606 W.Wang.J.Zhao Appl.Math.Comput.147 (2004)601-606 Since Kl2,we have IA应SK①x()JAKE 匹2 IKI If K(L)=1(it cannot be less),then the bound(21)reduces to I△Lle≤lL-1l2 AKIF (22) References [1]J.Zhao,The generalized Cholesky factorization method for solving saddle point problems, Appl.Math.Comput.92 (1998)49-58. [2]G.H.Golub,C.F.Vanloan,Matrix Computation,third ed.,The Johns Hopkins University Press,Baltimore,MD,1996. [3]J.Sun,Perturbation bounds for the Cholesky and QR factorizations,BIT 31(1991)341-352. [4]P.E.Gill,M.A.Saunders,J.R.Shinnerl,On the stability of Cholesky factorization for symmetric quasidefinite systems,SIAM J.Matrix Anal.Appl.17(1)(1996)35-46. [5]G.W.Stewart.On the Perturbation of LU and Cholesky factors,IMA J.Numer.Anal.17 (1997)1-6. [6]X.Chang,C.C.Paige,G.W.Stewart,New perturbation analyses for the Cholesky fac- torization.IMA J.Numer.Anal.16(1996)457-484. [7]A.Forsgren,P.E.Gill,J.R.Shinnerl,Stability of symmetric ill-conditioned systems arising in interior methods for constrained optimization,SIAM J.Matrix Anal.17(1)(1996)187-211. [8]J.Stoer,R.Bulirsch,Introduction to Numerical Analysis,second ed.,Springer-Verlag.New York,1993. [9]G.W.Stewart,Perturbation bounds for the QR factorization of matrix,SIAM J.Numer.Anal. 14(1977509-518. [10]K.Veslic,Perturbation theory for the eigenvalues of factorised symmetric matrices,Linear Algebra Appl.309(2000)85-102. [11]A.Bjorck,C.C.Paige,Solution of augmented linear systems using orthogonal factorizations. BIT34(1994)1-24. [12]A.Bjorck,Numerical stability of methods for solving augmented systems,Contemp.Math. 204(1997)51-60.Since kKk2 6 kLk 2 2, we have kD fLkF kLk2 6 jðbLÞjðLÞ kDKkF kKk2 : If KðbLÞ ¼ 1(it cannot be less), then the bound (21) reduces to kD fLkF 6 kL
1 k2kDKkF: ð22Þ References [1] J. Zhao, The generalized Cholesky factorization method for solving saddle point problems, Appl. Math. Comput. 92 (1998) 49–58. [2] G.H. Golub, C.F. Vanloan, Matrix Computation, third ed., The Johns Hopkins University Press, Baltimore, MD, 1996. [3] J. Sun, Perturbation bounds for the Cholesky and QR factorizations, BIT 31 (1991) 341–352. [4] P.E. Gill, M.A. Saunders, J.R. Shinnerl, On the stability of Cholesky factorization for symmetric quasidefinite systems, SIAM J. Matrix Anal. Appl. 17 (1) (1996) 35–46. [5] G.W. Stewart, On the Perturbation of LU and Cholesky factors, IMA J. Numer. Anal. 17 (1997) 1–6. [6] X. Chang, C.C. Paige, G.W. Stewart, New perturbation analyses for the Cholesky fac￾torization, IMA J. Numer. Anal. 16 (1996) 457–484. [7] A. Forsgren, P.E. Gill, J.R. Shinnerl, Stability of symmetric ill-conditioned systems arising in interior methods for constrained optimization, SIAM J. Matrix Anal. 17 (1) (1996) 187–211. [8] J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, second ed., Springer-Verlag, New York, 1993. [9] G.W. Stewart, Perturbation bounds for the QR factorization of matrix, SIAM J. Numer. Anal. 14 (1977) 509–518. [10] K. Vesli
c, Perturbation theory for the eigenvalues of factorised symmetric matrices, Linear Algebra Appl. 309 (2000) 85–102. [11] A. Bj orck, C.C. Paige, Solution of augmented linear systems using orthogonal factorizations, € BIT 34 (1994) 1–24. [12] A. Bj orck, Numerical stability of methods for solving augmented systems, Contemp. Math. € 204 (1997) 51–60. 606 W. Wang, J. Zhao / Appl. Math. Comput. 147 (2004) 601–606