matrix multiplication with rotations

mm_rot.R

# 4 elem vector
w <- c(10, 20, 30, 40)

# 4*3 matrix
#      [,1] [,2] [,3]
# [1,]    1    2    3
# [2,]    4    5    6
# [3,]    7    8    9
# [4,]   10   11   12
x <- matrix(1:12, ncol=3, byrow=T)

# do vectorised matrix multiplation with only element-wise ops to get:
# > w %*% x
#      [,1] [,2] [,3]
# [1,]  700  800  900

# we can do left/right vector rotations:
# > rot(c(1,2,3,4), 1)
# [1] 2 3 4 1
# > rot(c(1,2,3,4), 2)
# [1] 3 4 1 2
rot <- function(x, n) c(tail(x, -n), head(x, n))

# 1st, multiply the w vector element-wise with each of the columns in x:
m1 <- w*x[, 1]; m2 <- w*x[, 2]; m3 <- w*x[, 3]

# 2nd, take each vector obtained above and continuously sum rotations such that
# the first element gets added with all others.
# e.g.,
# > c(1,2,3) + c(2,3,1) + c(3,1,2)
# [1] 6 6 6
s1 <- m1 + rot(m1, 1) + rot(m1, 2) + rot(m1, 3)
s2 <- m2 + rot(m2, 1) + rot(m2, 2) + rot(m2, 3)
s3 <- m3 + rot(m3, 1) + rot(m3, 2) + rot(m3, 3)

# now, use a single element mask (matrix diagonal) to add element-wise to get
# final result: [ 700 800 900   0]
mask <- c(1, 0, 0, 0)
s1*mask + s2*rot(mask, -1) + s3*rot(mask, -2)

# note that if the length of the vector is a power of 2, can do this
# nice trick!
x <- 1:128
sum(x)
# [1] 8256
x <- x + rot(x, 1)
x <- x + rot(x, 2)
x <- x + rot(x, 4)
x <- x + rot(x, 8)
x <- x + rot(x, 16)
x <- x + rot(x, 32)
x <- x + rot(x, 64)
head(x, 1)
# [1] 8256