Replicate a function more efficiently

Question

Suppose I have the following matrices.

x <- matrix(seq(1:4), 2, 2)
y <- matrix(seq(1:4), 2, 2)

I want to do the following.

for(i in 1:5)
{
  x <- x %*% y
}

However, this is a simple example. I usually have large matrices for X and Y and i is also a large number. So, use of for-loop might be too time consuming.

Does anyone aware of using lapply or apply function for these types.

Thanks.

Answer 1

library(expm)
x %*% (y %^% 5)
#     [,1]  [,2]
#[1,] 5743 12555
#[2,] 8370 18298

Benchmarks:

set.seed(42)
x <- matrix(rnorm(1e4), 1e2, 1e2)
y <- matrix(rnorm(1e4), 1e2, 1e2)

fun1 <- function(x, y, j) {
  for(i in 1:j)
  {
    x <- x %*% y
  }
  x
}

fun2 <- function(x, y, i) {
  x %*% (y %^% i)
}

fun3 <- function(x, y, i) {
  Reduce("%*%", c(list(x), rep(list(y), i)))
}


library(expm)
all.equal(fun1(x,y,5), fun2(x,y,5))
#[1] TRUE
all.equal(fun1(x,y,5), fun3(x,y,5))
#[1] TRUE

library(microbenchmark)
microbenchmark(fun1(x,y,30), 
               fun2(x,y,30), 
               fun3(x,y,30), times=10)


#Unit: milliseconds
#          expr       min        lq    median        uq        max neval
#fun1(x, y, 30) 21.317917 21.908592 22.103380 22.182989 141.933427    10
#fun2(x, y, 30)  5.899368  6.068441  6.235974  6.345301   6.477417    10
#fun3(x, y, 30) 21.385668 21.896274 22.023001 22.086904  22.269527    10

Answer 2

Reduce("%*%", c(list(x), rep(list(y), 5)))
#      [,1]  [,2]
# [1,] 5743 12555
# [2,] 8370 18298

will do the trick.

Answer 3

Just for fun, here is a solution using RcppEigen:

C++ code:

// [[Rcpp::depends(RcppEigen)]]
#include <RcppEigen.h>
using namespace Rcpp;
using   Eigen::Map;
using   Eigen::MatrixXd;
typedef  Map<MatrixXd>  MapMatd;

// [[Rcpp::export]]
NumericMatrix XYpow(NumericMatrix A, NumericMatrix B, const int j) {
   const MapMatd  X(as<MapMatd>(A)), Y(as<MapMatd>(B));
   MatrixXd X1(X);
   for (int i = 0; i < j; ++i) X1 = X1 * Y;
    return wrap(X1);
}

Then in R:

all.equal(fun2(x,y,5), XYpow(x,y,5))
#[1] TRUE

microbenchmark(fun2(x,y,30), 
               XYpow(x,y,30), times=10)
#Unit: milliseconds
#            expr      min       lq   median       uq      max neval
#  fun2(x, y, 30) 5.726292 5.768792 5.948027 6.041340 6.276624    10
# XYpow(x, y, 30) 6.926737 7.032061 7.232238 7.512486 7.617502    10