We don't give information on what SVD algorithm we use, look up the LAPACK library for detailed descriptions. For practical purposes, you can assume the complexity of doing
[U, S, V] = svd(X, 'econ')
is O(m*n*min(m,n)), while the complexity of doing
[U, S, V] = svd(X)
This wouldn't be easy to proof, since the SVD computation involves an iteration that needs to converge to each of the singular values, but the complexities above are the ones that dominate in all practical cases.