From: Ravi Varadhan <rvaradha_at_jhsph.edu>

Date: Sat 07 May 2005 - 00:59:02 EST

deriv <- function(x, fun, h=NULL, order=1, accur=4) { macheps <- .Machine$double.eps

sin(10*x) - exp(-x)

}

> curve(func1,from=0,to=5)

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@jhmi.edu

R-help@stat.math.ethz.ch mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Received on Sat May 07 01:12:10 2005

Date: Sat 07 May 2005 - 00:59:02 EST

Hi,

There is a nice article by Fornberg and Sloan (Acta Numerica 1994) on various order accuracy (Taylor-series based) approximations for different order derivatives. I had coded a couple of these in R for first and second order derivatives, with truncation errors of orders 2 and 4.

Here is the code and a simple example demonstrating its accuracy for a sharply oscillating function:

############################################# A function to compute highly accurate first- and second-order derivatives # From Fornberg and Sloan (Acta Numerica, 1994, p. 203-267; Table 1, page 213)

deriv <- function(x, fun, h=NULL, order=1, accur=4) { macheps <- .Machine$double.eps

if (order==1) {

if(is.null(h)) h <- macheps^(1/3)* abs(x)
ifelse (accur==2, w <- c(-1/2,1/2), w <- c(1/12,-2/3, 2/3,-1/12))
ifelse (accur==2, xd <- x + h*c(-1,1), xd <- x + h*c(-2,-1,1,2))
return(sum(w*fun(xd))/h)

}

else if (order==2) {

if(is.null(h)) h <- macheps^(1/4)* abs(x)
ifelse (accur==2, w <- c(1,-2,1), w <- c(-1/12,4/3,-5/2,4/3,-1/12))
ifelse (accur==2, xd <- x + h*c(-1,0,1), xd <- x + h*c(-2,-1,0,1,2))
return(sum(w*fun(xd))/h^2)

}

}

############################################func1 <- function(x){

sin(10*x) - exp(-x)

}

#############################################

> curve(func1,from=0,to=5)

> x <- 2.04

# first order derivative

*> numd1 <- deriv(x,f=func1)
*

> exact <- 10*cos(10*x) + exp(-x)

> c(numd1,exact,numd1/exact-1)

[1] 3.335371e-01 3.335371e-01 1.981793e-11
# second order derivative

*> numd2 <- deriv(x,f=func1,order=2)
*

> exact <- -100*sin(10*x) - exp(-x)

> c(numd2,exact,numd2/exact-1)

[1] -1.001093e+02 -1.001093e+02 -2.300948e-11

Hope this is helpful.

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@jhmi.edu

> -----Original Message-----

*> From: r-help-bounces@stat.math.ethz.ch [mailto:r-help-
**> bounces@stat.math.ethz.ch] On Behalf Of Peter Dalgaard
**> Sent: Thursday, May 05, 2005 8:51 PM
**> To: Uzuner, Tolga
**> Cc: 'r-help@stat.math.ethz.ch'; 'Berton Gunter'
**> Subject: Re: [R] Numerical Derivative / Numerical Differentiation of unkno
**> wnfunct ion
**>
**> "Uzuner, Tolga" <tolga.uzuner@csfb.com> writes:
**>
**> > Ah... I searched for half an hour for this function... you know, the
**> > help function in R could really be a lot better...
**> >
**> > But wait a minute... looking at this, it appears you have to pass in
**> > an expression. What if it is an unknown function, where you only
**> > have a handle to the function, but you cannot see it's
**> > implementation ? Will this work then ?
**> >
**> > -----Original Message-----
**> > From: Berton Gunter [mailto:gunter.berton@gene.com]
**> > Sent: 05 May 2005 23:34
**> > To: 'Uzuner, Tolga'; r-help@stat.math.ethz.ch
**> > Subject: RE: [R] Numerical Derivative / Numerical Differentiation of
**> > unknown funct ion
**> >
**> >
**> > But...
**> >
**> > See ?numericDeriv which already does it via a C call and hence is much
**> > faster (and probably more accurate,too).
**> >
**>
**> The expression passed to numericDeriv can easily be a call to .C or
**> similar.
**>
**> Actually, numericDeriv can get you in trouble if the function is not
**> smooth enough. It basically just calculates (f(a+d)-f(a))/d where d is
**> on the order of 1e-7 * a for each parameter. Sometimes a larger d and
**> a higher order approximation is need to avoid getting stuck in the
**> rough.
**>
**> (Yes, Bill, I do remember that you wanted an R News Programmer's Niche
**> item from me on this...)
**>
**> --
**> O__ ---- Peter Dalgaard Blegdamsvej 3
**> c/ /'_ --- Dept. of Biostatistics 2200 Cph. N
**> (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918
**> ~~~~~~~~~~ - (p.dalgaard@biostat.ku.dk) FAX: (+45) 35327907
**>
**> ______________________________________________
**> R-help@stat.math.ethz.ch mailing list
**> https://stat.ethz.ch/mailman/listinfo/r-help
**> PLEASE do read the posting guide! http://www.R-project.org/posting-
**> guide.html
*

R-help@stat.math.ethz.ch mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Received on Sat May 07 01:12:10 2005

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