I am late to this discussion so forgive me if I am being redundant. It
appears to me that the integral from 0 to infinity of log-gamma may
diverge to infinity. Equation 6.1.50 of Abramowitz & Stegun shows that a
re-expression of lnGamma(x) and the Stirling approximation involves
(x-1/2)*log(x) -x along with other terms. This term appears to dominate
the integral and itself diverge. It is worth checking out.
> integrate(lgamma, lower = 0, upper = 10)
43.24636 with absolute error < 1.6e-11
> integrate(lgamma, lower = 0, upper = 100)
15438.12 with absolute error < 1.4
> integrate(lgamma, lower = 0, upper = 1000)
2701843 with absolute error < 254
> integrate(lgamma, lower = 0, upper = 10000)
385485116 with absolute error < 18464
> gterm <- function(x){(x-.5)*log(x)-x}
> integrate(gterm,lower=0,upper=10)
33.61633 with absolute error < 1.1e-05
> integrate(gterm,lower=0,upper=100)
15345.59 with absolute error < 1.2
> integrate(gterm,lower=0,upper=1000)
2700923 with absolute error < 252
> integrate(gterm,lower=0,upper=10000)
385475926 with absolute error < 18462
>
Joseph F. Lucke
Senior Statistician
Research Institute on Addictions
University at Buffalo
State University of New York
1021 Main Street
Buffalo, NY 14203-1016
Office: 716-887-6807
Fax: 716-887-2510
http://www.ria.buffalo.edu/profiles/lucke.html
"Andreas Wittmann" <andreas_wittm...@gmx.de>
Sent by: r-help-boun...@r-project.org
04/22/2009 03:28 AM
To
r-help@r-project.org
cc
Subject
[R] integrate lgamma from 0 to Inf
Dear R users,
i try to integrate lgamma from 0 to Inf. But here i get the message
"roundoff error is detected in the extrapolation table", if i use 1.0e120
instead of Inf the computation works, but this is against the suggestion
of integrates help information to use Inf explicitly. Using stirlings
approximation doesnt bring the solution too.
## Stirlings approximation
lgammaApprox <- function(x)
{
0.5*log(2*pi)+(x-(1/2))*log(x)-x
}
integrate(lgamma, lower = 0, upper = 1.0e120)
integrate(lgammaApprox, lower = 0, upper = 1.0e120)
> integrate(lgamma, lower = 0, upper = 1.0e120)
1.374051e+242 with absolute error < 3.2e+235
> integrate(lgammaApprox, lower = 0, upper = 1.0e120)
1.374051e+242 with absolute error < 3.2e+235
integrate(lgamma, lower = 0, upper = Inf)
integrate(lgammaApprox, lower = 0, upper = Inf)
> integrate(lgamma, lower = 0, upper = Inf)
Fehler in integrate(lgamma, lower = 0, upper = Inf) :
roundoff error is detected in the extrapolation table
> integrate(lgammaApprox, lower = 0, upper = Inf)
Fehler in integrate(lgammaApprox, lower = 0, upper = Inf) :
roundoff error is detected in the extrapolation table
Many thanks if you have any advice for me!
best regards
Andreas
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