Numeric.SpecFunctions.Internal

Error function.

\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} \exp(-t^2) dt

Function limits are:

\begin{aligned} &\operatorname{erf}(-\infty) &=& -1 \\ &\operatorname{erf}(0) &=& \phantom{-}\,0 \\ &\operatorname{erf}(+\infty) &=& \phantom{-}\,1 \\ \end{aligned}

Complementary error function.

\operatorname{erfc}(x) = 1 - \operatorname{erf}(x)

Function limits are:

\begin{aligned} &\operatorname{erf}(-\infty) &=&\, 2 \\ &\operatorname{erf}(0) &=&\, 1 \\ &\operatorname{erf}(+\infty) &=&\, 0 \\ \end{aligned}

Inverse of erf.

Inverse of erfc.

Compute the logarithm of the gamma function, Γ(x).

\Gamma(x) = \int_0^{\infty}t^{x-1}e^{-t}\,dt = (x - 1)!

This implementation uses Lanczos approximation. It gives 14 or more significant decimal digits, except around x = 1 and x = 2, where the function goes to zero.

Returns ∞ if the input is outside of the range (0 < x ≤ 1e305).

Synonym for logGamma. Retained for compatibility

Compute the normalized lower incomplete gamma function γ(z,x). Normalization means that γ(z,∞)=1

\gamma(z,x) = \frac{1}{\Gamma(z)}\int_0^{x}t^{z-1}e^{-t}\,dt

Uses Algorithm AS 239 by Shea.

Inverse incomplete gamma function. It's approximately inverse of incompleteGamma for the same z. So following equality approximately holds:

invIncompleteGamma z . incompleteGamma z ≈ id

Compute ψ(x), the first logarithmic derivative of the gamma function.

\psi(x) = \frac{d}{dx} \ln \left(\Gamma(x)\right) = \frac{\Gamma'(x)}{\Gamma(x)}

Uses Algorithm AS 103 by Bernardo, based on Minka's C implementation.

Compute the natural logarithm of the beta function.

B(a,b) = \int_0^1 t^{a-1}(1-t)^{b-1}\,dt = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}

Regularized incomplete beta function.

I(x;a,b) = \frac{1}{B(a,b)} \int_0^x t^{a-1}(1-t)^{b-1}\,dt

Uses algorithm AS63 by Majumder and Bhattachrjee and quadrature approximation for large p and q.

Regularized incomplete beta function. Same as incompleteBeta but also takes logarithm of beta function as parameter.

Compute inverse of regularized incomplete beta function. Uses initial approximation from AS109, AS64 and Halley method to solve equation.

Compute sinc function sin(x)/x

Compute log(1+x)-x:

O(log n) Compute the logarithm in base 2 of the given value.

Compute the factorial function n!. Returns +∞ if the input is above 170 (above which the result cannot be represented by a 64-bit Double).

Compute the natural logarithm of the factorial function. Gives 16 decimal digits of precision.

Calculate the error term of the Stirling approximation. This is only defined for non-negative values.

\operatorname{stirlingError}(n) = \log(n!) - \log(\sqrt{2\pi n}\frac{n}{e}^n)

Compute the binomial coefficient n `choose` k. For values of k > 50, this uses an approximation for performance reasons. The approximation is accurate to 12 decimal places in the worst case

Example:

7 `choose` 3 == 35

Compute logarithm of the binomial coefficient.

Calculate binomial coefficient using exact formula

Quickly compute the natural logarithm of n choose k, with no checking.

Less numerically stable:

exp $ lg (n+1) - lg (k+1) - lg (n-k+1)
  where lg = logGamma . fromIntegral

Compute the log gamma correction factor for Stirling approximation for x ≥ 10. This correction factor is suitable for an alternate (but less numerically accurate) definition of logGamma:

\log\Gamma(x) = \frac{1}{2}\log(2\pi) + (x-\frac{1}{2})\log x - x + \operatorname{logGammaCorrection}(x)

log1p x computes log (1 + x), but provides more precise results for small (absolute) values of x if possible.

expm1 x computes exp x - 1, but provides more precise results for small (absolute) values of x if possible.