Computing the small singular values of large matrices

We consider the difficult problem of computing the smallest singular values and corresponding singular vectors of large matrices, matrices of order several hundred to several thousand. We compute these quantities by computing corresponding eigenvalues and eigenvectors of related symmetric matrices. We discuss several mechanisms for accelerating the computations which may also prove useful in other types of computations. A combination singular value algorithm that makes use of two symmetric matrices associated with the singular values of the given matrix is proposed.