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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

Stability of Two Direct Methods for Bidiagonalization and Partial Least Squares