Perturbation analysis for the hyperbolic QR factorization

The hyperbolic QR factorization is a generalization of the classical QR factorization and can be regarded as the triangular case of the indefinite QR factorization proposed by Sanja Singer and Sasa Singer. In this paper, the perturbation analysis for this factorization is considered using the classical matrix equation approach, the refined matrix equation approach, and the matrix-vector equation approach. The first order and rigorous normwise perturbation bounds with normwise or componentwise perturbations in the given matrix are derived. The obtained first order bounds can be much tighter than the corresponding existing ones. Each of the obtained rigorous bounds is composed of a small constant multiple of the corresponding first order bound and an additional term with simple form. In particular, for square matrix, the rigorous bounds for the factor R are just the 6+3 multiple of the corresponding first order bounds. These rigorous bounds can be used safely for all cases in comparison to the first order bounds.

[1]  Fang Xi-ming.  The Generalized Cholesky Factorization and Perturbation of the Real Symmetric Matrices , 2008 .

[2]  Xiao-Wen Chang,et al.  Perturbation analysis of some matrix factorizations , 1997 .

[3]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[4]  Nicholas J. Higham,et al.  Solving the Indefinite Least Squares Problem by Hyperbolic QR Factorization , 2002, SIAM J. Matrix Anal. Appl..

[5]  A. Bunse-Gerstner.  An analysis of the HR algorithm for computing the eigenvalues of a matrix , 1981 .

[6]  Damien Stehlé,et al.  Rigorous Perturbation Bounds of Some Matrix Factorizations , 2010, SIAM J. Matrix Anal. Appl..

[7]  G. Stewart,et al.  Perturbation Analyses for the QR Factorization , 1997, SIAM J. Matrix Anal. Appl..

[8]  R. Bhatia.  Matrix factorizations and their perturbations , 1994 .

[9]  H. Zha.  A componentwise perturbation analysis of the QR decomposition , 1993 .

[10]  Mei Han An,et al.  accuracy and stability of numerical algorithms , 1991 .

[11]  M. Berhanu.  The Polynomial Eigenvalue Problem , 2005 .

[12]  Nicholas J. Higham,et al.  J-Orthogonal Matrices: Properties and Generation , 2003, SIAM Rev..

[13]  Sanja Singer.  Indefinite QR Factorization , 2006 .

[14]  Damien Stehlé,et al.  Perturbation Analysis of the QR factor R in the context of LLL lattice basis reduction , 2012, Math. Comput..

[15]  Sanja Singer,et al.  Rounding-error and perturbation bounds for the indefinite QR factorization , 2000 .

[16]  Xiao-Wen Chang,et al.  On the sensitivity of the SR decomposition , 1998 .

[17]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[18]  Xiao-Wen Chang,et al.  Componentwise perturbation analyses for the QR factorization , 2001, Numerische Mathematik.

[19]  Xiao-Wen Chang,et al.  Multiplicative perturbation analysis for QR factorizations , 2011 .