In terminological/tidy-info-stats: Functions for manipulating information statistics

knitr::opts_chunk$set(echo = TRUE)

Euler–Mascheroni constant ($\lambda$)

$\lambda = 0.577215664901532$

N.B. for integers:

$digamma(M) = harmonic(M-1) - \lambda$

$harmonic(N) \approx log(N) + \lambda + 1/(2N) - 1/(12N^2)$

digamma2 = function(m) {
  n = m-1
  log(n)+1/(2*n)-1/(12*n^2)
}

First 10 terms of:

$harmonic(N) \approx log(N) + \lambda + 1/(2N) - \sum_{k=1}^{\infty} \frac{B_2k}{2kn^{2k}}$

B2k = c(1/6,-1/30,1/42,-1/30,5/66,-691/2730,7/6,-3617/510, 43867/798, -174611/330)
k = c(1:10)

digamma10 = function(m) {
  n = m-1
  log(n)+1/(2*n)-sapply(n,function(n) sum(B2k/(2*k*n^(2*k))))
}

Deltas between first 60 using the 10 term estimate using R digamma function

n = c(2:60)

digamma10(n)-base::digamma(n)

And using the 2 term estimate

digamma2(n)-base::digamma(n)

In conclusion 2 term estimate if pretty good after 50 terms.

terminological/tidy-info-stats documentation built on Nov. 19, 2022, 11:23 p.m.