/// <summary>
/// Returns the value "x" such that: p == GammaP(a, x);
/// </summary>
/// <param name="a">Requires a > 0</param>
/// <param name="p">Requires 0 ≤ p ≤ 1</param>
public static double GammaPInv(double a, double p)
{
if ((!(a > 0) || double.IsInfinity(a)) ||
(!(p >= 0 && p <= 1)))
{
Policies.ReportDomainError("GammaPInv(a: {0}, p: {1}): Requires finite a > 0; p in [0,1]", a, p);
return(double.NaN);
}
if (p == 1)
{
return(double.PositiveInfinity);
}
if (p == 0)
{
return(0);
}
// Since: P(1,x) = p = 1.0 - Math.Exp(-x), therefore: x = ln(1-p)
if (a == 1.0)
{
return(-Math2.Log1p(-p));
}
// Since P(1/2, x) = p = erf(sqrt(x)), therefore x = ErfInv(p)^2
if (a == 0.5)
{
return(Squared(Math2.ErfInv(p)));
}
const double cutoff = 0.8125; // chosen through experimentation
if (p > cutoff)
{
return(Math2.GammaQInv(a, 1.0 - p));
}
double guess = _IgammaInv.GammaPInvGuess(a, p, 1 - p, out bool has10Digits);
// set the upper and lower limits
var(lower, upper) = _IgammaInv.GammaPInvLimits(a, p);
// there can be some numerical uncertainties in the limits,
// particularly for denormalized values, so... adjust the limits
if (upper == 0 && lower == 0)
{
return(0.0);
}
if (guess <= DoubleLimits.MinNormalValue)
{
return(guess);
}
if (guess <= lower)
{
lower = DoubleLimits.MinNormalValue;
}
if (guess > upper)
{
upper = guess;
}
return(_IgammaInv.SolveGivenAP(a, p, guess, lower, upper));
}