188 W.Wang.J.Zhao I Linear Algebra and its Applications 291 (1999)185-199 Algorithm 1 (RowMR). INPUT:An n-vector b,a nonsingular n x n-matrix D,and the weight matrix M=diag(41,,4n,na+i】 if=0 then d=0,2=I. else Put u=1/llbll2.and solve DTd ub. Compute d=--+aMd.Let d=(d.) en OUTPUT:=I-2MddT/dTMd having the property that the first row of OB is all zeros. Algorithm RowMR will have good numerical properties as long as d is solved by a numerically stable method. In Section 3 we consider the problem of annihilating r rows.This is easily done by applying a sequence of M-invariant reflections described above as 0=2Q-1…Q1. 2.2.Row hyperbolic M-invariant reflections LetΦ=diag(士l),and assume that2∈Rmxm and M∈Rmxm,then is said to be hyperbolic M-invariant if it is nonsingular and OMor=M(see Ref. [10]) Lemma2.3.Assume that Q=中-2cdr,dTMΦd≠0，whereΦ=diag(士l),and O is hyperbolic M-invariant with M nonsingular.Then 2=Φ-2Mddr/d'Mpd,with O2=Φ， i.e.,O is a hyperbolic reflector.We call o a hyperbolic M-invariant reflection. Theorem 2.4.Let B be an (n+1)x n matrix of the form B D》 where D is nonsingular,and M=diag(,....)with >0.Assume that DTM,D-1/u bb>0 then there is a hyperbolic M-invariant reflection 2=Φ-2cd,whereΦ=diag(-l,l,,I),such that QB= (8) (7) where DE Raxn