190 W.Wang,J.Zhao Linear Algebra and its Applications 291 (1999)185-199 Compute di =u-udMid. Let d=(di,dT). end OUTPUT:=-2Mdd/dMd that has the property that the first row of OB is all zeros. 3.Inverse updating In this section we will solve the updating problem by M-invariant matrix methods. Consider the weighted linear least squares problem below m9aM2(s-Kwl2, (11) where Mi diag(,...u),u>0 and X is an m x n matrix with rank(X)=n. Let X=OR,where o is an M-invariant matrix with columns, R1=(0 ∈代mxn and R is an n x n upper triangular matrix.Then the solution to (11)is given by w=(R10)2's. Suppose k new observations (YT u),where YT e Rex",and ueR,be added to the dating defining the weighted linear least squares problem(11).We then show how the solution w to (11)can be updated to the solution w to m-a(a)-()儿 (12) where M=diag(M,M2). Lemma 3.1.Let ()-(8) (13) where is an M-invariant matrix,R.R is upper triangular,then ()=() (1 where ET E Rkxn