public static double TgammaLowerImp(double a, double x)
{
const Routine routine = Routine.Lower;
// special values
if (x == 0.0)
{
return(0.0);
}
if (double.IsInfinity(x))
{
return(Math2.Tgamma(a));
}
if (a == 1.0)
{
return(-Math2.Expm1(-x)); // 1.0 - Math.Exp(-x);
}
if (a == 0.5)
{
return(Constants.SqrtPI * Math2.Erf(Math.Sqrt(x)));
}
// Process small a, large x with asymptotic series, which is faster than CF
if (x >= Asym_MinLargeZ(a))
{
return(Math2.Tgamma(a) - Asym_SeriesLargeZ(a, x) * Prefix(a, x) / x);
}
// is a small
if ((a < 30) && (x >= a + 1) && (x < DoubleLimits.MaxLogValue))
{
double frac = a - Math.Floor(a);
if (frac == 0)
{
return(IntegerA(routine, a, x));
}
else if (frac == 0.5)
{
return(HalfIntegerA(routine, a, x));
}
}
if (x < (a + 1))
{
if (a < 1)
{
return(SmallA(routine, a, x));
}
return(Prefix(a, x, 1 / a) * LowerSeries(a, x));
}
// Use CF for x > a+1
return(TgammaMinusUpperFraction(a, x));
}
/// <summary>
/// Returns (x^y)-1 with improved accuracy when x is close to 1, or when y is very small
/// </summary>
/// <param name="x">Powm1 function value</param>
/// <param name="y">Powm1 function exponent</param>
/// <returns></returns>
/// <remarks>
/// Note: The .NET treatment of Math.Pow for NaN parameters is different from that of ISO C.
/// Math.Pow: for any parameter that is NaN, return NaN.
/// ISO C: 1^y == 1, x^0 = 1, even if x,y are NaN
/// see: http://msdn.microsoft.com/en-us/library/system.math.pow.aspx
/// see also: http://pubs.opengroup.org/onlinepubs/009695399/functions/pow.html
/// for the conditions on Math.Pow.
/// This implementation treats NaN similarly to .NET - treat every NaN as an error condition.
/// </remarks>
public static double Powm1(double x, double y)
{
if (double.IsNaN(x) || double.IsNaN(y))
{
Policies.ReportDomainError("Powm1(x: {0}, y: {1}): Requires x, y not NaN", x, y);
return(double.NaN);
}
if (y == 0 || x == 1.0)
{
return(0); // x^0 = 1, 1^y = 1
}
if (x == 0)
{
return((y > 0) ? -1 : double.PositiveInfinity);
}
double xOrig = x;
double yOrig = y;
if (x < 0)
{
if (!IsInteger(y))
{
Policies.ReportDomainError("Powm1(x: {0}, y: {1}): Requires integer y when x < 0", x, y);
return(double.NaN);
}
if (IsOdd(y))
{
return(Math.Pow(x, y) - 1);
}
x = -x;
}
Debug.Assert(x > 0, "Expecting x > 0");
// log(x) = log((z+1)/(z-1)) = 2z * (1+z^2/3 .. ), where z = (x-1)/(x+1)
// see https://en.wikipedia.org/wiki/Natural_logarithm
// Series converges fastest when z is closest to 1
if ((x - 1) * y <= 0.25 * (x + 1))
{
double p = Math.Log(x) * y;
if (p <= 1)
{
return(Math2.Expm1(p));
}
// otherwise fall though:
}
return(Math.Pow(x, y) - 1);
}
/// <summary>
/// Returns Sqrt(x+1)-1 with improved accuracy for |x| ≤ 0.75
/// </summary>
/// <param name="x">The function argument</param>
public static double Sqrt1pm1(double x)
{
if (!(x >= -1))
{
Policies.ReportDomainError("Sqrt1pm1(x: {0}): Requires x >= -1", x);
return(double.NaN);
}
if (double.IsInfinity(x))
{
return(x);
}
if (Math.Abs(x) <= 0.75)
{
return(Math2.Expm1(Math2.Log1p(x) / 2));
}
return(Math.Sqrt(1 + x) - 1);
}
/// <summary>
/// Returns Γ(x + 1) - 1 with accuracy improvements in [-0.5, 2]
/// </summary>
/// <param name="x">Tgamma1pm1 function argument</param>
public static double Tgamma1pm1(double x)
{
if (double.IsNaN(x))
{
Policies.ReportDomainError("Tgamma1pm1(x: {0}): NaN not allowed", x);
return(double.NaN);
}
if (double.IsInfinity(x))
{
if (x < 0)
{
Policies.ReportDomainError("Tgamma1pm1(x: {0}): Requires x != -Infinity", x);
return(double.NaN);
}
return(double.PositiveInfinity);
}
double result;
if (x < -0.5 || x >= 2)
{
return(Math2.Tgamma(1 + x) - 1); // Best method is simply to subtract 1 from tgamma:
}
if (Math.Abs(x) < GammaSmall.UpperLimit)
{
return(GammaSmall.Tgamma1pm1(x));
}
if (x < 0)
{
// compute exp( log( Γ(x+2)/(1+x) ) ) - 1
result = Math2.Expm1(-Math2.Log1p(x) + _Lgamma.Rational_1_3(x + 2, x + 1, x));
}
else
{
// compute exp( log(Γ(x+1)) ) - 1
result = Math2.Expm1(_Lgamma.Rational_1_3(x + 1, x, x - 1));
}
return(result);
}
public static double GammaPImp(double a, double x)
{
const Routine routine = Routine.P;
// special values
if (x == 0.0)
{
return(0.0);
}
if (double.IsInfinity(x))
{
return(1.0);
}
if (a == 0)
{
return(1);
}
if (a == 1.0)
{
return(-Math2.Expm1(-x)); // 1.0 - Math.Exp(-x)
}
if (a == 0.5)
{
return(Math2.Erf(Math.Sqrt(x)));
}
// Process small a, large x with asymptotic series, which is faster than CF
if (x >= Asym_MinLargeZ(a))
{
return(1 - Asym_SeriesLargeZ(a, x) * PrefixRegularized(a, x) / x);
}
// is a small
if ((a < 30) && (x >= a + 1) && (x < DoubleLimits.MaxLogValue))
{
double frac = a - Math.Floor(a);
if (frac == 0)
{
return(IntegerA(routine, a, x));
}
else if (frac == 0.5)
{
return(HalfIntegerA(routine, a, x));
}
}
//
// Begin by testing whether we're in the "bad" zone
// where the result will be near 0.5 and the usual
// series and continued fractions are slow to converge:
//
if (a > 20)
{
// This second limit below is chosen so that we use Temme's expansion
// only if the result would be larger than about 10^-6.
// Below that the regular series and continued fractions
// converge OK, and if we use Temme's method we get increasing
// errors from the dominant erfc term as it's (inexact) argument
// increases in magnitude.
double sigma = Math.Abs((x - a) / a);
if (sigma < 0.4 || (a > 200 && 20 / a > sigma * sigma))
{
double t = TemmeSymmetricAsym.GammaP(a, x);
return((x >= a) ? 1 - t : t);
}
}
if (x < (a + 1))
{
if (a < 1)
{
return(SmallA(routine, a, x));
}
return(PrefixRegularized(a, x) * LowerSeries(a, x) / a);
}
// Use CF for x > a+1
return(1.0 - PrefixRegularized(a, x) * UpperFraction(a, x));
}
public static double ImpP(double z, double p, double q)
{
Debug.Assert(p > 0 && p < 0.5);
// Set outside limits
// Since P(a, z) >= 1/2 for a >= z, so minA < z
double minA = z;
double maxA = double.MaxValue;
double aGuess = 0;
double step = 0.5;
if (z <= 1)
{
var e = -Math2.Expm1(-z);
if (p >= e)
{
if (p == e)
{
return(1);
}
// This is one of the outside limits of the GammaP/GammaQ
// From: http://dlmf.nist.gov/8.10#E11
// If a >= 1, P(a, x) <= (1-e^-x)^a
// If a <= 1, P(a, x) >= (1-e^-x)^a
// not a bad guess, for small z and small a
// but gets much worse as x increases or a >> 1
maxA = 1;
var limit = Math.Log(p) / Math.Log(e);
if (limit < 1)
{
minA = Math.Max(minA, limit);
}
// could use bisection, but the root often lies nearer to minA
step = (1 - minA) * 0.125;
aGuess = minA * (1 + step);
var r = SolveA(z, p, q, aGuess, step, minA, maxA);
ExtraDebugLog(z, p, q, r.Result, aGuess, step, minA, maxA, r.Iterations, "");
return(r.Result);
}
else
{
minA = 1;
var limit = Math.Log(p) / Math.Log(e);
if (limit > 1)
{
maxA = Math.Min(maxA, limit);
}
}
}
aGuess = 0.5 + InversePoissonCornishFisher(z, q, p);
if (aGuess > 100)
{
step = 0.01;
}
else if (aGuess > 10)
{
step = 0.01;
}
else
{
step = 0.05;
}
#if EXTRA_DEBUG
double og = aGuess;
#endif
aGuess = Math.Max(aGuess, minA);
aGuess = Math.Min(aGuess, maxA);
if (p < DoubleLimits.MachineEpsilon)
{
step *= 10;
}
#if EXTRA_DEBUG
var gs = new Stack <double>();
if (og != aGuess)
{
gs.Push(og);
}
#endif
// When p is very small, our Inverse Poisson Cornish Fisher can be significantly off.
// Refining the guess here can significantly reduce iterations later
int refineIter = 0;
if (aGuess >= 5 && z / (1 + aGuess) <= 0.7)
{
#if EXTRA_DEBUG
gs.Push(aGuess);
#endif
(aGuess, refineIter) = RefineP_LargeA_SmallZ(z, p, q, aGuess, ref minA, ref maxA);
step = aGuess * 1e-6;
aGuess = Math.Max(aGuess, minA);
aGuess = Math.Min(aGuess, maxA);
}
var rr = SolveA(z, p, q, aGuess, step, minA, maxA);
#if EXTRA_DEBUG
if (rr.Iterations + refineIter > 6)
{
string refString = (refineIter == 0) ? string.Empty : "(" + rr.Iterations + "+" + refineIter + ")";
string guesses = (gs.Count == 0) ? string.Empty : ", og: (" + string.Join(", ", gs) + ")";
Debug.WriteLine($"PInvA(z: {z}, p: {p}) = {rr.Result}, g: {aGuess}, TIter = {rr.Iterations + refineIter}{refString}{guesses}");
}
#endif
return(rr.Result);
}
public static double ImpQ(double z, double p, double q)
{
Debug.Assert(q <= 0.5);
if (q == 0)
{
return(DoubleLimits.MinNormalValue);
}
#if EXTRA_DEBUG
var gs = new Stack <double>();
#endif
// Set outside limits
// Since Q(a, z) > 1/2 for a >= z+1, so maxA < z+1
double minA = DoubleLimits.MinNormalValue;
double maxA = z + 1;
if (maxA == z)
{
maxA = Math2.FloatNext(z);
}
double step = 0.5;
double aGuess = 0;
// We can use the relationship between the incomplete
// gamma function and the poisson distribution to
// calculate an approximate inverse.
// Mostly it is pretty accurate, except when a is small or q is tiny.
// Case 1: a <= 1 : Inverse Cornish Fisher is unreliable in this area
var e = Math.Exp(-z);
if (q <= e)
{
if (q == e)
{
return(1);
}
maxA = 1;
// If a <= 1, P(a, x) = 1 - Q(a, x) >= (1-e^-x)^a
var den = (z > 0.5) ? Math2.Log1p(-Math.Exp(-z)) : Math.Log(-Math2.Expm1(-z));
var limit = Math2.Log1p(-q) / den;
if (limit < 1)
{
minA = Math.Max(minA, limit);
}
step = (1 - minA) * 0.125;
aGuess = minA * (1 + step);
if (minA < 0.20)
{
// Use a first order approximation Q(a, z) at a=0
// Mathematica: Assuming[ a >= 0 && z > 0, FunctionExpand[Normal[Series[GammaRegularized[a, z], {a, 0, 1}]]]]
// q = -a * ExpIntegralEi[-z]
// The smaller a is the better the guess
// as z->0 term2/term1->1/2*Log[z], which is adequate for double precision
aGuess = -q / Math2.Expint(-z);
if (aGuess <= DoubleLimits.MachineEpsilon)
{
ExtraDebugLog(z, p, q, aGuess, aGuess, step, minA, maxA, 0, "");
return(aGuess);
}
step = aGuess * 0.125;
}
else if (minA >= 0.40)
{
// Use a first order approximation Q(a, z) at a=1
// Mathematica: Assuming[ a >= 0 && z > 0, FunctionExpand[Normal[Series[GammaRegularized[a, z], {a, 1, 1}]]]]
var d = Math.Exp(-z) * (Math.Log(z) + Constants.EulerMascheroni) + Math2.Expint(1, z);
aGuess = (q - Math.Exp(-z)) / d + 1;
}
aGuess = Math.Max(aGuess, minA);
aGuess = Math.Min(aGuess, maxA);
var r = SolveA(z, p, q, aGuess, step, minA, maxA);
ExtraDebugLog(z, p, q, r.Result, aGuess, step, minA, maxA, r.Iterations, "");
return(r.Result);
}
// Case 2: a > 1 : Inverse Poisson Corning Fisher approximation improves as a->Infinity
minA = 1;
aGuess = 0.5 + InversePoissonCornishFisher(z, q, p);
if (aGuess > 100)
{
step = 0.01;
}
else if (aGuess > 10)
{
step = 0.01;
}
else
{
// our poisson approximation is weak.
step = 0.05;
}
#if EXTRA_DEBUG
double og = aGuess;
#endif
aGuess = Math.Max(aGuess, minA);
aGuess = Math.Min(aGuess, maxA);
#if EXTRA_DEBUG
if (og != aGuess)
{
gs.Push(og);
}
#endif
// For small values of q, our Inverse Poisson Cornish Fisher guess can be way off
// Try to come up with a better guess
if (q < DoubleLimits.MachineEpsilon)
{
step *= 10;
if (Math.Abs((aGuess - 1) / z) < 1.0 / 8)
{
#if EXTRA_DEBUG
double old = aGuess;
#endif
(aGuess, _) = RefineQ_SmallA_LargeZ(z, p, q, aGuess);
#if EXTRA_DEBUG
if (old != aGuess)
{
gs.Push(old);
}
#endif
}
}
var rr = SolveA(z, p, q, aGuess, step, minA, maxA);
#if EXTRA_DEBUG
if (rr.Iterations > 6)
{
string guesses = (gs.Count == 0) ? string.Empty : ", og: (" + string.Join(", ", gs) + ")";
ExtraDebugLog(z, p, q, rr.Result, aGuess, step, minA, maxA, rr.Iterations, guesses);
}
#endif
return(rr.Result);
}
// get the upper and lower limits on x for the GammaInv functions
static (double upperLimit, double lowerLimit) GammaInvLimits(double a, double p, double q, bool getQ)
{
Debug.Assert(a > 0 && a != 1);
Debug.Assert(p >= 0 && p <= 1);
Debug.Assert(q >= 0 && q <= 1);
// set the upper and lower limits
double lower = 0;
double upper = double.MaxValue;
// Find the upper or lower limits using http://dlmf.nist.gov/8.10.E11
// For Q, the limits are:
// limit1 = -ln(1-(1-q)^(1/a))
// limit2 = limit1 * (Gamma(1+a)^(1/a))
// For P, the limits are
// limit1 = -ln(1-p^(1/a))
// limit2 = limit1 * (Gamma(1+a)^(1/a))
double limit1;
if (getQ)
{
// if (1-q)^(1/a) > 1/2
double y = Math2.Log1p(-q) / a;
if (y > -Constants.Ln2)
{
limit1 = -Math.Log(-Math2.Expm1(y));
}
else
{
limit1 = -Math2.Log1p(-Math.Exp(y));
}
}
else
{
// double y = Math.Exp(Math.Log(p)/a);
double y = Math.Pow(p, 1 / a);
if (y > 0.5)
{
limit1 = -Math.Log(-Math2.Expm1(Math.Log(p) / a));
}
else
{
limit1 = -Math2.Log1p(-y);
}
}
if (a < 1)
{
upper = limit1;
lower = limit1 * Math.Exp(Math2.Log1p(Math2.Tgamma1pm1(a)) / a);
}
else if (a > 1)
{
lower = limit1;
upper = limit1 * Math.Exp(Math2.Lgamma(1 + a) / a);
}
Debug.Assert(upper >= lower, "Upper < lower");
// Our computations for upper and lower can be slightly off in the last few digits
// which can be problematic when the solution is close to max or min
// So, give ourselves some wiggle room
const double factor = 1.125;
return(lower / factor, upper *factor);
}
