[fgdelcueto]

It depends a lot of the problem. For a general least squares solution of the Ax=b problem, is your A matrix dense, sparse, structured? Do you need to solve it very accurately or just approximately? Are you going to be using this over and over with the same A but different b's? In that case, you may profit from a one time Cholesky factorization that can be used over and over very quickly. Is your linear operator A explicit (as in you can actually inspect the matrix) or is it a linear operator function in which you can only see the result of A(x) (and its adjoint A_adj(y)) ? In this case, you cannot use common factorizations (QR, cholesky, svd) but will have to use iterative methods (ie. Krylov space methods).

There's a whole branch in applied math that's called numerical linear algebra and deals with all these kinds of situations. One of the most important results is that you should almost never do an explicit computation of the inverse of a matrix. It is expensive and can be highly unstable. If you ever see a A\b or pinv(A)*b in somebody's code, it should raise a big red flag.

[unishark]

A little defense of the pseudoinverse here, it's often a fine tool, far more so than the true inverse.

If the matrix is not full rank the psuedoinverse provides the least squares solution.

It can also be done efficiently and may well already be implemented using whatever numerical method one prefers (SVD, QR).

And the implementation probably allows for regularization via a singular value cutoff.

I'm not sure what matlab's backslash operator does these days, but it looks like a more efficient way to get a least-squares solution (compared to the pseudoinverse), possibly with sparsity imposed slightly.