Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Lanczos approximation is a method for computing the Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
In mathematics, Stirling's approximation is an approximation for large factorials. It is named after James Stirling.The formula as typically used in applications is\ln n! = n\ln n - n +O\...
for calculating the Gamma function with fixed precision.
In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition...
that may be chosen arbitrarily subject to the restriction that Re(z+g+1/2) > 0. 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
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
In mathematics, a reflection formula or reflection relation for a function f is a relationship between f and f...
,
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
Double precision
In computing, double precision is a computer number format that occupies two adjacent storage locations in computer memory. A double-precision number, sometimes simply called a double, may be defined to be an integer, fixed point, or floating point .Modern computers with 32-bit storage locations...
In computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...
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
Numerical Recipes
Numerical Recipes is the generic title of a series of books on algorithms and numerical analysis by William H. Press, Saul Teukolsky, William Vetterling and Brian Flannery. In various editions, the books have been in print since 1986...
, 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
GNU Scientific Library
In computing, the GNU Scientific Library is a software library written in the C programming language for numerical calculations in applied mathematics and science...
.
Coefficients
The coefficients are given by
with denoting the (i, j)th element of the Chebyshev polynomial coefficient matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
from the identities
Paul Godfrey describes how to obtain the coefficients and also the value of the truncated series A as a matrix product
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
works for complex arguments and typically gives 15 correct decimal places:
from cmath import *
Coefficients used by the GNU Scientific Library
g = 7
p = [0.99999999999980993, 676.5203681218851, -1259.1392167224028,
771.32342877765313, -176.61502916214059, 12.507343278686905,
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7]
def gamma(z):
z = complex(z)
# Reflection formula
if z.real < 0.5:
return pi / (sin(pi*z)*gamma(1-z))
else:
z -= 1
x = p[0]
for i in range(1, g+2):
x += p[i]/(z+i)
t = z + g + 0.5
return sqrt(2*pi) * t**(z+0.5) * exp(-t) * x
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denoting the (i, j)th element of the Chebyshev polynomial coefficient matrix
