/// <summary>
/// Implements the Exponential Series for integer n
/// <para>Sum = ((-z)^(n-1)/(n-1)!) * (ψ(n)-Log(z)) - Σ( (k==n-1) ? 0 : ((-1)^k * z^k)/((k-n+1) * k!)) k={0,∞}</para>
/// </summary>
/// <param name="n"></param>
/// <param name="z"></param>
/// <returns></returns>
/// <see href="http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/06/01/04/01/02/0005/"/>
public static double En_Series(int n, double z)
{
Debug.Assert(n > 0);
const double DefaultTolerance = 2 * DoubleLimits.MachineEpsilon;
// the sign of the summation is negative
// so negate these
double sum = (n > 1) ? -1.0 / (1.0 - n) : 0;
double term = -1.0;
int k = 1;
for (; k < n - 1; k++)
{
term *= -z / k;
if (term == 0)
{
break;
}
sum += term / (k - n + 1);
}
sum += Math.Pow(-z, n - 1) * (Math2.Digamma(n) - Math.Log(z)) / Math2.Factorial(n - 1);
if (term == 0)
{
return(sum);
}
// skip n-1 term
term *= -z / k;
for (int i = 0; i < Policies.MaxSeriesIterations; i++)
{
double prevSum = sum;
k++;
term *= -z / k;
double delta = term / (k - n + 1);
sum += delta;
if (Math.Abs(delta) <= Math.Abs(prevSum) * DefaultTolerance)
{
return(sum);
}
}
Policies.ReportConvergenceError("Series did not converge in {0} iterations", Policies.MaxSeriesIterations);
return(double.NaN);
}
public static (double Result, int Iterations) RefineQ_SmallA_LargeZ(double z, double p, double q, double aGuess)
{
Debug.Assert(z >= 0, "Requires z >= 0");
Debug.Assert(aGuess >= 0 && aGuess <= double.MaxValue, "Requires finite aGuess >= 0");
Debug.Assert(q <= 0.5, "Requires q <= 0.5");
Debug.Assert(aGuess < z + 1);
// f = Log(z^a/Gamma(a))-Log(q);
// f/f' = (Log(z^a/Gamma(a))-Log(q))/ (Log(z) - Polygamma(0, a))
//
// http://functions.wolfram.com/GammaBetaErf/GammaRegularized/06/02/02/0001/
// For z->Infinity, Q(a,z) ~= z^(a-1)*e^-z/Gamma(a) * (1 - (1-a)/z ...)
// Find a quick estimate of a
// where Log[q] = Log[z^(a-1)*e^-z/Gamma(a)] to get into the ballpark
double lz = Math.Log(z);
double lq = Math.Log(q);
int i = 0;
for (; i < 1; i++)
{
const double eps = 1.0 / 2;
double f = (aGuess - 1) * lz - z - Math2.Lgamma(aGuess) - lq;
if (f == 0)
{
break;
}
var delta = f / (lz - Math2.Digamma(aGuess));
double aLast = aGuess;
aGuess -= delta;
if (Math.Abs(delta) <= eps * Math.Abs(aLast))
{
break;
}
}
return(aGuess, i);
}
