602 W.Wang.J.Zhao Appl.Math.Comput.147 (2004)601-606 In [1],the generalized Cholesky factorization is presented and this method inherits the advantage of Cholesky factorization with small storage and low computation costs.We first derive the generalized Cholesky factorization theorem. Theorem 1.1 [1].Given any symmetric indefinite matrix K= -C (2) where A,B and C are the same as that defined in (1).Then we have K=LJLT, (3) =(e)J-(-) (4) where Ln E Rmxm and L2 E Raxh are lower triangular,La E R"xm,Im and In are identity matrices. Letr=K+△be a perturbation of K in which△K is symmetric.If△Kis sufficiently small,then K also has a generalized Cholesky factorization: K+△K=(L+△L)J(L+△L)T (5) There have been several results dealing with the perturbation analysis for the Cholesky factor (see [3,8,9)).They obtained the first-order perturbation result. The result is sharpened in [5,6]. About (5),K.Veslic [10]has given some eigenvalue perturbation results.In this paper we derive the first-order perturbation bound and the rigorous per- turbation bound for the generalized Cholesky factorization. 2.Perturbation theorems for the generalized Cholesky factorization In the section,two perturbation theorems on the generalized Cholesky factor will be given.The symbols ll2 and llE will be used for the spectral norm and the Frobenius norm,respectively. We need a lemma controlling the triangular indefinite decomposition of J+N for small N. Lemma 2.1 [10].Let N be a Hermitian matrix withN<1/2,then there exists a unique lower triangular matrix such that J+N=(I+)J(I+*),In [1], the generalized Cholesky factorization is presented and this method inherits the advantage of Cholesky factorization with small storage and low computation costs. We first derive the generalized Cholesky factorization theorem. Theorem 1.1 [1]. Given any symmetric indefinite matrix K ¼ A BT B
C
; ð2Þ where A, B and C are the same as that defined in (1). Then we have K ¼ LJLT; ð3Þ L ¼ L11 L21 L22
; J ¼ Im
In
; ð4Þ where L11 2 Rmm and L22 2 Rnn are lower triangular, L21 2 Rnm; Im and In are identity matrices. Let Ke ¼ K þ DK be a perturbation of K in which DK is symmetric. If DK is sufficiently small, then Ke also has a generalized Cholesky factorization: K þ DK ¼ ðL þ DLÞJðL þ DLÞ T : ð5Þ There have been several results dealing with the perturbation analysis for the Cholesky factor (see [3,8,9]). They obtained the first-order perturbation result. The result is sharpened in [5,6]. About (5), K. Vesli
c [10] has given some eigenvalue perturbation results. In this paper we derive the first-order perturbation bound and the rigorous per￾turbation bound for the generalized Cholesky factorization. 2. Perturbation theorems for the generalized Cholesky factorization In the section, two perturbation theorems on the generalized Cholesky factor will be given. The symbols kk2 and kkF will be used for the spectral norm and the Frobenius norm, respectively. We need a lemma controlling the triangular indefinite decomposition of J þ N for small N. Lemma 2.1 [10]. Let N be a Hermitian matrix with kNk < 1=2, then there exists a unique lower triangular matrix such that J þ N ¼ ðI þ CÞJðI þ C Þ; 602 W. Wang, J. Zhao / Appl. Math. Comput. 147 (2004) 601–606