STABILITY OF TWO DIRECT METHODS FOR BIDIAGONALIZATION AND PARTIAL LEAST SQUARES

The partial least squares (PLS) method computes a sequence of approximate solutions x(k) is an element of K-k (A(T) A, A(T) b), k = 1, 2, ..., to the least squares problem min(x) parallel to Ax - b parallel to(2). If carried out to completion, the method always terminates with the pseudoinverse solution x(dagger) = A(dagger)b. Two direct PLS algorithms are analyzed. The first uses the Golub-Kahan Householder algorithm for reducing A to upper bidiagonal form. The second is the NIPALS PLS algorithm, due to Wold et al., which is based on rank-reducing orthogonal projections. The Householder algorithm is known to be mixed forward-backward stable. Numerical results are given, that support the conjecture that the NIPALS PLS algorithm shares this stability property. We draw attention to a flaw in some descriptions and implementations of this algorithm, related to a similar problem in Gram-Schmidt orthogonalization, that spoils its otherwise excellent stability. For large-scale sparse or structured problems, the iterative algorithm LSQR is an attractive alternative, provided an implementation with reorthogonalization is used.