Lanczos approximation

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, the Lanczos approximation is a method for computing the gamma function numerically, published by Cornelius Lanczos in 1964. It is a practical alternative to the more popular Stirling's approximation for calculating the gamma function with fixed precision.


The Lanczos approximation consists of the formula

for the gamma function, with

Here g is a constant that may be chosen arbitrarily subject to the restriction that Re(z) > 1/2.[1] The coefficients p, which depend on g, are slightly more difficult to calculate (see below). Although the formula as stated here is only valid for arguments in the right complex half-plane, it can be extended to the entire complex plane by the reflection formula,

The series A is convergent, and may be truncated to obtain an approximation with the desired precision. By choosing an appropriate g (typically a small integer), only some 5–10 terms of the series are needed to compute the Gamma function with typical single or double floating-point precision. If a fixed g is chosen, the coefficients can be calculated in advance and the sum is recast into the following form:

Thus computing the gamma function becomes a matter of evaluating only a small number of elementary functions and multiplying by stored constants. The Lanczos approximation was popularized by Numerical Recipes, according to which computing the Gamma function becomes "not much more difficult than other built-in functions that we take for granted, such as sin x or ex". The method is also implemented in the GNU Scientific Library.


The coefficients are given by

with C(i, j) denoting the (i, j)th element of the Chebyshev polynomial coefficient matrix which can be calculated recursively from the identities

Paul Godfrey describes how to obtain the coefficients and also the value of the truncated series A as a matrix product.


Lanczos derived the formula from Leonhard Euler's integral

performing a sequence of basic manipulations to obtain

and deriving a series for the integral.

Simple implementation[edit]

The following implementation in the Python programming language works for complex arguments and typically gives 15 correct decimal places:

from cmath import sin, sqrt, pi, exp

p = [676.5203681218851

EPSILON = 1e-07  
def drop_imag(z):
    if abs(z.imag) <= EPSILON:
        z = z.real
    return z
def gamma(z):
    z = complex(z)
    if z.real < 0.5:
        y = pi / (sin(pi*z) * gamma(1-z)) ### Reflection formula 
        z -= 1
        x = 0.99999999999980993
        for (i, pval) in enumerate(p):
            x += pval / (z+i+1)
        t = z + len(p) - 0.5
        y = sqrt(2*pi) * t**(z+0.5) * exp(-t) * x
    return drop_imag(y)
The above use of the reflection (thus the if-else structure) seems unnecessary
and just adds more code to execute. It calls itself again, so it still needs
to execute the same "for" loop yet has an extra calculation at the end)

print gamma(1)
print gamma(5)    
print gamma(0.5)

See also[edit]


  1. ^ Pugh thesis