[R] Gaussian Adaptive Quadrature

[Ravi Varadhan]

Hi,

The function "integrate" is based on QUADPACK Fortran package by Piessens.
It does indeed perform adaptive Gauss-Kronrod quadrature, where Kronrod's
method allows the re-use of abscissa from the previous iteration, thus
enabling the estimation of the quadrature error and its control.  In
contrast, in the standard Gaussian quadrature methods this is not feasible. 

The terminology "Gaussian quadrature" is not restricted to Gauss-Hermite
quadrature (where exp(-x^2) is the weight function), but applies more
broadly to Gauss-Legendre, Gauss-Laguerre, etc., where the abscissa are
chosen from Legendre, Laguerre polynomials.

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 stat.math.ethz.ch
[mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Douglas Bates
Sent: Wednesday, March 21, 2007 9:25 AM
To: Doran, Harold
Cc: Help mailing list - R
Subject: Re: [R] Gaussian Adaptive Quadrature

On 3/21/07, Doran, Harold <HDoran at air.org> wrote:
> The function integrate() uses AGQ. There are other functions for
> gaussian quadrature in the statmod() package that I really like.

I think that integrate does adaptive quadrature but not adaptive
Gaussian quadrature (which probably should have been called adaptive
Gauss-Hermite quadrature to be more specific).  In the first case the
"adaptive" refers to a choice of mesh size.  In the second case one is
integrating a function that is close to a multivariate Gaussian
density by first finding the conditional optimum of the integrand and
using a quadratic approximation to the log-integrand to establish the
location of the Gauss-Hermite quadrature points.

The Laplace approximation to the log-likelihood for a generalized
linear mixed model is a 1-point adaptive Gauss-Hermite quadrature
evaluation.

>
> > -----Original Message-----
> > From: r-help-bounces at stat.math.ethz.ch
> > [mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Caio
> > Lucidius Naberezny Azevedo
> > Sent: Wednesday, March 21, 2007 5:55 AM
> > To: Help mailing list - R
> > Subject: [R] Gaussian Adaptive Quadrature
> >
> > Hi all,
> >
> >   Does anybody know any function that performs gaussian
> > adapative quadrature integration of univariate functions?
> >
> >   Thanks in advance,
> >
> >   Regards,
> >
> > Caio
> >
> >  __________________________________________________
> >
> >
> >       [[alternative HTML version deleted]]
> >
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> >
>
> ______________________________________________
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