How to quickly solve several equations in R?

Question

I am trying to quickly solve equations (forms x% *% res = y) for a large array in R CRAN.
I have data x and y and you want to calculate res. How can this be done best, i.e. Quickly? Many thanks!

Here is an example and some approaches: (does it seem that the “solve” is the fastest?)

# setup:
p   = 20 # dimension of matrix to solve
nmkt= 3000 # length of array, i.e., number of equations to solve
res = matrix(0,p,nmkt) # result matrix
x   = array(rnorm(p*p*nmkt),c(p,p,nmkt)) # data
# make x symetric  and invertible
for(i in 1:nmkt){ x[, , i]= crossprod(x[, , i])+diag(p)*0.01}
y  = matrix(rnorm(p*nmkt),nmkt,p) # data

# computation and test:
R=100  # number of replications (actually much larger than 100 in my application R=1e5 or 1e7)
system.time(for(r in 1:R){ for(i in 1:nmkt){res[,i] = qr.solve(x[, , i], y[i,], tol = 1e-7)}})
system.time(for(r in 1:R){ for(i in 1:nmkt){res[,i] = solve(x[, , i], y[i,], tol = 1e-7)}})
system.time(for(r in 1:R){ for(i in 1:nmkt){res[,i] = crossprod(  chol2inv(chol( x[, , i] ))  , y[i,] )}})

Is looping through an array a good solution?

Answer 1

Or use a sparse matrix?

require(Matrix)
j = c(matrix(1:(p*nmkt),p,p*nmkt,byrow=TRUE))
i = c(aperm( array(j,c(p,p,nmkt)), c(2,1,3)))    
system.time(for(r in 1:R){  res=    solve(sparseMatrix(i=i, j=j, x = c(x)), c(t(y)), tol = 1e-7)} )