Vectorised computation of sums of squares

Author

Enrico Schumann

Keywords

R, Matlab

Review Status

In many applications the objective function is a sum of squares, that is for some vector of length we need to compute

(1)

In vector-oriented languages like R and Matlab, matrix computations are often much more efficient than directly squaring the elements. Hence, replacing Equation (1) by

(2)

we get the same result (up to numerical precision), but generally the computation is much faster.

R test

The following code was tested with R 2.8.1.

trials <- 10000
lenV <- 1000
aux <- rnorm(lenV)
#
t1 <- system.time(
    for (i in 1:trials)
    {
        z1 <- sum(aux^2)
    }
)
t2 <- system.time(
    for (i in 1:trials)
    {
        z2 <- aux %*% aux
    }
)
#
t3 <- system.time(
    for (i in 1:trials)
    {
        z3 <- crossprod(aux)
    }
)

Computing the inner product (method 2) already gives a considerable speedup, which rises to more than an order of magnitude with the function crossprod.

Matlab test

The following code was tested with Matlab 2007a.

trials = 10000; lenV = 1000; aux = randn(lenV,1);

tic
for i = 1:trials
    z1 = sum(aux .^ 2);
end
t1 = toc;

tic
for i = 1:trials
    z2 = aux' * aux;
end
t2 = toc;

tic
for i = 1:trials
    z3 = dot(aux,aux);
end
t3 = toc;

Here, the function dot (which is not a built-in function) is actually slowest.

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References

1. R Development Core Team (2008). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. http://www.R-project.org.

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