[Rd] Inaccurate complex arithmetic of R (Matlab is accurate)
Martin Becker
martin.becker at mx.uni-saarland.de
Tue Aug 4 13:34:17 CEST 2009
Dear Ravi,
I suspect that, in general, you may be facing the limitations of machine
accuracy (more precisely, IEEE 754 arithmetics on [64-bit] doubles) in
your application. This is not a problem of R itself, but rather a
problem of standard arithmetics provided by underlying C compilers/CPUs.
In fact, every operation in IEEE arithmetics (so, this is not really a
problem only for complex numbers) may suffer from inexactness, a
particularly difficult one is addition/subtraction. Consider the
following example for real numbers (I know, it is not a very good one...):
The two functions
badfn <- function(x) 1-(1+x)*(1-x)
goodfn <- function(x) x^2
both calculate x^2 (theoretically, given perfect arithmetic). So, as you
want to allow the user to 'specify the mathematical function ... in
"any" form the user chooses', both functions should be ok.
But, unfortunately:
> badfn(1e-8)
[1] 2.220446049250313e-16
> goodfn(1e-8)
[1] 1e-16
I don't know what happens in matlab/octave/scilab for this example. They
may do better, but probably at some cost (dropping IEEE arithmetic/do
"clever" calculations should result in massive speed penalties, try
evaluating hypergeom([1,-99.9],[-49.9-24.9*I],(1+1.71*I)/2); in
Maple...).
Now, you have some options:
- assume, that the user is aware of the numerical inexactness of ieee
arithmetics and that he is able to supply some "robust" version of the
mathematical function.
- use some other software (eg., matlab) for the critical calculations
(there is a R <-> Matlab interface, see package R.matlab on CRAN), if
you are sure, that this helps.
- implement/use multiple precision arithmetics within R (Martin
Maechler's Rmpfr package may be very useful:
http://r-forge.r-project.org/projects/rmpfr/ , but this will slow down
calculations considerably)
All in all, I think it is unfair just to blame R here. Of course, it
would be great if there was a simple trigger to turn on multiple
precision arithmetics in R. Packages such as Rmpfr may provide a good
step in this direction, since operator overloading via S4 classes allows
for easy code adaption. But Rmpfr is still declared "beta", and it
relies on some external library, which could be problematic on Windows
systems. Maybe someone else has other/better suggestions, but I do not
think that there is an easy solution for the "general" problem.
Best wishes,
Martin
Ravi Varadhan wrote:
> Dear Martin,
>
> Thank you for this useful trick. However, we are interested in a "general"
> approach for exact derivative computation. This approach should allow the
> user to specify the mathematical function that needs to be differentiated in
> "any" form that the user chooses. So, your trick will be difficult to
> implement there. Furthermore, do we know for sure that `exponentiation' is
> the only operation that results in inaccuracy? Are there other operations
> that also yield inaccurate results for complex arithmetic?
>
> Hans Borchers also checked the computations with other free numerical
> software, such as Octave, Scilab, Euler, and they all return exactly the
> same results as Matlab. It would be a shame if R could not do the same.
>
> It would be great if the R core could address the "fundamental" issue.
>
> Thank you.
>
> Best regards,
> 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_personal_pages/Varadhan.h
> tml
>
>
>
> ----------------------------------------------------------------------------
> --------
>
>
> -----Original Message-----
> From: Martin Becker [mailto:martin.becker at mx.uni-saarland.de]
> Sent: Monday, August 03, 2009 5:50 AM
> To: Ravi Varadhan
> Cc: r-devel at stat.math.ethz.ch; hwborchers at googlemail.com
> Subject: Re: [Rd] Inaccurate complex arithmetic of R (Matlab is accurate)
>
> Dear Ravi,
>
> the inaccuracy seems to creep in when powers are calculated. Apparently,
> some quite general function is called to calculate the squares, and one can
> avoid the error by reformulating the example as follows:
>
> rosen <- function(x) {
> n <- length(x)
> x1 <- x[2:n]
> x2 <- x[1:(n-1)]
> sum(100*(x1-x2*x2)*(x1-x2*x2) + (1-x2)*(1-x2)) }
>
> x0 <- c(0.0094, 0.7146, 0.2179, 0.6883, 0.5757, 0.9549, 0.7136, 0.0849,
> 0.4147, 0.4540) h <- c(1.e-15*1i, 0, 0, 0, 0, 0, 0, 0, 0, 0) xh <- x0 + h
>
> rx <- rosen(xh)
> Re(rx)
> Im (rx)
>
>
> I don't know which arithmetics are involved in the application you
> mentioned, but writing some auxiliary function for the calculation of x^n
> when x is complex and n is (a not too large) integer may solve some of the
> numerical issues. A simple version is:
>
> powN <- function(x,n) sapply(x,function(x) prod(rep(x,n)))
>
> The corresponding summation in 'rosen' would then read:
>
> sum(100*powN(x1-powN(x2,2),2) + powN(1-x2,2))
>
>
> HTH,
>
> Martin
>
>
> Ravi Varadhan wrote:
>
>> Dear All,
>>
>> Hans Borchers and I have been trying to compute "exact" derivatives in R
>>
> using the idea of complex-step derivatives that Hans has proposed. This is
> a really, really cool idea. It gives "exact" derivatives with only a
> minimal effort (same as that involved in computing first-order
> forward-difference derivative).
>
>> Unfortunately, we cannot implement this in R as the "complex arithmetic"
>>
> in R appears to be inaccurate.
>
>> Here is an example:
>>
>> #-- Classical Rosenbrock function in n variables rosen <- function(x)
>> { n <- length(x)
>> x1 <- x[2:n]
>> x2 <- x[1:(n-1)]
>> sum(100*(x1-x2^2)^2 + (1-x2)^2)
>> }
>>
>>
>> x0 <- c(0.0094, 0.7146, 0.2179, 0.6883, 0.5757, 0.9549, 0.7136,
>> 0.0849, 0.4147, 0.4540) h <- c(1.e-15*1i, 0, 0, 0, 0, 0, 0, 0, 0, 0)
>> xh <- x0 + h
>>
>> rx <- rosen(xh)
>> Re(rx)
>> Im (rx)
>>
>> # rx = 190.3079796814885 - 12.13915588266717e-15 i # incorrect
>> imaginary part in R
>>
>> However, the imaginary part of the above answer is inaccurate. The
>>
> correct imaginary part (from Matlab) is:
>
>> 190.3079796814886 - 4.66776376640000e-15 i # correct imaginary part
>> from Matlab
>>
>> This inaccuracy is serious enough to affect the acuracy of the compex-step
>>
> gradient drastically.
>
>> Hans and I were wondering if there is a way to obtain accurate "small"
>>
> imaginary part for complex arithmetic.
>
>> I am using Windows XP operating system.
>>
>> Thanks for taking a look at this.
>>
>> Best regards,
>> Ravi.
>>
>>
>> ____________________________________________________________________
>>
>> Ravi Varadhan, Ph.D.
>> Assistant Professor,
>> Division of Geriatric Medicine and Gerontology School of Medicine
>> Johns Hopkins University
>>
>> Ph. (410) 502-2619
>> email: rvaradhan at jhmi.edu
>>
>> ______________________________________________
>> R-devel at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-devel
>>
>>
>
>
> --
> Dr. Martin Becker
> Statistics and Econometrics
> Saarland University
> Campus C3 1, Room 206
> 66123 Saarbruecken
> Germany
>
>
--
Dr. Martin Becker
Statistics and Econometrics
Saarland University
Campus C3 1, Room 206
66123 Saarbruecken
Germany
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