194 W.Wang.J.Zhao Linear Algebra and its Applications 291 (1999)185-199 W=w-E元=w-EDT(u-YTw).▣ (21) This algorithm computes w,where w solves Eq.(12). Algorithm 3. INPUT:An upper triangular matrix R-T and w,where QX=R and w solves (11).A new set of k observations (YT u)and M=diag(,...,um). 1.Compute V=-R-TY. 2.Find -T=OTOr..Of.where 2 is a row M-invariant reflection, such that o()-(8) 3.Update R-T to R-T,i.e, (0)-() 4.Update w to w,i.e, w=w-ED-T(u-YTw). The cost in flops for each step is:1.kn2/2,2.15k2n,3.kn2,4.2+2kn with a total cost for Algorithm 3 as kn2+15k2n+2kn+flops.A straightforward implementation of the rank-1 method of Pan and Plemmons [4]would require 3kn2+O(kn)multiplications.Thus,roughly speaking,Algorithm 3 requires less flops when n≥l5k. 4.Inverse downdating Let the matrix Y and the vector s be given by the partition =(@)x=() where 2T∈Rxn,d∈R,Then the problem min llM 2(s-w)ll2 (22) is our downdating problem.Thus,we assume that we have the solution to(11) where Mi=diag(M3,M4)and want the the solution to (22)by our row hy- perbolic M-invariant method. Assume that =O.,where are hyperbolic M-invariant reflections. Then we define o=1...O to be used in the sequel. We have the following lemma that is used to construct the downdating al- gorithm