/// <summary>
/// Use the root finder to solve Q(a,x) == q, for x
/// </summary>
/// <param name="a"></param>
/// <param name="q"></param>
/// <param name="guess"></param>
/// <param name="min"></param>
/// <param name="max"></param>
/// <returns></returns>
public static double SolveGivenAQ(double a, double q, double guess, double min, double max)
{
Func <double, ValueTuple <double, double, double> > halleyF = (double x) => {
// Calculate Q(x) - q and the first two derivatives
double f = Math2.GammaQ(a, x) - q;
double f1 = -Math2.GammaPDerivative(a, x);
double f2 = -f1 * (1.0 + (1.0 - a) / x);
return(f, f1, f2);
};
const double relTolerance = RootFinder.DefaultRelTolerance;
const double absTolerance = 0;
RootResults rr = RootFinder.Halley(halleyF, guess, min, max, relTolerance, absTolerance, Policies.MaxRootIterations);
if (rr == null)
{
Policies.ReportRootNotFoundError("Invalid parameter in root solver");
return(double.NaN);
}
#if EXTRA_DEBUG
Debug.WriteLine("Halley iterations: {0}", rr.Iterations);
#endif
if (!rr.Success)
{
Policies.ReportRootNotFoundError("Root not found after {0} iterations", rr.Iterations);
return(double.NaN);
}
return(rr.SolutionX);
}
public static (double Result, int Iterations) SolveA(double z, double p, double q, double aGuess, double step, double min, double max)
{
Debug.Assert(aGuess >= min && aGuess <= max, "Guess out of range");
Func <double, double> f;
if (p < q)
{
f = a => Math2.GammaP(a, z) - p;
}
else
{
f = a => Math2.GammaQ(a, z) - q;
}
//
// Use our generic derivative-free root finding procedure.
// We could use Newton steps here, taking the PDF of the
// Poisson distribution as our derivative, but that's
// even worse performance-wise than the generic method :-(
//
// dQ/da is increasing, dP/da is decreasing
FunctionShape fShape = (p < q) ? FunctionShape.Decreasing : FunctionShape.Increasing;
RootResults rr = RootFinder.Toms748Bracket(f, aGuess, step, fShape, min, max);
if (rr == null)
{
Policies.ReportRootNotFoundError($"Invalid Parameter: PQInvA(z: {z}, p: {p}, q: {q}) [{min}, {max}] g: {aGuess}");
return(double.NaN, 0);
}
if (!rr.Success)
{
Policies.ReportRootNotFoundError("Root not found after {0} iterations", rr.Iterations);
return(double.NaN, rr.Iterations);
}
return(rr.SolutionX, rr.Iterations);
}
