In this paper a proof is given that only one of either the X- or the Y-matrix in PLS algorithms needs to be deflated during the sequential process of computing latent vectors. With the aid of this proof the original kernel algorithm developed by Lindgren et al. (J. Chemometrics, 7, 45 (1993)) is modified to provide two faster and more economical algorithms. The performances of these new algorithms are compared with that of De Jong and Ter Braak's (J. Chemometrics, 8, 169 (1994)) modified kernel algorithm in terms of speed and the new algorithms are shown to be much faster. A very fast kernel algorithm for updating PLS models in a recursive manner and for exponentially discounting past data is also presented. © 1997 John Wiley & Sons, Ltd.