Lanczos approximation: Difference between revisions

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{{Short description|Numerical method for calculating the gamma function}}

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.

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:<math>p_k(g) = \frac{\sqrt{2\,}}{\pi} \sum_{\ell=0}^k C_{2k+1,\,2\ell+1} \left(\ell - \tfrac{1}{2} \right)! {\left(\ell + g + \tfrac{1}{2} \right)}^{-(\ell+1/2)} e^{\ell + g + 1/2 }</math>

where <math>C_{n,m}</math> represents the (''n'', ''m'')th element of the [[matrix (mathematics)|matrix]] of coefficients for the [[Chebyshev polynomialpolynomials]]s, which can be calculated [[recursion|recursively]] from these identities:

:<math>\begin{align}

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| pages = 86&ndash;96

| year = 1964

| issnissue = 0887-459X1

| issn = 0887-459X

| doi= 10.1137/0701008

| bibcode = 1964SJNA....1...86L