How to do R multiplication with Nx1 1xM for Matrix NxM?

Question

I want to do a simple column (Nx1) times row (1xM) multiplication, resulting in (NxM) matrix. Code where I create a row by sequence, and column by transposing a similar seq

Answer 1

Here's a comparison of the execution speed for the three methods when the vectors being used are of length 100. The fastest is tcrossprod, with x%*%t(y) taking 17% longer and outer(x,y) taking 45% longer (in median time). In the table, neval is the number of times the function was evaluated to get the benchmark scores.

> x <- runif(100,0,100)
> y <- runif(100,0,100)
> microbenchmark(tcrossprod(x,y), x%*%t(y), outer(x,y), times=5000)
Unit: microseconds
             expr    min      lq     mean  median      uq       max neval
 tcrossprod(x, y) 11.404 16.6140 50.42392 17.7300 18.7555  5590.103  5000
       x %*% t(y) 13.878 19.4315 48.80170 20.5405 21.7310  4459.517  5000
      outer(x, y) 19.238 24.0810 72.05250 25.3595 26.8920 89861.855  5000

To get the the following graph, have

library("ggplot2")
bench <- microbenchmark(tcrossprod(x,y), x%*%t(y), outer(x,y), times=5000)
autplot(bench)

Edit: The performance depends on the size of x and y, and of course the machine running the code. I originally did the benchmark with vectors of length 100 because that's what Masi asked about. However, it appears the three methods have very similar performance for larger vectors. For vectors of length 1000, the median times of the three methods are within 5% of each other on my machine.

> x <- runif(1000)
> y <- runif(1000)
> microbenchmark(tcrossprod(x,y),x%*%t(y),outer(x,y),times=2000)
Unit: milliseconds
             expr      min       lq     mean   median       uq       max neval
 tcrossprod(x, y) 1.870282 2.030541 4.721175 2.916133 4.482346  75.77459  2000
       x %*% t(y) 1.861947 2.067908 4.921061 3.067670 4.527197 105.60500  2000
      outer(x, y) 1.886348 2.078958 5.114886 3.033927 4.556067  93.93450  2000

Answer 2

In this case using outer would be a more natural choice

outer(1:6, 1:6)

In general for two numerical vectors x and y, the matrix rank-1 operation can be computed as

outer(x, y)

If you want to resort to real matrix multiplication routines, use tcrossprod:

tcrossprod(x, y)

If either of your x and y is a matrix with dimension, use as.numeric to cast it as a vector first.

It is not recommended to use general matrix multiplication operation "%*%" for this. But if you want, make sure you get comformable dimension: x is a one-column matrix and y is a one-row matrix, so x %*% y.

---

Can you say anything about efficiency?

Matrix rank-1 operation is known to be memory-bound. So make sure we use gc() for garbage collection to tell R to release memory from heap after every replicate (otherwise your system will stall):

x <- runif(500)
y <- runif(500)
xx <- matrix(x, ncol = 1)
yy <- matrix(y, nrow = 1)

system.time(replicate(200, {outer(x,y); gc();}))
#   user  system elapsed 
#  4.484   0.324   4.837 

system.time(replicate(200, {tcrossprod(x,y); gc();}))
#   user  system elapsed 
#  4.320   0.324   4.653 

system.time(replicate(200, {xx %*% yy; gc();}))
#   user  system elapsed 
#  4.372   0.324   4.708 

In terms of performance, they are all very alike.

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Follow-up

When I came back I saw another answer with a different benchmark. Well, the thing is, it depends on the problem size. If you just try a small example you can not eliminate function interpretation / calling overhead for all three functions. If you do

x <- y <- runif(500)
microbenchmark(tcrossprod(x,y), x %*% t(y), outer(x,y), times = 200)

you will see roughly identical performance again.

#Unit: milliseconds
#             expr     min      lq     mean  median      uq      max neval cld
# tcrossprod(x, y) 2.09644 2.42466 3.402483 2.60424 3.94238 35.52176   200   a
#       x %*% t(y) 2.22520 2.55678 3.707261 2.66722 4.05046 37.11660   200   a
#      outer(x, y) 2.08496 2.55424 3.695660 2.69512 4.08938 35.41044   200   a

Answer 3

An easy way to look at this is to transform your vectors to a matrix

row1.mat = matrix(row1)
col1.mat = matrix(col1)

and then use dim to see the dimension of the matrices:

dim(row1.mat)
dim(col1.mat)

If you want the product to work for this you need a 6*1 matrix, multiplied by a 1*6 matrix. so you need to transpose the col1.mat using t(col1.mat).

And as you might know the matrix product is %*%

row1.mat %*% t(col1.mat)

Comparison of this method to others

library("microbenchmark")
x <- runif(1000)
y <- runif(1000)
xx = matrix(x)
yy = matrix(y)
microbenchmark(tcrossprod(x,y),x%*%t(y),outer(x,y), xx %*% t(yy), times=2000)

Unit: milliseconds
             expr      min       lq     mean   median       uq      max neval
 tcrossprod(x, y) 2.829099 3.243785 6.015880 4.801640 5.040636 77.87932  2000
       x %*% t(y) 2.847175 3.251414 5.942841 4.810261 5.049474 86.53374  2000
      outer(x, y) 2.886059 3.277811 5.983455 4.788054 5.074997 96.12442  2000
     xx %*% t(yy) 2.868185 3.255833 6.126183 4.699884 5.056234 87.80024  2000