Converting nested summation to R type matrix multiplication

Question

I am trying to implement a nested summation in R. The for loop implementation is:

 sum = 0
  for(i in 1:n){
    for(j in 1:n){
      for(k in 1:n){
        sum = sum + w[i,j]*w[j,k]
      }
    }
  }

where w is a symmetric square matrix and n is the number of rows (or columns). Please see the formula I am trying to implement. (SO did not allow me to write Latex nor add the image here.)

The nested for loops above takes forever. How do I implement this efficiently the R way?

Answer 1

Try this:

Sum2 <- sum(w %*% w)

all.equal(Sum, Sum2)
## [1] TRUE

Note

We used for comparison:

# input
set.seed(123)
n <- 5
w <- matrix(rnorm(n^2), n)

# from question
Sum = 0
  for(i in 1:n){
    for(j in 1:n){
      for(k in 1:n){
        Sum = Sum + w[i,j]*w[j,k]
      }
    }
  }