[Rd] the incredible lightness of crossprod
Prof Brian Ripley
ripley at stats.ox.ac.uk
Thu Jan 27 21:23:39 CET 2005
On Thu, 27 Jan 2005, Paul Gilbert wrote:
> A few weeks ago I noticed
>> z <- matrix(rnorm(20000),10000,2)
>> system.time(for (i in 1:1000) apply(z,2,sum))
>  13.44 0.48 14.08 0.00 0.00
>> system.time(for (i in 1:1000) rep(1,10000) %*% z)
>  6.46 0.11 6.84 0.00 0.00
So both run in a few milliseconds for rather large problems.
> which seemed completely contrary to all my childhood teachings. Now
>> system.time(for (i in 1:1000) crossprod(rep(1,10000), z))
>  1.90 0.12 2.24 0.00 0.00
> makes sense because it is suppose to be faster than %*% , but why is apply so
`so slow' sic: what are you going to do in the 7ms you saved?
> (And should I go back and change apply in my code everywhere or is this
> likely to reverse again?)
It's not likely. apply is an R-level loop, and %*% is a C-level one.
However, %*% is not supposed to be much slower than crossprod, and the
devil is in the details of how the BLAS is implemented: the code is very
That %*% was faster than apply has been true in all my (17 years) of S/R
experience. Your childhood may predate S3, of course.
I still think one should use row/colSums for clarity with acceptable
efficiency. It must be very unusual for these operations to be a dominant
part of a calculation, so let's not lose proportion here.
> Paul Gilbert
> Patrick Burns wrote:
>> The following is at least as much out of intellectual curiosity
>> as for practical reasons.
>> On reviewing some code written by novices to R, I came
>> crossprod(x, y)[1,1]
>> I thought, "That isn't a very S way of saying that, I wonder
>> what the penalty is for using 'crossprod'." To my surprise the
>> penalty was substantially negative. Handily the client had S-PLUS
>> as well -- there the sign of the penalty was as I had expected, but
>> the order of magnitude was off.
>> Here are the timings of 1 million computations on vectors of
>> length 1000. This is under Windows, R version 1.9.1 and S-PLUS
>> 6.2 (on the same machine).
>> Command R S-PLUS
>> sum(x * y) 28.61 97.6
>> crossprod(x, y)[1,1] 6.77 2256.2
>> Another example is when computing the sums of the columns of a
>> matrix. For example:
>> jjm <- matrix(rnorm(600), 5)
>> Timings for this under Windows 2000 with R version 2.0.1 (on an
>> old chip running at about 0.7Ghz) for 100,000 computations are:
>> apply(jjm, 2, sum) 536.59
>> colSums(jjm) 18.26
>> rep(1,5) %*% jjm 15.41
>> crossprod(rep(1,5), jjm) 13.16
>> (These timings seem to be stable across R versions and on at least
>> one Linux platform.)
>> Andy Liaw showed another example of 'crossprod' being fast a couple
>> days ago on R-help.
>> Questions for those with a more global picture of the code:
>> * Is the speed advantage of 'crossprod' inherent, or is it because
>> more care has been taken with its implementation than the other
>> * Is 'crossprod' faster than 'sum(x * y)' because 'crossprod' is
>> going to BLAS while 'sum' can't?
>> * Would it make sense to (essentially) use 'crossprod' in
>> 'colSums' and its friends at least for the special case of matrices?
>> Patrick Burns
>> Burns Statistics
>> patrick at burns-stat.com
>> +44 (0)20 8525 0696
>> (home of S Poetry and "A Guide for the Unwilling S User")
>> R-devel at stat.math.ethz.ch mailing list
> R-devel at stat.math.ethz.ch mailing list
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
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