R help - Mar 2008 - 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

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).

-- 
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

 

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

______________________________________________
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.