J.Zhuo Appl.Muth.Comput.92 (1998)49.58 51 where andL1∈Rmwi8 low triangular,.Ln∈is low triangular..Lg∈R×m. Proof.Since 4 is symmetric positive definite,there always exists the Cholesky factorization A=LiLi where LE Ris nonsingular low triangular.Take La B(LI) (51 so that B=LgLi It follows that Le is full row rank because B is full row rank.Because C is sym- metric semi-positive definite,the matrix C+LaL must be symmetric positive definite.Hence we have the Cholesky factorization LuL=C+LaLg (6) also let 4=(2)=(日) Thus we have L1L=（ Remark 1.From Theorem 1.we set that matrices Li and L can be obtained conveniently as long as submatrices L.Lg and L have been computed.This is why our method is as fast as the classical Cholesky factorization for symmetric positive definite. We now discuss the realization of the generalized Cholesky factorization (4) of G.Let A=a∈Rmxm.L=l, Using the Cholesky factorization of A,the elements of L can be computed from