[R] Numerical Integration in 1D

[Ravi Varadhan]

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.


----------------------------------------------------------------------------
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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

 

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-----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

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