Reorthogonalization for the Golub-Kahan-Lanczos bidiagonal reduction

The Golub–Kahan–Lanczos (GKL) bidiagonal reduction generates, by recurrence, the matrix factorization of X ∈ Rm×n,m ≥ n, given by X = U BV T where U ∈ Rm×n is left orthogonal, V ∈ Rn×n is orthogonal, and B ∈ Rn×n is bidiagonal. When the GKL recurrence is implemented in finite precision arithmetic, the columns of U and V tend to lose orthogonality, making a reorthogonalization strategy necessary to preserve convergence of the singular values. The use of an approach started by Simon and Zha (SIAM J Sci Stat Comput, 21:2257–2274, 2000) that reorthogonalizes only one of the two left orthogonal matrices U and V is shown to be very effective by the results presented here. Supposing that V is the matrix reorthogonalized, the reorthogonalized GKL algorithm proposed here is modeled as the Householder Q–R factorization of ( 0n×k X Vk ) where Vk = V ( : , 1 : k). That model is used to show that if εM is the machine unit and η̄ = ‖tril(I − V T V )‖F , where tril(·) is the strictly lower triangular part of the contents, then: (1) the GKL recurrence produces Krylov spaces generated by a nearby matrix X + δX , The research of Jesse L. Barlow was supported by the National Science Foundation under Grant No. CCF-0429481 and CCF-1115704. J. L. Barlow (B) Department of Computer Science and Engineering, The Pennsylvania State University, University Park, PA 16802-6822, USA e-mail: [email protected]