6 messages · Ravi Varadhan, Max, Brian Ripley
Dear UseRs, I'm curious about the derivative of n!. We know that Gamma(n+1)=n! So when on takes the derivative of Gamma(n+1) we get Int(ln(x)*exp(-x)*x^n,x=0..Inf). I've tried code like
integrand<-function(x) {log(x)*exp(x)*x^n}
integrate(integrand,lower=0,upper=Inf)
It seems that R doesn't like to integrate for any n, and I was wondering if anyone knew a way around this? -Max
Dear UseRs, I'm curious about the derivative of n!. We know that Gamma(n+1)=n! So when on takes the derivative of Gamma(n+1) we get Int(ln(x)*exp(-x)*x^n,x=0..Inf). I've tried code like
integrand<-function(x) {log(x)*exp(x)*x^n}
integrate(integrand,lower=0,upper=Inf)
It seems that R doesn't like to integrate for any n, and I was wondering if anyone knew a way around this?
ln(x) e^x x^n is not integrable on (0, Inf). You presumably slipped over a minus sign, but your definition of gamma(n) is wrong -- see ?gamma. integrate(function(x) exp(-x)*x^n, lower=0, upper=Inf) will work for gamma(n+1).
Brian D. Ripley, ripley at stats.ox.ac.uk Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/ University of Oxford, Tel: +44 1865 272861 (self) 1 South Parks Road, +44 1865 272866 (PA) Oxford OX1 3TG, UK Fax: +44 1865 272595
Hi Max,
The analytic integral \int _0 ^\Inf exp(-t) t^n log(t) might not converge
because the integrand tends to -Inf as t -> 0.
So, here is a numerical approach to estimating the derivative of the gamma
function:
library(numDeriv)
fx <- function(x, n) exp(-x) * x^n
gf <- function(n) {integrate(fx, lower=0, upper=Inf, n=n)$val}
grad(x=3, func=gf)
[1] 7.536706
grad(x=10, func=gf)
[1] 8534040
Best, Ravi. ---------------------------------------------------------------------------- ------- Ravi Varadhan, Ph.D. Assistant Professor, The Center on Aging and Health Division of Geriatric Medicine and Gerontology Johns Hopkins University Ph: (410) 502-2619 Fax: (410) 614-9625 Email: rvaradhan at jhmi.edu Webpage: http://www.jhsph.edu/agingandhealth/People/Faculty/Varadhan.html ---------------------------------------------------------------------------- -------- -----Original Message----- From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On Behalf Of Max Sent: Friday, March 07, 2008 1:41 PM To: r-help at stat.math.ethz.ch Subject: [R] Numerical Integration in 1D Dear UseRs, I'm curious about the derivative of n!. We know that Gamma(n+1)=n! So when on takes the derivative of Gamma(n+1) we get Int(ln(x)*exp(-x)*x^n,x=0..Inf). I've tried code like
integrand<-function(x) {log(x)*exp(x)*x^n}
integrate(integrand,lower=0,upper=Inf)
It seems that R doesn't like to integrate for any n, and I was wondering if anyone knew a way around this? -Max ______________________________________________ R-help at r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Hi max,
Prof. Ripley is right. Your problem is that you missed a (-) sign in the
exponential. Here is a demonstration showing the agreement between
numerical and analytical results:
gx <- function(x, n) exp(-x) * x^n * log(x)
df <- function(n) {integrate(gx, lower=0, upper=Inf, n=n)$val}
library(numDeriv)
fx <- function(x, n) exp(-x) * x^n
gf <- function(n) {integrate(fx, lower=0, upper=Inf, n=n)$val}
grad(x=6, func=gf)
[1] 1348.405
df(6)
[1] 1348.405
Ravi. ---------------------------------------------------------------------------- ------- Ravi Varadhan, Ph.D. Assistant Professor, The Center on Aging and Health Division of Geriatric Medicine and Gerontology Johns Hopkins University Ph: (410) 502-2619 Fax: (410) 614-9625 Email: rvaradhan at jhmi.edu Webpage: http://www.jhsph.edu/agingandhealth/People/Faculty/Varadhan.html ---------------------------------------------------------------------------- -------- -----Original Message----- From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On Behalf Of Max Sent: Friday, March 07, 2008 1:41 PM To: r-help at stat.math.ethz.ch Subject: [R] Numerical Integration in 1D Dear UseRs, I'm curious about the derivative of n!. We know that Gamma(n+1)=n! So when on takes the derivative of Gamma(n+1) we get Int(ln(x)*exp(-x)*x^n,x=0..Inf). I've tried code like
integrand<-function(x) {log(x)*exp(x)*x^n}
integrate(integrand,lower=0,upper=Inf)
It seems that R doesn't like to integrate for any n, and I was wondering if anyone knew a way around this? -Max ______________________________________________ R-help at r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Prof Brian Ripley formulated on Friday :
On Fri, 7 Mar 2008, Max wrote:
Dear UseRs, I'm curious about the derivative of n!. We know that Gamma(n+1)=n! So when on takes the derivative of Gamma(n+1) we get Int(ln(x)*exp(-x)*x^n,x=0..Inf). I've tried code like
integrand<-function(x) {log(x)*exp(x)*x^n}
integrate(integrand,lower=0,upper=Inf)
It seems that R doesn't like to integrate for any n, and I was wondering if anyone knew a way around this?
ln(x) e^x x^n is not integrable on (0, Inf). You presumably slipped over a minus sign, but your definition of gamma(n) is wrong -- see ?gamma. integrate(function(x) exp(-x)*x^n, lower=0, upper=Inf) will work for gamma(n+1).
I did miss a minus sign in the integration, which explains part of my problems. The function of interest is the derivative of Gamma(n+1) with respect to n, but obviously integrated over x from 0 to Infinity. What happens now is:
integrand<-function(x) {log(x)*exp(-x)*x^n}
integrate(integrand,lower=0,upper=Inf)
Error in f(x, ...) : object "n" not found Any ideas on how to get around this error?
Prof Brian Ripley formulated on Friday :
On Fri, 7 Mar 2008, Max wrote:
Dear UseRs, I'm curious about the derivative of n!. We know that Gamma(n+1)=n! So when on takes the derivative of Gamma(n+1) we get Int(ln(x)*exp(-x)*x^n,x=0..Inf). I've tried code like
integrand<-function(x) {log(x)*exp(x)*x^n}
integrate(integrand,lower=0,upper=Inf)
It seems that R doesn't like to integrate for any n, and I was wondering if anyone knew a way around this?
ln(x) e^x x^n is not integrable on (0, Inf). You presumably slipped over a minus sign, but your definition of gamma(n) is wrong -- see ?gamma. integrate(function(x) exp(-x)*x^n, lower=0, upper=Inf) will work for gamma(n+1).
I did miss a minus sign in the integration, which explains part of my problems. The function of interest is the derivative of Gamma(n+1) with respect to n, but obviously integrated over x from 0 to Infinity.
And you said n!, so n must be integer and you cannot differentiate a function of a integer argument. If you are interested in the derivative of gamma(x), check out ?digamma.
What happens now is:
integrand<-function(x) {log(x)*exp(-x)*x^n}
integrate(integrand,lower=0,upper=Inf)
Error in f(x, ...) : object "n" not found Any ideas on how to get around this error?
Set 'n' in the evaluation environment. E.g.
n <- 3 integrate(integrand, lower=0, upper=Inf)
7.536706 with absolute error < 4.7e-06
digamma(4)*gamma(4)
[1] 7.536706
Brian D. Ripley, ripley at stats.ox.ac.uk Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/ University of Oxford, Tel: +44 1865 272861 (self) 1 South Parks Road, +44 1865 272866 (PA) Oxford OX1 3TG, UK Fax: +44 1865 272595