[R] crossprod vs %*% timing

[Dimitris Rizopoulos]

Hi Robin,

I some cases you could benefit from special features of the matrices 
at hand (I don't know if this is applicable in your case). For 
instance, in Bayesian computations I often face the quadratic form 
t(y)%*%solve(Sigma)%*%y, where Sigma is a covariance matrix. In this 
case you could gain by using the positive definiteness of Sigma i.e.,

library(MASS)
y <- rnorm(100)
Sigma <- var(mvrnorm(1000, rep(0,100), diag(100)))
###
system.time( for(i in 1:1000) c(t(y)%*%solve(Sigma)%*%y) )
system.time( for(i in 1:1000) c(crossprod(y, solve(Sigma))%*%y) )
system.time( for(i in 1:1000) c(t(y)%*%solve(Sigma, y)) )
system.time( for(i in 1:1000) quadform(y, Sigma) )

where

quadform <- function(y, Sigma){
    # Warning: The code does not check for symmetry of Sigma
    stopifnot(is.vector(y), length(y)==nrow(Sigma))
    chol.Sigma <- chol(Sigma)
    x <- forwardsolve(t(chol.Sigma), y)
    sum(x*x)
}

I hope this helps.

Best,
Dimitris

----
Dimitris Rizopoulos
Ph.D. Student
Biostatistical Centre
School of Public Health
Catholic University of Leuven

Address: Kapucijnenvoer 35, Leuven, Belgium
Tel: +32/16/396887
Fax: +32/16/337015
Web: http://www.med.kuleuven.ac.be/biostat/
     http://www.student.kuleuven.ac.be/~m0390867/dimitris.htm



----- Original Message ----- 
From: "Robin Hankin" <rksh at soc.soton.ac.uk>
To: <R-help at stat.math.ethz.ch>
Sent: Wednesday, October 06, 2004 10:09 AM
Subject: [R] crossprod vs %*% timing


> Hi
>
> the manpage says that crossprod(x,y) is formally equivalent to, but
> faster than, the call 't(x) %*% y'.
>
> I have a vector 'a' and a matrix 'A', and need to evaluate 't(a) %*% 
> A
> %*% a' many many times, and performance is becoming crucial.  With
>
> f1 <- function(a,X){ ignore <- t(a) %*% X %*% a               }
> f2 <- function(a,X){ ignore <- crossprod(t(crossprod(a,X)),a) }
> f3 <- function(a,X){ ignore <- crossprod(a,X) %*% a           }
>
> a <- rnorm(100)
> X <- matrix(rnorm(10000),100,100)
>
> print(system.time( for(i in 1:10000){ f1(a,X)}))
> print(system.time( for(i in 1:10000){ f2(a,X)}))
> print(system.time( for(i in 1:10000){ f3(a,X)}))
>
>
> I get something like:
>
> [1] 2.68 0.05 2.66 0.00 0.00
> [1] 0.48 0.00 0.49 0.00 0.00
> [1] 0.29 0.00 0.31 0.00 0.00
>
> with quite low variability from run to run.  What surprises me is 
> the
> third figure: about 40% faster than the second one, the extra time
> possibly related to the call to t() (and Rprof shows about 35% of
> total time in t() for my application).
>
> So it looks like f3() is the winner hands down, at least for this
> task.  What is a good way of thinking about such issues?  Anyone got
> any performance tips?
>
> I quite often need things like 'a %*% X %*% t(Y) %*% Z %*% t(b)' 
> which
> would be something like
> crossprod(t(crossprod(t(crossprod(t(crossprod(a,X)),t(Y))),Z)),t(b))
> (I think).
>
> (R-1.9.1, 2GHz G5 PowerPC, MacOSX10.3.5)
>
> -- 
> Robin Hankin
> Uncertainty Analyst
> Southampton Oceanography Centre
> SO14 3ZH
> tel +44(0)23-8059-7743
> initialDOTsurname at soc.soton.ac.uk (edit in obvious way; spam 
> precaution)
>
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