The paper presents an algorithm for computing the singular values of a real upper tridiagonal matrix by extending the Golub-Kahan algorithm. A relation is found for left and right singular vectors that help to prove monotonic convergence of singular values and vectors. The algorithm is not suited for dense matrices, but helps understand the Golub-Kahan algorithm for singular values of bidiagonal matrices and the QR algorithm for eigenvalues of symmetric matrices.
The algorithm proceeds in two phases: first the large singular values are separated from the small ones; then convergence sets in. This is used to propose a divide-and-conquer algorithm for finding the singular values of dense matrices.