Available at www.EIsevierMathematics.com APPLIED MATHEMATICS POWERED BY●cI■Nc■@DIn■cT同 AN COMP风TATION ELSEVIER Applied Mathematics and Computation 147(2004)601-606 www.elsevier.com/locate/amc Perturbation analysis for the generalized Cholesky factorization Wei-guo Wang a,Jin-xi Zhao b.* Department of Mathematics.Ocean University of Qingdao,Shandong 266071.PR China bState Key Laboratory for Novel Software Technology.Nanjing University. Nanjing 210093,PR China Abstract Let K be a symmetric indefinite matrix.Suppose that K=LILT is the generalized Cholesky factorization of K.In this paper we present perturbation analysis for the generalized Cholesky factorization.We obtain the first-order bound on the norm of the perturbation in the generalized Cholesky factor.Also,we give rigorous perturbation bounds. 2002 Elsevier Inc.All rights reserved. Keywords:Generalized Cholesky factorization;Perturbation bounds;Augmented matrix 1.Introduction Consider the problem of solving the structured linear system (日)()=() (1) for x and y,where A E Rmxm is symmetric positive definite matrix,B E Rmx",x, b E Rm,and y,d E R",C E R"x#.This system is called an augmented system,or an equilibrium system.The system(1)has been investigated by many authors for numerical algorithms.(See,[1,2,4,7,11,12]) 'Corresponding author.Address:Department of Computer Science and Technology,Nanjing University,Nanjing 210008,China. E-mail address:jxzhao@nju.edu.cn (J.Zhao). 0096-3003/S-see front matter 2002 Elsevier Inc.All rights reserved. doi10.1016/S0096-3003(02)00798-1Perturbation analysis for the generalized Cholesky factorization Wei-guo Wang a , Jin-xi Zhao b,* a Department of Mathematics, Ocean University of Qingdao, Shandong 266071, PR China b State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing 210093, PR China Abstract Let K be a symmetric indefinite matrix. Suppose that K ¼ LJLT is the generalized Cholesky factorization of K. In this paper we present perturbation analysis for the generalized Cholesky factorization. We obtain the first-order bound on the norm of the perturbation in the generalized Cholesky factor. Also, we give rigorous perturbation bounds.
2002 Elsevier Inc. All rights reserved. Keywords: Generalized Cholesky factorization; Perturbation bounds; Augmented matrix 1. Introduction Consider the problem of solving the structured linear system A BT B
C
x y
¼ b d
; ð1Þ for x and y, where A 2 Rmm is symmetric positive definite matrix, B 2 Rmn, x, b 2 Rm, and y; d 2 Rn, C 2 Rnn. This system is called an augmented system, or an equilibrium system. The system (1) has been investigated by many authors for numerical algorithms. (See, [1,2,4,7,11,12]) * Corresponding author. Address: Department of Computer Science and Technology, Nanjing University, Nanjing 210008, China. E-mail address: jxzhao@nju.edu.cn (J. Zhao). 0096-3003/$ - see front matter
2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00798-1 Applied Mathematics and Computation 147 (2004) 601–606 www.elsevier.com/locate/amc