) / x;
double next = fact * current - prev;
if (double.IsInfinity(next))
{
int exp = 0;
current = Math2.Frexp(current, out exp);
prev = Math2.Ldexp(prev, -exp);
scale += exp;
next = fact * current - prev;
}
prev = current;
current = next;
}
if (scale != 0)
{
current = Math2.Ldexp(current, scale);
prev = Math2.Ldexp(prev, scale);
}
return(current, prev);
}
/// <summary>
/// Compute (J, Y) using Steeds method
/// </summary>
/// <param name="v"></param>
/// <param name="x"></param>
/// <param name="needJ"></param>
/// <param name="needY"></param>
/// <returns></returns>
static (double J, DoubleX Y) JY_Steed(double v, double x, bool needJ, bool needY)
{
Debug.Assert(x > 0 && v >= 0);
Debug.Assert(needJ || needY);
double J = double.NaN;
DoubleX Y = DoubleX.NaN;
int n = (int)Math.Floor(v + 0.5);
double u = v - n;
Debug.Assert(u >= -0.5 && u < 0.5); // Ensure u in [-1/2, 1/2)
// compute J{v+1}(x) / J{v}(x)
var(fv, s) = J_CF1(v, x);
var(Jvmn, Jvmnp1, scale) = Recurrence.BackwardJY_B(v, x, n, s, fv * s);
double ratio = s / Jvmn; // normalization
double fu = Jvmnp1 / Jvmn;
// prev/current, can also call CF1_jy() to get fu, not much difference in precision
//double fuCF;
//int sfu;
//Bessel.CF1_jy(u,x,fuCF,sfu);
//fu = fuCF;
var(p, q) = JY_CF2(u, x); // continued fraction JY_CF2
double t = u / x - fu; // t = J'/J
double gamma = (p - t) / q;
//
// We can't allow gamma to cancel out to zero competely as it messes up
// the subsequent logic. So pretend that one bit didn't cancel out
// and set to a suitably small value. The only test case we've been able to
// find for this, is when v = 8.5 and x = 4*PI.
//
if (gamma == 0)
{
gamma = u * DoubleLimits.MachineEpsilon / x;
}
double W = (2 / Math.PI) / x; // Wronskian
double Ju = Math.Sign(Jvmn) * Math.Sqrt(W / (q + gamma * (p - t)));
double Jv = Ju * ratio; // normalization
J = Math2.Ldexp(Jv, -scale);
if (needY)
{
double Yu = gamma * Ju;
double Yu1 = Yu * (u / x - p - q / gamma);
var(JYvpn, JYvpnm1, YScale) = Recurrence.ForwardJY_B(u + 1, x, n, Yu1, Yu);
Y = DoubleX.Ldexp(JYvpnm1, YScale);
}
return(J, Y);
}
/// <summary>
/// Computes the Jacobi elliptic functions using the arithmetic-geometric mean
/// </summary>
/// <param name="k">The modulus</param>
/// <param name="u">The argument</param>
/// <param name="anm1"></param>
/// <param name="bnm1"></param>
/// <param name="N"></param>
/// <param name="pTn"></param>
/// <returns></returns>
/// <see href="http://dlmf.nist.gov/22.20#ii"/>
private static double Jacobi_Recurse(double k, double u, double anm1, double bnm1, int N, out double pTn)
{
++N;
double Tn;
double cn = (anm1 - bnm1) / 2;
double an = (anm1 + bnm1) / 2;
if (cn < DoubleLimits.MachineEpsilon)
{
Tn = Math2.Ldexp(1, N) * u * an;
}
else
{
Tn = Jacobi_Recurse(k, u, an, Math.Sqrt(anm1 * bnm1), N, out double unused);
}
pTn = Tn;
return((Tn + Math.Asin((cn / an) * Math.Sin(Tn))) / 2);
}
/// <summary>
/// Returns the double factorial = n!!
/// <para>If n is odd, returns: n * (n-2) * ... * 5 * 3 * 1</para>
/// <para>If n is even, returns: n * (n-2) * ... * 6 * 4 * 2</para>
/// <para>If n = 0, -1, returns 1</para>
/// </summary>
/// <param name="n">Requires n ≥ -1</param>
public static double Factorial2(int n)
{
if (n < -1)
{
Policies.ReportDomainError("Factorial2(n: {0}): Requires n >= -1", n);
return(double.NaN);
}
// common values
if (n == -1 || n == 0)
{
return(1);
}
double result;
if (IsOdd(n))
{
// odd n:
if (n < FactorialTable.Length)
{
int k = (n - 1) / 2;
return(Math.Ceiling(Math2.Ldexp(FactorialTable[n] / FactorialTable[k], -k) - 0.5));
}
//
// Fallthrough: n is too large to use table lookup, try the
// gamma function instead.
//
result = Constants.RecipSqrtPI * Math2.Tgamma(0.5 * n + 1.0);
result = Math.Ceiling(Math2.Ldexp(result, (n + 1) / 2) - 0.5);
}
else
{
// even n:
int k = n / 2;
result = Math2.Ldexp(Factorial(k), k);
}
return(result);
}
//
// Spherical
//
/// <summary>
/// Returns the associated legendre polynomial with angle parameterization P{l,m}( cos(theta) )
/// </summary>
/// <param name="n"></param>
/// <param name="m"></param>
/// <param name="theta"></param>
/// <returns></returns>
static double LegendrePAngle(int n, int m, double theta)
{
if (n < 0)
{
n = -(n + 1); // P{n,m}(x) = P{-n-1,m}(x)
}
if (m == 0)
{
return(Math2.LegendreP(n, Math.Cos(theta)));
}
// if ( abs(m) < n ) return 0 by definition; handling int.MinValue too
if (m < -n || m > n)
{
return(0);
}
// P{n,-m} = (-1)^m * (n-m)!/(n+m)! * P{n,m}
if (m < 0)
{
double legP = LegendrePAngle(n, -m, theta);
if (legP == 0)
{
return(legP);
}
if (IsOdd(m))
{
legP = -legP;
}
int MaxFactorial = Math2.FactorialTable.Length - 1;
if (n - m <= MaxFactorial)
{
double ratio = Math2.FactorialTable[n + m] / Math2.FactorialTable[n - m];
return(ratio * legP);
}
if (n - m <= 300)
{
// break up the Tgamma ratio into two separate factors
// to prevent an underflow: a!/b! = (a!/170!) * (170!/b!)
// 170!/300! ~= 2.37e-308 > double.MinNormalizedValue
return((legP * (Math2.FactorialTable[n + m] / Math2.FactorialTable[MaxFactorial])) * Math2.TgammaRatio(MaxFactorial + 1, n + 1 - m));
}
#if false
// since we are setting Lmax = 128, so Mmax = 128, and Lmax+Mmax = 256
// so this should never be called, but left here for completeness
// 170!/300! ~= 2.37e-308 > double.MinNormalizedValue
double ratio = Math2.TgammaRatio(l + m + 1, l + 1 - m);
if (ratio >= DoubleLimits.MinNormalizedValue)
{
return(ratio * legP);
}
double value = Math.Exp(Math2.Lgamma(l + m + 1) - Math2.Lgamma(l + 1 - m) + Math.Log(Math.Abs(legP)));
return(Math.Sign(legP) * value);
#else
Policies.ReportNotImplementedError("LegendrePAngle(l: {0}, m: {1}, theta: {2}): Not implemented for |l| >= 128", n, m, theta);
return(double.NaN);
#endif
}
// at this point n >= 0 && 0 <= m <= n
Debug.Assert(n >= 0 && (m >= 0 && m <= n));
// use the recurrence relations
// P{n, n} = (-1)^n * (2*n - 1)!! * (1 - x^2)^(n/2)
// P{n+1, n} = (2*n + 1) * x * P{n, n}
// (n-m+1)*P{n+1,m} = (2*n + 1)* x * P{n,m} - (n+m)*P{n-1, m}
// see: http://en.wikipedia.org/wiki/Associated_Legendre_polynomials
double x = Math.Cos(theta);
double sin_theta = Math.Sin(theta);
// the following computes (2m-1)!! * sin_theta^m
// taking special care not to underflow too soon
// double pmm = Math.Pow(sin_theta, m) * Math2.Factorial2(2 * m - 1)
int exp = 0;
double mant = Math2.Frexp(sin_theta, out exp);
double pmmScaled = Math.Pow(mant, m) * Math2.Factorial2(2 * m - 1);
double pmm = Math2.Ldexp(pmmScaled, m * exp);
if (IsOdd(m))
{
pmm = -pmm;
}
if (n == m)
{
return(pmm);
}
// p0 - current
// p1 - next
double p0 = pmm;
double p1 = (2.0 * m + 1.0) * x * p0; //P{m+1,m}(x)
for (uint k = (uint)m + 1; k < (uint)n; k++)
{
double next = ((2 * k + 1) * x * p1 - (k + (uint)m) * p0) / (k - (uint)m + 1);
p0 = p1;
p1 = next;
}
return(p1);
}
/// <summary>
/// Returns the Associated Legendre Polynomial of the first kind
/// </summary>
/// <param name="n">Polynomial degree. Limited to |<paramref name="n"/>| ≤ 127</param>
/// <param name="m">Polynomial order</param>
/// <param name="x">Requires |<paramref name="x"/>| ≤ 1</param>
public static double LegendreP(int n, int m, double x)
{
if (!(x >= -1 && x <= 1))
{
Policies.ReportDomainError("LegendreP(n: {0}, m: {1}, x: {2}): Requires |x| <= 1", n, m, x);
return(double.NaN);
}
if (n <= -128 || n >= 128)
{
Policies.ReportNotImplementedError("LegendreP(n: {0}, m: {1}, x: {2}): Not implemented for |n| >= 128", n, m, x);
return(double.NaN);
}
// P{l,m}(x) == P{-l-1,m}(x); watch overflow
if (n < 0)
{
n = -(n + 1);
}
if (m == 0)
{
return(Math2.LegendreP(n, x));
}
// by definition P{n,m}(x) == 0 when |m| > l
if (m < -n || m > n)
{
return(0);
}
// P{n,-m} = (-1)^m * (n-m)!/(n+m)! * P{n,m}
if (m < 0)
{
double legP = LegendreP(n, -m, x);
if (legP == 0)
{
return(legP);
}
if (IsOdd(m))
{
legP = -legP;
}
int MaxFactorial = Math2.FactorialTable.Length - 1;
if (n - m <= MaxFactorial)
{
double ratio = Math2.FactorialTable[n + m] / Math2.FactorialTable[n - m];
return(ratio * legP);
}
if (n - m <= 300)
{
// break up the Tgamma ratio into two separate factors
// to prevent an underflow: a!/b! = (a!/170!) * (170!/b!)
// 170!/300! ~= 2.37e-308 > double.MinNormalizedValue
return((legP * (Math2.FactorialTable[n + m] / Math2.FactorialTable[MaxFactorial])) * Math2.TgammaRatio(MaxFactorial + 1, n + 1 - m));
}
#if false
// since we are setting Nmax = 128, so Mmax = 128, and Nmax+Mmax = 256
// so this should never be called, but left here for completeness
// 170!/300! ~= 2.37e-308 > double.MinNormalizedValue
double ratioL = Math2.TgammaRatio(n + m + 1, n + 1 - m);
if (ratioL >= DoubleLimits.MinNormalizedValue)
{
return(ratioL * legP);
}
double value = Math.Exp(Math2.Lgamma(n + m + 1) - Math2.Lgamma(n + 1 - m) + Math.Log(Math.Abs(legP)));
return(Math.Sign(legP) * value);
#else
Policies.ReportNotImplementedError("LegendreP(n: {0}, m: {1}, x: {2}): Not implemented for |n| >= 128", n, m, x);
return(double.NaN);
#endif
}
// at this point n >= 0 && 0 <= m <= n
Debug.Assert(n >= 0 && (m >= 0 && m <= n));
// use the following recurrence relations:
// P{n, n} = (-1)^n * (2*n - 1)!! * (1 - x^2)^(n/2)
// P{n+1, n} = (2*n + 1) * x * P{n, n}
// (n-m+1)*P{n+1,m} = (2*n + 1)* x * P{n,m} - (n+m)*P{n-1, m}
// see: http://en.wikipedia.org/wiki/Associated_Legendre_polynomials
// the following is:
// pmm = Math2.Factorial2(2 * m - 1) * Math.Pow(1.0 - x*x, m/2.0);
// being careful not to underflow too soon
double pmm = 0;
if (Math.Abs(x) > Constants.RecipSqrt2)
{
double inner = (1 - x) * (1 + x);
int exp = 0;
double mant = Math2.Frexp(inner, out exp);
// careful: use integer division
double pmm_scaled = Math.Pow(mant, m / 2) * Math2.Factorial2(2 * m - 1);
pmm = Math2.Ldexp(pmm_scaled, (m / 2) * exp);
if (IsOdd(m))
{
pmm *= Math.Sqrt(inner);
}
}
else
{
double p_sin_theta = Math.Exp(m / 2.0 * Math2.Log1p(-x * x)); //Math.Pow(1.0 - x*x, m/2.0);
pmm = Math2.Factorial2(2 * m - 1) * p_sin_theta;
}
if (IsOdd(m))
{
pmm = -pmm;
}
if (n == m)
{
return(pmm);
}
// p0 - current
// p1 - next
double p0 = pmm;
double p1 = (2.0 * m + 1.0) * x * p0; //P{m+1,m}(x)
for (uint k = (uint)m + 1; k < (uint)n; k++)
{
double next = ((2 * k + 1) * x * p1 - (k + (uint)m) * p0) / (k - (uint)m + 1);
p0 = p1;
p1 = next;
}
return(p1);
}
/// <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>
/// 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);
}
/// <summary>
/// Computes J{v}(x), Y{v}(x) using a combination of asymptotic approximation and recurrence
/// </summary>
/// <param name="v"></param>
/// <param name="x"></param>
/// <param name="needJ"></param>
/// <param name="needY"></param>
/// <returns></returns>
static (double J, DoubleX Y) JY_AsymRecurrence(double v, double x, bool needJ, bool needY)
{
Debug.Assert(v >= 0 && x >= 1, "Requires positive values for v,x");
Debug.Assert(v < int.MaxValue, "v too large: v = " + v);
var J = double.NaN;
var Y = DoubleX.NaN;
int nPos = (int)Math.Floor(v);
double uPos = v - nPos;
// Using Hankel, find:
// J{v-Floor(v)}(x) and J{v-Floor(v)+1}(x)
// Y{v-Floor(v)}(x) and Y{v-Floor(v)+1}(x)
// then use recurrence to find J{v}(x), Y{v}(x)
double u0, u1;;
int n;
if (x >= 9)
{
// set the start of the recurrence near sqrt(x)
double maxV = Math.Floor(Math.Sqrt(x));
u1 = (maxV - 1) + uPos;
u0 = u1 - 1;
n = (int)Math.Floor(v - u1 + 0.5);
Debug.Assert(n >= 0);
}
else
{
u0 = uPos;
u1 = uPos + 1;
n = nPos - 1;
}
Debug.Assert(x >= HankelAsym.JYMinX(u1), "x is too small for HankelAsym");
var(Ju, Yu) = HankelAsym.JY(u0, x);
var(Jup1, Yup1) = HankelAsym.JY(u1, x);
if (needJ)
{
if (v < x)
{
J = Recurrence.ForwardJY(u1, x, n, Jup1, Ju).JYvpn;
}
else
{
// Use fv = J{v+1}(x) / J{v}(x)
// and backward recurrence to find (J{v-n+1}/J{v-n})
var(fv, s) = J_CF1(v, x);
var(Jvmn, Jvmnp1, scale) = Recurrence.BackwardJY_B(v, x, n, s, fv * s);
var Jv = Math2.Ldexp(Jup1 / Jvmn, -scale);
J = s * Jv; // normalization
}
}
if (needY)
{
var(Yv, Yvm1, YScale) = Recurrence.ForwardJY_B(u1, x, n, Yup1, Yu);
Y = DoubleX.Ldexp(Yv, YScale);
}
return(J, Y);
}
