/// <summary>
/// Returns the Beta function
/// <para>B(a,b) = Γ(a)*Γ(b)/Γ(a+b)</para>
/// </summary>
/// <param name="a">Requires finite a > 0</param>
/// <param name="b">Requires finite b > 0</param>
public static double Beta(double a, double b)
{
double c = a + b;
if ((!(a > 0) || double.IsInfinity(a))
|| (!(b > 0) || double.IsInfinity(b))) {
Policies.ReportDomainError("Beta(a: {0}, b: {1}): Requires finite a,b > 0", a, b);
return double.NaN;
}
if (double.IsInfinity(c)) {
Policies.ReportDomainError("Beta(a: {0}, b: {1}): Requires finite c == a+b: c = {2}", a, b, c);
return double.NaN;
}
// Special cases:
if ((c == a) && (b < DoubleLimits.MachineEpsilon))
return Math2.Tgamma(b);
if ((c == b) && (a < DoubleLimits.MachineEpsilon))
return Math2.Tgamma(a);
if (b == 1)
return 1 / a;
if (a == 1)
return 1 / b;
// B(a,b) == B(b, a)
if (a < b)
Utility.Swap(ref a, ref b);
// from this point a >= b
if (a < 1) {
// When x < TinyX, Γ(x) ~ 1/x
const double SmallX = DoubleLimits.MachineEpsilon / Constants.EulerMascheroni;
if ( a < SmallX ) {
if (c < SmallX)
return (c / a) / b;
Debug.Assert(c <= GammaSmall.UpperLimit);
return 1/(GammaSmall.Tgamma(c) * a * b);
}
if ( b < SmallX )
return Tgamma(a) / (b * Tgamma(c));
return Tgamma(a) * ( Tgamma(b)/Tgamma(c) );
}
// Our Stirling series is more accurate than Lanczos
if (a >= StirlingGamma.LowerLimit ) {
if (b >= StirlingGamma.LowerLimit)
return StirlingGamma.Beta(a, b);
// for large a, Γ(a)/Γ(a + b) ~ a^-b, so don't underflow too soon
if (b < 1 || b * Math.Log(a) <= -DoubleLimits.MinLogValue)
return Tgamma(b) * StirlingGamma.TgammaDeltaRatio(a, b);
// fall through for very large a small 1 < b < LowerLimit
}
return Lanczos.Beta(a, b);
}
/// <summary>
/// Returns Log(Beta(a,b))
/// </summary>
/// <param name="a">Requires finite a > 0</param>
/// <param name="b">Requires finite b > 0</param>
public static double LogBeta(double a, double b)
{
double c = a + b;
if ((!(a > 0) || double.IsInfinity(a))
|| (!(b > 0) || double.IsInfinity(b))) {
Policies.ReportDomainError("Beta(a: {0}, b: {1}): Requires finite a,b > 0", a, b);
return double.NaN;
}
if (double.IsInfinity(c)) {
Policies.ReportDomainError("Beta(a: {0}, b: {1}): Requires finite c == a+b: c = {2}", a, b, c);
return double.NaN;
}
// Special cases:
if ((c == a) && (b < DoubleLimits.MachineEpsilon))
return Math2.Lgamma(b);
if ((c == b) && (a < DoubleLimits.MachineEpsilon))
return Math2.Lgamma(a);
if (b == 1)
return -Math.Log(a);
if (a == 1)
return -Math.Log(b);
// B(a,b) == B(b, a)
if (a < b)
Utility.Swap(ref a, ref b);
// from this point a >= b
if (a < 1) {
// When x < TinyX, Γ(x) ~ 1/x
const double SmallX = DoubleLimits.MachineEpsilon / Constants.EulerMascheroni;
if (a < SmallX) {
if (c < SmallX) {
// In this range, Beta(a, b) ~= 1/a + 1/b
// so, we won't have an argument overflow as long as
// the min(a, b) >= 2*DoubleLimits.MinNormalValue
if (b < 2 * DoubleLimits.MinNormalValue)
return Math.Log(c / a) - Math.Log(b);
return Math.Log((c / a) / b);
}
Debug.Assert(c <= GammaSmall.UpperLimit);
return -Math.Log((GammaSmall.Tgamma(c) * a * b));
}
if (b < SmallX)
return Math.Log(Tgamma(a) / (b * Tgamma(c)));
return Math.Log((Tgamma(a) / Tgamma(c)) * Tgamma(b));
} else if (a >= StirlingGamma.LowerLimit) {
if ( b >= StirlingGamma.LowerLimit )
return StirlingGamma.LogBeta(a, b);
return StirlingGamma.LgammaDelta(a, b) + Lgamma(b);
}
return Lanczos.LogBeta(a, b);
}
/// <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);
}