  In this paper a vectorized algorithm for simultaneously computing up to eight
singular value decompositions (SVDs, each of the form $A=U\Sigma V^{\ast}$) of
real or complex matrices of order two is proposed. The algorithm extends to a
batch of matrices of an arbitrary length $n$, that arises, for example, in the
annihilation part of the parallel Kogbetliantz algorithm for the SVD of a
square matrix of order $2n$. The SVD algorithm for a single matrix of order two
is derived first. It scales, in most instances error-free, the input matrix $A$
such that its singular values $\Sigma_{ii}$ cannot overflow whenever its
elements are finite, and then computes the URV factorization of the scaled
matrix, followed by the SVD of a non-negative upper-triangular middle factor. A
vector-friendly data layout for the batch is then introduced, where the
same-indexed elements of each of the input and the output matrices form
vectors, and the algorithm's steps over such vectors are described. The
vectorized approach is then shown to be about three times faster than
processing each matrix in isolation, while slightly improving accuracy over the
straightforward method for the $2\times 2$ SVD.
