LINEAR ALGEBRA AND ITS APPLICATIONS ELSEVIER Linear Algebra and its Applications 291(1999)185-199 Inverse updating and downdating for weighted linear least squares using M-invariant reflections Weiguo Wang a, Jinxi Zhao b. * Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, People's Republic of China b Department of Mathematics, Nanjing University, Nanjing 210008, People's Republic of China Received 23 May 1996; accepted 30 November 1998 Submitted by L. Elsner Abstract A new method for the weighted linear least squares problem min, M-12(b - Ax), is presented by introducing a row M-invariant matrix (i.e., QMQ' = M). Our purpose in this paper is to introduce new row M-invariant and row hyperbolic M-invariant re- flections. We then show how these row M-invariant reflections can be used to design efficient sliding-date-window recursive weighted linear least squares covariance algo- rithms, which are based upon rank-k modifications to the inverse like-Cholesky factor R of the covariance matrix. The algorithms are rich in matrix-matrix BLAS-3 com- putations. We also provide computational experiments indicating the numerical stability of the methods. 1999 Elsevier Science Inc. All rights reserved. AMS classification: 65F25; 65F35; 65G05 Keywords:Linear weighted least squares;Row M-invariant reflection;Updating:Downdating 1.Introduction Consider the weighted linear least squares problem [1,7-9] min(b-Ax)'M1(b-Ax), (1) x∈R Corresponding author. Project supported by the State 863-high science technology plan of China. 0024-3795/99/S- see front matter 1999 Elsevier Science Inc. All rights reserved. PII:S0024-3795(99)00007-5