trade-off between speed and storage in matrix multiplications

-----Original Message-----
From: r-help-bounces at r-project.org 
[mailto:r-help-bounces at r-project.org] On Behalf Of Camarda, 
Carlo Giovanni
Sent: Monday, February 23, 2009 1:18 PM
To: r-help at stat.math.ethz.ch
Subject: [R] trade-off between speed and storage in matrix 
multiplications

Dear R-users,

	I coded two equivalent ways to perform (in a simplified 
version) some matrix multiplications I would like to use in a 
more general framework.
In the first case I used Kronecker product and vectorization 
of a certain matrix. This approach takes less time, but, as 
you may guess, I run out of memory when dimensions are large.
In the second approach, I profited of sparseness and 
structure of the matrices and use outer-functions for 
performing operations. Here it takes more time, but I have 
not problem of memory.

Is there any way to enhance the second approach for 
speeding-up the whole process? Or, in computing, this is a 
well-known trade-off between speed and storage-spaces which 
I'm not aware (sorry for that).

Please have a look to the example below.

Needless to say that I'd appreciate any suggestion.

Best,
Carlo Giovanni


# dimensions
m <- 10
n <- 15
# A-matrix
rnA <- runif(m*m)
A <- matrix(rnA, m, m)
# vector
v <- runif(n)
# B-matrix
rnB <- runif(m*n)
B <- matrix(rnB, m, n)

# first solution: vectorize B + kronecker product => faster 
but storage issues system.time( for(i in 1:100){
    b <- c(B)
    vKron.A <- kronecker(diag(v), A)
    SOL1 <- vKron.A %*% b
})

# second solution: orig. dims + apply + mapply => slower, but 
w/o storage issues system.time( for(i in 1:100){
    Av <- outer(A, v, FUN="*")
    Av.df1 <- apply(Av, 3, as.data.frame)
    Av.df2 <- lapply(X=Av.df1, FUN=as.matrix, nrow=m, ncol=m)
    SOL2 <- mapply(FUN="%*%", Av.df2, as.data.frame(B)) # as a matrix
})





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