[R] Numerical Derivative / Numerical Differentiation of unkno wnfunct ion

[Ravi Varadhan]

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 at jhmi.edu
--------------------------------------------------------------------------

> -----Original Message-----
> From: r-help-bounces at stat.math.ethz.ch [mailto:r-help-
> bounces at stat.math.ethz.ch] On Behalf Of Peter Dalgaard
> Sent: Thursday, May 05, 2005 8:51 PM
> To: Uzuner, Tolga
> Cc: 'r-help at stat.math.ethz.ch'; 'Berton Gunter'
> Subject: Re: [R] Numerical Derivative / Numerical Differentiation of unkno
> wnfunct ion
> 
> "Uzuner, Tolga" <tolga.uzuner at 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 at gene.com]
> > Sent: 05 May 2005 23:34
> > To: 'Uzuner, Tolga'; r-help at 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 at biostat.ku.dk)             FAX: (+45) 35327907
> 
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