[R] eliminate t() and %*% using crossprod() and solve(A,b)

[Prof Brian Ripley]

On Wed, 5 Oct 2005, Robin Hankin wrote:

> I have a square matrix Ainv of size N-by-N where N ~  1000
> I have a rectangular matrix H of size N by n where n ~ 4.
> I have a vector d of length N.
>
> I need   X  = solve(t(H) %*% Ainv %*% H) %*% t(H) %*% Ainv %*% d
>
> and
>
> H %*% X.
>
> It is possible to rewrite X in the recommended crossprod way:
>
> X <-  solve(quad.form(Ainv, H), crossprod(crossprod(Ainv, H), d))
>
> where quad.form() is a little function that evaluates a quadratic form:
>
>  quad.form <- function (M, x){
>         jj <- crossprod(M, x)
>         return(drop(crossprod(jj, jj)))
> }

That is not the same thing: it gives t(H) %*% Ainv %*% t(Ainv) %*% H .

> QUESTION:
>
> how  to calculate
>
> H %*% X
>
> in the recommended crossprod way?  (I don't want to take a transpose
> because t() is expensive, and I know that %*% is slow).

Have you some data to support your claims?  Here I find (for random 
matrices of the dimensions given on a machine with a fast BLAS)

> system.time(for(i in 1:100) t(H) %*% Ainv)
[1] 2.19 0.01 2.21 0.00 0.00
> system.time(for(i in 1:100) crossprod(H, Ainv))
[1] 1.33 0.00 1.33 0.00 0.00

so each is quite fast and the difference is not great.  However, that is 
not comparing %*% with crossprod, but t & %*% with crossprod.

I get

> system.time(for(i in 1:1000) H %*% X)
[1] 0.05 0.01 0.06 0.00 0.00

which is hardly 'slow' (60 us for %*%), especially compared to forming X 
in

> system.time({X  = solve(t(H) %*% Ainv %*% H) %*% t(H) %*% Ainv %*% d})
[1] 0.04 0.00 0.04 0.00 0.00

I would probably have written

> system.time({X <- solve(crossprod(H, Ainv %*% H), crossprod(crossprod(Ainv, H), d))})
1] 0.03 0.00 0.03 0.00 0.00

which is faster and does give the same answer.

[BTW, I used 2.2.0-beta which defaults to gcFirst=TRUE.]

-- 
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