[1] Li Xinxiu, Nie Xiaobing,. Rice condition numbers of QR and Cholesky factorizations [J]. Journal of Southeast University (English Edition), 2004, 20 (1): 130-134. [doi:10.3969/j.issn.1003-7985.2004.01.027]

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Rice condition numbers of QR and Cholesky factorizations()

QR分解和Cholesky分解的Rice条件数

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

20

Issue:

2004 1

Page:

130-134

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2004-03-30

Info

Title:

Rice condition numbers of QR and Cholesky factorizations

QR分解和Cholesky分解的Rice条件数

Author(s):

Li Xinxiu¹;  Nie Xiaobing²

¹Department of Mathematics and Physics, Nanjing University of Posts and Telecommunications, Nanjing 210003, China
²Department of Mathematics, Southeast University, Nanjing 210096, China

李新秀¹;  聂小兵²

¹南京邮电学院应用数理系, 南京 210003; ²东南大学数学系, 南京 210096

Keywords:

Rice condition number;  Cholesky factorization;  QR decomposition

Rice条件数;  Cholesky 分解;  QR分解

PACS:

O241.6

DOI:

10.3969/j.issn.1003-7985.2004.01.027

Abstract:

A condition number is an amplification coefficient due to errors in computing. Thus the theory of condition numbers plays an important role in error analysis. In this paper, following the approach of Rice, condition numbers are defined for factors of some matrix factorizations such as the Cholesky factorization of a symmetric positive definite matrix and QR factorization of a general matrix. The condition numbers are derived by a technique of analytic expansion of the factor dependent on one parameter and matrix-vector equation. Condition numbers of the Cholesky and QR factors are different from the ones previously introduced by other authors, but similar to Chang’s results. In Cholesky factorization, corresponding with the condition number of the factor matrix L, K_L is a low bound of Stewart’s condition number K.

条件数是在计算过程中由于误差引起的放大系数, 所以条件数理论在误差分析中占有非常重要的地位. 本文运用Rice关于条件数的一般理论, 采取一种统一的方式, 在单参数扰动的情况下, 定义了与正定对称矩阵的Cholesky 分解和一般矩阵的QR分解有关的一些矩阵因子的条件数.利用解析展开和矩阵向量方程的方法, 求出了用Frobenius 范数所定义的Rice条件数的具体表达式.所得结果与常小文的结果类似.在Cholesky分解情况下, 与因子矩阵L 相对应的条件数 K_L是 Stewart条件数K的一个下界.

References:

[1] Rice J R. A theory of condition [J]. SIAM J Numer Anal, 1966(3): 287-310.
[2] Sun J G. Perturbation analysis of singular subspaces and deflating subspaces [J]. Numer Math, 1996, 173: 235-263.
[3] Sun J G. Perturbation analysis of system Hessenberg and Hessenberg-triangular forms [J].Lin Alg Appl, 1996, 241-243: 811-849.
[4] Stewart G W. On the perturbation of LU, Cholesky and QR factorizations [J]. SIAM J Matrix Anal Appl, 1993, 14: 1141-1145.
[5] Wilkinson J H. The algebraic eigenvalue problems [M]. London, New York: Oxford University Press, 1965.
[6] Chang Xiaowen, Paige C C. New perturbation analysis for the Cholesky factorization [J]. IMA J Numer Anal, 1996, 16: 457-484.
[7] Chang Xiaowen, Paige C C. Sensitivity analysis for factorizations of sparse or structured matrices [J]. Lin Alg Appl, 1998284: 53-72.
[8] Sun J G. Eigenvalues and eigenvectors of a matrix dependent on several parameters [J]. J Comp Math, 1985(3): 351-364.
[9] Chang Xiaowen, Paige C C, Stewart G W. Perturbation analysis for the QR factorization [J]. SIAM J Matrix Anal Appl, 1997, 18: 775-791.

Memo

Memo:

Biography: Li Xinxiu(1976—), female, master, lixinxiu410@163.com.

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