/// <summary>
/// Returns the value of the Riemann Zeta function at <paramref name="x"/>
/// <para>ζ(x) = Σ n^-x, n={1, ∞}</para>
/// </summary>
/// <param name="x">The function argument</param>
public static double Zeta(double x)
{
if (double.IsNaN(x))
{
Policies.ReportDomainError("Zeta(x: {0}): NaN not allowed", x);
return(double.NaN);
}
if (double.IsInfinity(x))
{
if (x > 0)
{
return(1);
}
Policies.ReportDomainError("Zeta(x: {0}): -Infinity not allowed", x);
return(double.NaN);
}
if (x == 1)
{
Policies.ReportPoleError("Zeta(x: {0}): Evaluation at pole", x);
return(double.NaN);
}
return(_Zeta.Imp(x, 1 - x));
}
/// <summary>
/// Returns Floor(Log(2,|<paramref name="x"/>|)).
/// Note: this is the exponent field of a double precision number.
/// <para>If x is NaN then x is returned</para>
/// <para>If x == ±∞ then +∞ is returned</para>
/// <para>If x == ±0 then -∞ is returned</para>
/// </summary>
/// <param name="x">Argument</param>
public static double Logb(double x)
{
IEEEDouble rep = new IEEEDouble(x);
// Check for +/- Inf or NaN
if (!rep.IsFinite) {
if ( rep.IsInfinity )
return double.PositiveInfinity;
Policies.ReportDomainError("Logb(x: {0}): NaNs not allowed", x);
return x;
}
if ( x == 0 ) {
Policies.ReportPoleError("Logb(x: {0}): Logb(0) == -∞", x);
return double.NegativeInfinity;
}
int exponent = 0;
if (rep.IsSubnormal) {
// Multiply by 2^53 to normalize the number
const double normalMult = (1L << IEEEDouble.MantissaBits);
rep = new IEEEDouble(x * normalMult);
exponent = -IEEEDouble.MantissaBits;
Debug.Assert(!rep.IsSubnormal);
}
exponent += rep.ExponentValue;
return exponent;
}
/// <summary>
/// Returns the value of the Riemann Zeta function at <paramref name="n"/>
/// <para>ζ(n) = Σ k^-n, k={1, ∞}</para>
/// </summary>
/// <param name="n">The function argument</param>
public static double Zeta(int n)
{
if (n == 1)
{
Policies.ReportPoleError("Zeta(n: {0}): Evaluation at pole", n);
return(double.NaN);
}
return(_Zeta.Imp(n));
}
/// <summary>
/// Returns the Exponential Integral
/// <para>E<sub>n</sub>(x) = ∫ e^(-x*t)/t^n dt, t={1,∞}</para>
/// </summary>
/// <param name="n">Requires n ≥ 0</param>
/// <param name="x">Requires x ≥ 0</param>
public static double Expint(int n, double x)
{
if (!(n >= 0) || !(x >= 0))
{
Policies.ReportDomainError("Expint(n: {0}, x: {1}): Requires n,x >= 0", n, x);
return(double.NaN);
}
// E_{n}(∞) = 0
// see: http://functions.wolfram.com/06.34.03.0014.01
if (double.IsInfinity(x))
{
return(0);
}
if (x == 0)
{
if (n <= 1)
{
Policies.ReportPoleError("Expint(n: {0}, x: {1}): Requires x > 0 when n <= 1", n, x);
return(double.NaN);
}
return(1.0 / (n - 1));
}
if (n == 0)
{
return(Math.Exp(-x) / x);
}
if (n == 1)
{
return(E1(x));
}
if (n < 3)
{
if (x < 0.5)
{
return(_Expint.En_Series(n, x));
}
return(_Expint.En_Fraction(n, x));
}
if (x < (n - 2.0) / (n - 1.0))
{
return(_Expint.En_Series(n, x));
}
return(_Expint.En_Fraction(n, x));
}
/// <summary>
/// Returns Γ(n)
/// </summary>
/// <param name="n">Argument</param>
public static double Tgamma(int n)
{
if (n <= 0)
{
Policies.ReportPoleError("Tgamma(n: {0}): Requires n > 0", n);
return(double.NaN);
}
if (n >= FactorialTable.Length + 1)
{
return(double.PositiveInfinity);
}
return(FactorialTable[n - 1]);
}
/// <summary>
/// Returns Tan(π * x)
/// </summary>
/// <param name="x">Argument</param>
public static double TanPI(double x)
{
if (double.IsNaN(x) || double.IsInfinity(x))
{
Policies.ReportDomainError("TanPI(x: {0}): Requires finite x", x);
return(double.NaN);
}
// save x for error messages
double originalX = x;
// tan(-x) == -tan(x)
bool neg = false;
if (x < 0)
{
x = -x;
neg = !neg;
}
// the period for tan(x) is π
if (x >= 1)
{
x -= Math.Floor(x);
}
Debug.Assert(x >= 0 && x < 1);
// shift x to [-0.5,0.5]*π
if (x > 0.5)
{
x -= 1.0;
}
if (x == 0.5)
{
Policies.ReportPoleError("TanPI(x: {0}): Requires x not a half integer", originalX);
return(double.NaN);
}
double result = Math.Tan(Math.PI * x);
return((neg) ? -result : result);
}
/// <summary>
/// Returns Cot(π * x)
/// </summary>
/// <param name="x">Argument</param>
public static double CotPI(double x)
{
if (double.IsNaN(x) || double.IsInfinity(x))
{
Policies.ReportDomainError("CotPI(x: {0}): Requires finite x", x);
return(double.NaN);
}
double absX = Math.Abs(x);
// the period for cot(x) is π
if (absX >= 1)
{
absX -= Math.Floor(absX);
}
if (absX == 0)
{
Policies.ReportPoleError("CotPI(x: {0}): Requires non-integer value for x", x);
return(double.NaN);
}
Debug.Assert(absX > 0 && absX < 1);
// cot(x) == tan(π/2 - x)
double result;
if (absX >= 0.5)
{
result = -Math.Tan((absX - 0.5) * Math.PI);
}
else
{
result = 1.0 / Math.Tan(absX * Math.PI);
}
// cot(-x) == -cot(x)
return((x < 0) ? -result : result);
}
/// <summary>
/// Returns the Digamma function
/// <para>ψ(x) = d/dx(ln(Γ(x))) = Γ'(x)/Γ(x)</para>
/// </summary>
/// <param name="x">Digamma function argument</param>
public static double Digamma(double x)
{
if (double.IsNaN(x))
{
Policies.ReportDomainError("Digamma(x: {0}): NaN not allowed", x);
return(double.NaN);
}
if (double.IsInfinity(x))
{
if (x < 0)
{
Policies.ReportDomainError("Digamma(x: {0}): Requires x != -Infinity", x);
return(double.NaN);
}
return(double.PositiveInfinity);
}
if (x <= 0)
{
if (IsInteger(x))
{
Policies.ReportPoleError("Digamma(x: {0}) Requires x is not an integer when x <= 0", x);
return(double.NaN);
}
// use the following equations to reflect
// ψ(1-x) - ψ(x) = π * cot(π*x)
// ψ(-x) = 1/x + ψ(x) + π * cot(π*x)
if (x < -1)
{
return(Digamma(1 - x) - Math.PI * Math2.CotPI(x));
}
// The following is the forward recurrence:
// -(1/x + 1/(x+1)) + Digamma(x + 2);
// with a little less cancellation error near Digamma root: x = -0.50408...
return(_Digamma.Rational_1_2(x + 2) - (2 * x + 1) / (x * (x + 1)));
}
// If we're above the lower-limit for the asymptotic expansion then use it:
if (x >= 10)
{
const double p0 = 0.083333333333333333333333333333333333333333333333333;
const double p1 = -0.0083333333333333333333333333333333333333333333333333;
const double p2 = 0.003968253968253968253968253968253968253968253968254;
const double p3 = -0.0041666666666666666666666666666666666666666666666667;
const double p4 = 0.0075757575757575757575757575757575757575757575757576;
const double p5 = -0.021092796092796092796092796092796092796092796092796;
const double p6 = 0.083333333333333333333333333333333333333333333333333;
const double p7 = -0.44325980392156862745098039215686274509803921568627;
double xm1 = x - 1;
double z = 1 / (xm1 * xm1);
double P = p0 + z * (p1 + z * (p2 + z * (p3 + z * (p4 + z * (p5 + z * (p6 + z * p7))))));
return(Math.Log(xm1) + 1.0 / (2 * xm1) - z * P);
}
double result = 0;
//
// If x > 2 reduce to the interval [1,2]:
//
while (x > 2)
{
x -= 1;
result += 1 / x;
}
//
// If x < 1 use recurrence to shift to > 1:
//
if (x < 1)
{
result = -1 / x;
x += 1;
}
result += _Digamma.Rational_1_2(x);
return(result);
}
/// <summary>
/// Returns the Trigamma function
/// <para>ψ'(x) = d/dx(ψ(x)) = d''/dx(ln(Γ(x)))</para>
/// </summary>
/// <param name="x">Trigamma function argument</param>
public static double Trigamma(double x)
{
if (double.IsNaN(x))
{
Policies.ReportDomainError("Trigamma(x: {0}): NaN not allowed", x);
return(double.NaN);
}
if (double.IsInfinity(x))
{
if (x < 0)
{
Policies.ReportDomainError("Trigamma(x: {0}): Requires x != -Infinity", x);
return(double.NaN);
}
return(0);
}
double result = 0;
//
// Check for negative arguments and use reflection:
//
if (x <= 0)
{
if (IsInteger(x))
{
Policies.ReportPoleError("Trigamma(x: {0}) Requires x is not an integer when x <= 0", x);
return(double.NaN);
}
// Reflect:
double z = 1 - x;
double s = Math.Abs(x) < Math.Abs(z) ? Math2.SinPI(x) : Math2.SinPI(z);
return(-Trigamma(z) + (Math.PI * Math.PI) / (s * s));
}
if (x < 1)
{
result = 1 / (x * x);
x += 1;
}
if (x <= 2)
{
// Max error in interpolated form: 3.736e-017
const double Y_1_2 = 2.1093254089355469;
const double p0 = -1.1093280605946045;
const double p1 = -3.8310674472619321;
const double p2 = -3.3703848401898283;
const double p3 = 0.28080574467981213;
const double p4 = 1.6638069578676164;
const double p5 = 0.64468386819102836;
const double q0 = 1.0;
const double q1 = 3.4535389668541151;
const double q2 = 4.5208926987851437;
const double q3 = 2.7012734178351534;
const double q4 = 0.64468798399785611;
const double q5 = -0.20314516859987728e-6;
double z = x;
double P = p0 + z * (p1 + z * (p2 + z * (p3 + z * (p4 + z * p5))));
double Q = q0 + z * (q1 + z * (q2 + z * (q3 + z * (q4 + z * q5))));
result += (Y_1_2 + P / Q) / (x * x);
}
else if (x <= 4)
{
// Max error in interpolated form: 1.159e-017
const double p0 = -0.13803835004508849e-7;
const double p1 = 0.50000049158540261;
const double p2 = 1.6077979838469348;
const double p3 = 2.5645435828098254;
const double p4 = 2.0534873203680393;
const double p5 = 0.74566981111565923;
const double q0 = 1.0;
const double q1 = 2.8822787662376169;
const double q2 = 4.1681660554090917;
const double q3 = 2.7853527819234466;
const double q4 = 0.74967671848044792;
const double q5 = -0.00057069112416246805;
double z = 1 / x;
double P = p0 + z * (p1 + z * (p2 + z * (p3 + z * (p4 + z * p5))));
double Q = q0 + z * (q1 + z * (q2 + z * (q3 + z * (q4 + z * q5))));
result += (1 + P / Q) / x;
}
else
{
// Maximum Deviation Found: 6.896e-018
// Expected Error Term : -6.895e-018
// Maximum Relative Change in Control Points : 8.497e-004
const double p0 = 0.68947581948701249e-17;
const double p1 = 0.49999999999998975;
const double p2 = 1.0177274392923795;
const double p3 = 2.498208511343429;
const double p4 = 2.1921221359427595;
const double p5 = 1.5897035272532764;
const double p6 = 0.40154388356961734;
const double q0 = 1.0;
const double q1 = 1.7021215452463932;
const double q2 = 4.4290431747556469;
const double q3 = 2.9745631894384922;
const double q4 = 2.3013614809773616;
const double q5 = 0.28360399799075752;
const double q6 = 0.022892987908906897;
double z = 1 / x;
double P = p0 + z * (p1 + z * (p2 + z * (p3 + z * (p4 + z * (p5 + z * p6)))));
double Q = q0 + z * (q1 + z * (q2 + z * (q3 + z * (q4 + z * (q5 + z * q6)))));
result += (1 + P / Q) / x;
}
return(result);
}
/// <summary>
/// Returns Y{0}(x)
/// </summary>
/// <param name="x"></param>
/// <returns></returns>
public static double Y0(double x)
{
if (x < 0)
{
Policies.ReportDomainError("BesselY(v: 0, x: {0}): Requires x >= 0; complex number result not supported", x);
return(double.NaN);
}
if (x == 0)
{
Policies.ReportPoleError("BesselY(v: 0, x: {0}): Overflow", x);
return(double.NegativeInfinity);
}
if (double.IsInfinity(x))
{
return(0);
}
double value, factor;
if (x < DoubleLimits.MachineEpsilon)
{
return((2 / Math.PI) * (Math.Log(x / 2) + Constants.EulerMascheroni));
}
// Bessel function of the second kind of order zero
// x <= 8, minimax rational approximations on root-bracketing intervals
// x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968
if (x <= 3) // x in [eps, 3]
{
const double Zero1 = 8.9357696627916752158e-01;
const double Zero1a = 3837883847.0 / 4294967296;
const double Zero1b = -8.66317862382940502854117587748531380699855129307710548e-11;
const double p0 = 1.0723538782003176831e+11;
const double p1 = -8.3716255451260504098e+09;
const double p2 = 2.0422274357376619816e+08;
const double p3 = -2.1287548474401797963e+06;
const double p4 = 1.0102532948020907590e+04;
const double p5 = -1.8402381979244993524e+01;
const double q0 = 5.8873865738997033405e+11;
const double q1 = 8.1617187777290363573e+09;
const double q2 = 5.5662956624278251596e+07;
const double q3 = 2.3889393209447253406e+05;
const double q4 = 6.6475986689240190091e+02;
const double q5 = 1.0;
double z = x * x;
double P = p0 + z * (p1 + z * (p2 + z * (p3 + z * (p4 + z * p5))));
double Q = q0 + z * (q1 + z * (q2 + z * (q3 + z * (q4 + z * q5))));
double y = (2 / Math.PI) * Math.Log(x / Zero1) * _Bessel.J0(x);
factor = (x + Zero1) * ((x - Zero1a) - Zero1b);
value = y + factor * (P / Q);
}
else if (x <= 5.5) // x in (3, 5.5]
{
const double Zero2 = 3.9576784193148578684e+00;
const double Zero2a = 16998099379.0 / 4294967296;
const double Zero2b = 9.846238317388612698651281418603765563630625507511794841e-12;
const double p0 = -2.2213976967566192242e+13;
const double p1 = -5.5107435206722644429e+11;
const double p2 = 4.3600098638603061642e+10;
const double p3 = -6.9590439394619619534e+08;
const double p4 = 4.6905288611678631510e+06;
const double p5 = -1.4566865832663635920e+04;
const double p6 = 1.7427031242901594547e+01;
const double q0 = 4.3386146580707264428e+14;
const double q1 = 5.4266824419412347550e+12;
const double q2 = 3.4015103849971240096e+10;
const double q3 = 1.3960202770986831075e+08;
const double q4 = 4.0669982352539552018e+05;
const double q5 = 8.3030857612070288823e+02;
const double q6 = 1.0;
double z = x * x;
double P = p0 + z * (p1 + z * (p2 + z * (p3 + z * (p4 + z * (p5 + z * p6)))));
double Q = q0 + z * (q1 + z * (q2 + z * (q3 + z * (q4 + z * (q5 + z * q6)))));
double y = (2 / Math.PI) * Math.Log(x / Zero2) * _Bessel.J0(x);
factor = (x + Zero2) * ((x - Zero2a) - Zero2b);
value = y + factor * (P / Q);
}
else if (x <= 8) // x in (5.5, 8]
{
const double Zero3 = 7.0860510603017726976e+00;
const double Zero3a = 15217178781.0 / 2147483648;
const double Zero3b = -5.07017534414388797421475310284896188222355301448323476e-11;
const double p0 = -8.0728726905150210443e+15;
const double p1 = 6.7016641869173237784e+14;
const double p2 = -1.2829912364088687306e+11;
const double p3 = -1.9363051266772083678e+11;
const double p4 = 2.1958827170518100757e+09;
const double p5 = -1.0085539923498211426e+07;
const double p6 = 2.1363534169313901632e+04;
const double p7 = -1.7439661319197499338e+01;
const double q0 = 3.4563724628846457519e+17;
const double q1 = 3.9272425569640309819e+15;
const double q2 = 2.2598377924042897629e+13;
const double q3 = 8.6926121104209825246e+10;
const double q4 = 2.4727219475672302327e+08;
const double q5 = 5.3924739209768057030e+05;
const double q6 = 8.7903362168128450017e+02;
const double q7 = 1.0;
double z = x * x;
double P = p0 + z * (p1 + z * (p2 + z * (p3 + z * (p4 + z * (p5 + z * (p6 + z * p7))))));
double Q = q0 + z * (q1 + z * (q2 + z * (q3 + z * (q4 + z * (q5 + z * (q6 + z * q7))))));
double y = (2 / Math.PI) * Math.Log(x / Zero3) * _Bessel.J0(x);
factor = (x + Zero3) * ((x - Zero3a) - Zero3b);
value = y + factor * (P / Q);
}
else // x in (8, infinity)
{
double y = 8 / x;
double z = y * y;
double rc, rs;
{
const double p0 = 2.2779090197304684302e+04;
const double p1 = 4.1345386639580765797e+04;
const double p2 = 2.1170523380864944322e+04;
const double p3 = 3.4806486443249270347e+03;
const double p4 = 1.5376201909008354296e+02;
const double p5 = 8.8961548424210455236e-01;
const double q0 = 2.2779090197304684318e+04;
const double q1 = 4.1370412495510416640e+04;
const double q2 = 2.1215350561880115730e+04;
const double q3 = 3.5028735138235608207e+03;
const double q4 = 1.5711159858080893649e+02;
const double q5 = 1.0;
double P = p0 + z * (p1 + z * (p2 + z * (p3 + z * (p4 + z * p5))));
double Q = q0 + z * (q1 + z * (q2 + z * (q3 + z * (q4 + z * q5))));
rc = P / Q;
}
{
const double p0 = -8.9226600200800094098e+01;
const double p1 = -1.8591953644342993800e+02;
const double p2 = -1.1183429920482737611e+02;
const double p3 = -2.2300261666214198472e+01;
const double p4 = -1.2441026745835638459e+00;
const double p5 = -8.8033303048680751817e-03;
const double q0 = 5.7105024128512061905e+03;
const double q1 = 1.1951131543434613647e+04;
const double q2 = 7.2642780169211018836e+03;
const double q3 = 1.4887231232283756582e+03;
const double q4 = 9.0593769594993125859e+01;
const double q5 = 1.0;
double P = p0 + z * (p1 + z * (p2 + z * (p3 + z * (p4 + z * p5))));
double Q = q0 + z * (q1 + z * (q2 + z * (q3 + z * (q4 + z * q5))));
rs = P / Q;
}
// The following code is:
//double z = x - 0.25f * Math.PI;
//value = (Constants.RecipSqrtHalfPI / Math.Sqrt(x)) * (rc * Math.Sin(z) + y * rs * Math.Cos(z));
// Which can be written as:
// (double, double) sincos = Math2.SinCos(x,-0.25);
// value = (Constants.RecipSqrtHalfPI / Math.Sqrt(x)) * (rc * sincos.Item1 + y * rs * sincos.Item2);
// Or, using trig addition rules, simplified to:
value = (Constants.RecipSqrtPI / Math.Sqrt(x)) * ((rc + y * rs) * Math.Sin(x) + (-rc + y * rs) * Math.Cos(x));
}
return(value);
}
/// <summary>
/// Returns the rising factorial:
/// <para>(x)_{n} = Γ(x+n)/Γ(x)</para>
/// </summary>
/// <returns>
/// <para> if n > 0, x*(x+1)*(x+2)...*(x+(n-1))</para>
/// <para> if n == 0, 1</para>
/// <para> if n < 0, 1/((x-1)*(x-2)*...*(x-n))</para>
/// </returns>
public static double FactorialRising(double x, int n)
{
// This section uses the following notation:
// falling factorial: (x)_{_n}
// rising factorial: (x)_{n}
// Standard eqn for the rising factorial:
// (x)_{n} = Γ(x+n)/Γ(x)
double result = double.NaN;
// (x)_{0} = 1
if (n == 0)
{
return(1);
}
if (double.IsNaN(x))
{
Policies.ReportDomainError("FactorialRising(x: {0}, n: {1}): Requires x not NaN", x, n);
return(double.NaN);
}
if (n == int.MinValue)
{
// avoid -int.MinValue error
// this will overflow/underflow, but here it is
if (x > 0 && IsInteger(x) && (x + n) <= 0)
{
Policies.ReportPoleError("FactorialRising(x: {0}, n: {1}): Divide by Zero", x, n);
return(double.NaN);
}
// (x)_{n} = (x)_{n+1}/(x+n)
return(FactorialRising(x, n + 1) / (x + n));
}
if (x == 0)
{
// (0)_{n} = 0, n > 0
if (n > 0)
{
return(0);
}
// (0)_{-n} = (-1)^n / n!
result = Math2.Factorial(-n);
if (IsOdd(n))
{
result = -result;
}
return(1 / result);
}
if (x < 0)
{
// For x less than zero, we really have a falling
// factorial, modulo a possible change of sign.
// Note: the falling factorial isn't defined for negative n
// so cover that first
if (n < -x)
{
// handle (n < 0 || ( n > 0 && n < -x)) directly
// (x)_{n} = Γ(1-x)/Γ(1-x-n)
result = Math2.TgammaDeltaRatio(1 - x, -n);
}
else
{
Debug.Assert(n > 0);
result = FactorialFalling(-x, n);
}
if (IsOdd(n))
{
result = -result;
}
return(result);
}
// x > 0
// standard case: (n > 0 || (n < 0 && x > -n))
// (x)_{n} = Γ(x+n)/Γ(x)
if (-x < n)
{
return(1 / Math2.TgammaDeltaRatio(x, n));
}
Debug.Assert((x + n) <= 0);
// (x)_{n} = (-1)^n * Γ(1-x)/Γ(1-x-n)
if (x < 1)
{
result = TgammaDeltaRatio(1 - x, -n);
if (IsOdd(n))
{
result = -result;
}
return(result);
}
// x >= 1
result = FactorialFalling(x - 1, -n);
if (result == 0)
{
if (IsInteger(x))
{
Policies.ReportPoleError("FactorialRising(x: {0}, n: {1}): Divide by Zero", x, n);
return(double.NaN);
}
return(double.PositiveInfinity);
}
return(1 / result);
}
/// <summary>
/// Returns the natural log of the Gamma Function = ln(|Γ(x)|).
/// Sets the sign = Sign(Γ(x)); 0 for poles or indeterminate.
/// </summary>
/// <param name="x">Lgamma function argument</param>
/// <param name="sign">Lgamma output value = Sign(Γ(x))</param>
public static double Lgamma(double x, out int sign)
{
sign = 0;
if (double.IsNaN(x))
{
Policies.ReportDomainError("Lgamma(x: {0}): NaN not allowed", x);
return(double.NaN);
}
if (double.IsInfinity(x))
{
if (x < 0)
{
Policies.ReportDomainError("Lgamma(x: {0}): Requires x != -Infinity", x);
return(double.PositiveInfinity);
}
sign = 1;
return(double.PositiveInfinity);
}
double result = 0;
sign = 1;
if (x <= 0)
{
if (IsInteger(x))
{
sign = 0;
Policies.ReportPoleError("Lgamma(x: {0}): Evaluation at zero, or a negative integer", x);
return(double.PositiveInfinity);
}
if (x > -GammaSmall.UpperLimit)
{
return(GammaSmall.Lgamma(x, out sign));
}
if (x <= -StirlingGamma.LowerLimit)
{
// Our Stirling routine does the reflection
// So no need to reflect here
result = StirlingGamma.Lgamma(x, out sign);
return(result);
}
else
{
double product = 1;
double xm2 = x - 2;
double xm1 = x - 1;
while (x < 1)
{
product *= x;
xm2 = xm1;
xm1 = x;
x += 1;
}
if (product < 0)
{
sign = -1;
product = -product;
}
result = -Math.Log(product) + _Lgamma.Rational_1_3(x, xm1, xm2);
return(result);
}
}
else if (x < 1)
{
if (x < GammaSmall.UpperLimit)
{
return(GammaSmall.Lgamma(x, out sign));
}
// Log(Γ(x)) = Log(Γ(x+1)/x)
result = -Math.Log(x) + _Lgamma.Rational_1_3(x + 1, x, x - 1);
}
else if (x < 3)
{
result = _Lgamma.Rational_1_3(x, x - 1, x - 2);
}
else if (x < 8)
{
// use recurrence to shift x to [2, 3)
double product = 1;
while (x >= 3)
{
product *= --x;
}
result = Math.Log(product) + _Lgamma.Rational_1_3(x, x - 1, x - 2);
}
else
{
// regular evaluation:
result = StirlingGamma.Lgamma(x);
}
return(result);
}
/// <summary>
/// Returns the Gamma Function = Γ(x)
/// </summary>
/// <param name="x">Tgamma function argument</param>
public static double Tgamma(double x)
{
if (double.IsNaN(x))
{
Policies.ReportDomainError("Tgamma(x: {0}): NaN not allowed", x);
return(double.NaN);
}
if (double.IsInfinity(x))
{
if (x < 0)
{
Policies.ReportDomainError("Tgamma(x: {0}): Requires x is not negative infinity", x);
return(double.NaN);
}
return(double.PositiveInfinity);
}
double result = 1;
if (x <= 0)
{
if (IsInteger(x))
{
Policies.ReportPoleError("Tgamma(x: {0}): Requires x != 0 and x be a non-negative integer", x);
return(double.NaN);
}
if (x >= -GammaSmall.UpperLimit)
{
return(GammaSmall.Tgamma(x));
}
if (x <= -StirlingGamma.LowerLimit)
{
const double MinX = -185;
if (x < MinX)
{
return(0);
}
// If we use the reflection formula directly for x < -NegMaxGamma, Tgamma will overflow;
// so, use recurrence to get x > -NegMaxGamma.
// The result is likely to be subnormal or zero.
double shiftedX = x;
double recurrenceMult = 1.0;
double NegMaxGamma = -(Math2.MaxFactorialIndex + 1);
while (shiftedX < NegMaxGamma)
{
recurrenceMult *= shiftedX;
shiftedX += 1;
}
result = -(Math.PI / StirlingGamma.Tgamma(-shiftedX)) / (shiftedX * recurrenceMult * Math2.SinPI(shiftedX));
return(result);
}
// shift x to > 0:
double product = 1;
while (x < 0)
{
product *= x++;
}
result /= product;
// fall through
}
if (x < 1)
{
if (x <= GammaSmall.UpperLimit)
{
result *= GammaSmall.Tgamma(x);
}
else
{
// Γ(x) = Exp(LogΓ(x+1))/x
result *= Math.Exp(_Lgamma.Rational_1_3(x + 1, x, x - 1)) / x;
}
}
else if (IsInteger(x) && (x <= Math2.FactorialTable.Length))
{
// Γ(n) = (n-1)!
result *= Math2.FactorialTable[(int)x - 1];
}
else if (x < StirlingGamma.LowerLimit)
{
// ensure x is in [1, 3)
// if x >= 3, use recurrence to shift x to [2, 3)
while (x >= 3)
{
result *= --x;
}
result *= Math.Exp(_Lgamma.Rational_1_3(x, x - 1, x - 2));
}
else
{
result *= StirlingGamma.Tgamma(x);
}
return(result);
}
