An efficient and reliable algorithm for computing the singular subspace of a matrix, associated with its smallest singular values

Abstract

In this paper, an improved algorithm PSVD for computing the singular subspace of a matrix corresponding to its smallest singular values is presented. As only a basis of the desired singular subspace is needed, the classical Singular Value Decomposition (SVD) algorithm can be modified in three ways. First, the Householder transformations of the bidiagonalization need only to be applied on the base vectors of the desired singular subspace. Second, the bidiagonal must only be partially diagonalized and third, the convergence rate of the iterative diagonalization can be improved by an appropriate choice between QR and QL iteration steps. An analysis of the operation counts, as well as computational results, show the relative efficiency of PSVD with respect to the classical SVD algorithm. Depending on the gap, the desired numerical accuracy and the dimension of the desired subspace, PSVD can be three times faster than the classical SVD algorithm while the same accuracy can be maintained. The new algorithm can be successfully used in total least squares applications, in the computation of the null space of a matrix and in solving (non) homogeneous linear equations. Based on PSVD a very efficient and reliable algorithm is also derived for solving nonhomogeneous equations.