Let A be a general real m-by-n matrix. The singular value decomposition (SVD) of A is the factorization , where U and V are orthogonal, and , , with . If A is complex, then its SVD is where U and V are unitary, and is as before with real diagonal elements. The are called the singular values, the first r columns of V the right singular vectors and the first r columns of U the left singular vectors.
The SVD and symmetric eigendecompositions are entirely analogous, so that any algorithm for one has a counterpart for the other. As soon as the final version of the symmetric eigenvalue algorithm has been developed, we will produce an SVD version. In the meantime, we plan to release an SVD code based on serial QR iteration, where each processor redundantly runs QR iteration on a bidiagonal matrix, but updates a subset of the rows of the U and V in an embarrassingly parallel fashion.