Portability | portable |
---|---|
Stability | experimental |
Maintainer | bos@serpentine.com |
Mathematical functions for statistics.
Functions
:: Double | Parameter of each function. |
-> Vector Double | Coefficients of each polynomial term, in increasing order. |
-> Double |
Evaluate a series of Chebyshev polynomials. Uses Clenshaw's algorithm.
Factorial functions
factorial :: Int -> DoubleSource
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 :: Int -> DoubleSource
Compute the natural logarithm of the factorial function. Gives 16 decimal digits of precision.
Gamma functions
Compute the incomplete gamma integral function γ(s,x). Uses Algorithm AS 239 by Shea.
logGamma :: Double -> DoubleSource
Compute the logarithm of the gamma function Γ(x). Uses Algorithm AS 245 by Macleod.
Gives an accuracy of 10–12 significant decimal digits, except
for small regions around x = 1 and x = 2, where the function
goes to zero. For greater accuracy, use logGammaL
.
Returns ∞ if the input is outside of the range (0 < x ≤ 1e305).
logGammaL :: Double -> DoubleSource
Compute the logarithm of the gamma function, Γ(x). Uses a Lanczos approximation.
This function is slower than logGamma
, but gives 14 or more
significant decimal digits of accuracy, 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).
References
- Clenshaw, C.W. (1962) Chebyshev series for mathematical functions. National Physical Laboratory Mathematical Tables 5, Her Majesty's Stationery Office, London.
- 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
- 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
- Shea, B. (1988) Algorithm AS 239: Chi-squared and incomplete gamma integral. Applied Statistics 37(3):466–473. http://www.jstor.org/stable/2347328