[Rd] the incredible lightness of crossprod

[Patrick Burns]

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
across:

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:

set.seed(1)
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
functions?

*  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
http://www.burns-stat.com
(home of S Poetry and "A Guide for the Unwilling S User")