# Difference between revisions of "Precision Models"

(Created page with "= Correction Factors = == Rayleigh correction factor == :<math>c_{R}(n) = 4^n \sqrt{\frac{n}{\pi}} \frac{ N!(N-1)!} {(2N)!}</math> To avoid overflows this is better calculate...") |
m |
||

Line 9: | Line 9: | ||

:<math>\frac{1}{c_{G}(n)} = \sqrt{\frac{2}{n-1}}\,\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)} \, = \, 1 - \frac{1}{4n} - \frac{7}{32n^2} - \frac{19}{128n^3} + O(n^{-4})</math> | :<math>\frac{1}{c_{G}(n)} = \sqrt{\frac{2}{n-1}}\,\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)} \, = \, 1 - \frac{1}{4n} - \frac{7}{32n^2} - \frac{19}{128n^3} + O(n^{-4})</math> | ||

The third-order approximation is adequate. The following spreadsheet formula gives a more direct calculation: | The third-order approximation is adequate. The following spreadsheet formula gives a more direct calculation: | ||

− | + | EXP(LN(SQRT(2/(N-1))) + GAMMALN(N/2) - GAMMALN((N-1)/2)) | |

== Bessel correction factor == | == Bessel correction factor == | ||

:<math>c_{B}(n) = \frac{n}{n-1}</math> | :<math>c_{B}(n) = \frac{n}{n-1}</math> |

## Revision as of 22:50, 19 November 2013

## Contents

# Correction Factors

## Rayleigh correction factor

\[c_{R}(n) = 4^n \sqrt{\frac{n}{\pi}} \frac{ N!(N-1)!} {(2N)!}\] To avoid overflows this is better calculated using log-gammas, as in the following spreadsheet formula:

EXP(LN(SQRT(N/PI())) + N*LN(4) + GAMMALN(N+1) + GAMMALN(N) - GAMMALN(2N+1))

## Gaussian correction factor

\[\frac{1}{c_{G}(n)} = \sqrt{\frac{2}{n-1}}\,\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)} \, = \, 1 - \frac{1}{4n} - \frac{7}{32n^2} - \frac{19}{128n^3} + O(n^{-4})\] The third-order approximation is adequate. The following spreadsheet formula gives a more direct calculation:

EXP(LN(SQRT(2/(N-1))) + GAMMALN(N/2) - GAMMALN((N-1)/2))

## Bessel correction factor

\[c_{B}(n) = \frac{n}{n-1}\]