Numeric.SpecFunctions

Special functions and factorials.

Error function

erf :: Double -> Double

Error function.

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

Function limits are:

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

erfc :: Double -> Double

Complementary error function.

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

Function limits are:

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

invErf

Arguments

:: Double

p ∈ [-1,1]

-> Double

Inverse of erf.

invErfc

Arguments

:: Double

p ∈ [0,2]

-> Double

Inverse of erfc.

Gamma function

logGamma :: Double -> Double

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).

logGammaL :: Double -> Double

Synonym for logGamma. Retained for compatibility

incompleteGamma

Arguments

:: Double

z ∈ (0,∞)

-> Double

x ∈ (0,∞)

-> Double

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.

invIncompleteGamma

Arguments

:: Double

z ∈ (0,∞)

-> Double

p ∈ [0,1]

-> Double

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

invIncompleteGamma z . incompleteGamma z ≈ id

digamma :: Double -> Double

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.

Beta function

logBeta

Arguments

:: Double

a > 0

-> Double

b > 0

-> Double

Compute the natural logarithm of the beta function.

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

incompleteBeta

Arguments

:: Double

a > 0

-> Double

b > 0

-> Double

x, must lie in [0,1] range

-> Double

Regularized incomplete beta function.

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

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

incompleteBeta_

Arguments

:: Double

logarithm of beta function for given p and q

-> Double

a > 0

-> Double

b > 0

-> Double

x, must lie in [0,1] range

-> Double

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

invIncompleteBeta

Arguments

:: Double

a > 0

-> Double

b > 0

-> Double

x ∈ [0,1]

-> Double

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

Sinc

sinc :: Double -> Double

Compute sinc function sin(x)/x

Logarithm

log1p :: Double -> Double

Compute the natural logarithm of 1 + x. This is accurate even for values of x near zero, where use of log(1+x) would lose precision.

log1pmx :: Double -> Double

Compute log(1+x)-x:

log2 :: Int -> Int

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

Exponent

expm1 :: Double -> Double

Compute exp x - 1 without loss of accuracy for x near zero.

Factorial

factorial :: Int -> Double

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

logFactorial :: Integral a => a -> Double

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

stirlingError :: Double -> Double

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

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

Combinatorics

choose :: Int -> Int -> Double

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

logChoose :: Int -> Int -> Double

Compute logarithm of the binomial coefficient.

References

• Bernardo, J. (1976) Algorithm AS 103: Psi (digamma) function. /Journal of the Royal Statistical Society. Series C (Applied Statistics)/ 25(3):315-317. http://www.jstor.org/stable/2347257
• Cran, G.W., Martin, K.J., Thomas, G.E. (1977) Remark AS R19 and Algorithm AS 109: A Remark on Algorithms: AS 63: The Incomplete Beta Integral AS 64: Inverse of the Incomplete Beta Function Ratio. /Journal of the Royal Statistical Society. Series C (Applied Statistics)/ Vol. 26, No. 1 (1977), pp. 111-114 http://www.jstor.org/pss/2346887
• Lanczos, C. (1964) A precision approximation of the gamma function. SIAM Journal on Numerical Analysis B 1:86–96. http://www.jstor.org/stable/2949767
• Loader, C. (2000) Fast and Accurate Computation of Binomial Probabilities. http://projects.scipy.org/scipy/raw-attachment/ticket/620/loader2000Fast.pdf
• Macleod, A.J. (1989) Algorithm AS 245: A robust and reliable algorithm for the logarithm of the gamma function. Journal of the Royal Statistical Society, Series C (Applied Statistics) 38(2):397–402. http://www.jstor.org/stable/2348078
• Majumder, K.L., Bhattacharjee, G.P. (1973) Algorithm AS 63: The Incomplete Beta Integral. /Journal of the Royal Statistical Society. Series C (Applied Statistics)/ Vol. 22, No. 3 (1973), pp. 409-411. http://www.jstor.org/pss/2346797
• Majumder, K.L., Bhattacharjee, G.P. (1973) Algorithm AS 64: Inverse of the Incomplete Beta Function Ratio. /Journal of the Royal Statistical Society. Series C (Applied Statistics)/ Vol. 22, No. 3 (1973), pp. 411-414 http://www.jstor.org/pss/2346798
• Temme, N.M. (1992) Asymptotic inversion of the incomplete beta function. /Journal of Computational and Applied Mathematics 41(1992) 145-157.
• Temme, N.M. (1994) A set of algorithms for the incomplete gamma functions. /Probability in the Engineering and Informational Sciences/, 8, 1994, 291-307. Printed in the U.S.A.