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## A Perturbation Analysis for R in the QR Factorization (1995)

Venue: | In preparation |

Citations: | 8 - 6 self |

### Citations

7717 |
Matrix Analysis
- Horn, Johnson
- 1990
(Show Context)
Citation Context ... \Gamma1sR ZsR and W \Gamma1 R ZR (see Appendix A for the zero/nonzero structure of W \Gamma1 R ZR ). In order to estimate kW \Gamma1 R ZsR ffi Tk 2 , we use the following lemma (see Theorem 5.5.3 in =-=[4]-=-). Lemma 3.1. Given any A; B 2 R m\Thetan we have kA ffi Bk 2sminfr(B); c(B)gkAk 2 ; where r(B) j max i ( P n j=1 jb ij j 2 ) 1 2 and c(B) j max j ( P m i=1 jb ij j 2 ) 1 2 . By the above lemma we hav... |

907 | Matrix Perturbation Theory - Stewart, Sun - 1990 |

100 |
Condition numbers and equilibration matrices,
- Sluis
- 1969
(Show Context)
Citation Context ... + 1)(4 n + 6n \Gamma 1)=6: The practical outcome of this analysis is that we now have an O(n 2 ) condition estimator for the R factor of the QR factorization. By a well known result of van der Sluis =-=[12]-=-,s2 (sR) will be nearly optimal when the rows ofsR are equilibrated. Thus the procedure is to choose D in R = D R so that the rows of R are equilibrated, and use a condition estimator, see for example... |

66 | A survey of condition number estimation for triangular matrices
- Higham
- 1987
(Show Context)
Citation Context ...s2 (sR) will be nearly optimal when the rows ofsR are equilibrated. Thus the procedure is to choose D in R = D R so that the rows of R are equilibrated, and use a condition estimator, see for example =-=[3]-=-, to estimates2 (sR) in 1sR (A)s~ R (A)s~ R (A; D) = p 2 minfr(T ); c(T )g 2 ( R): (3.8) We will use another approach to estimate R c (A) in a coming paper. 4. Numerical experiments. In Section 2 we p... |

35 |
Perturbation bounds for the QR factorization of a matrix
- Stewart
- 1977
(Show Context)
Citation Context ...e orthogonal factor, and R the triangular factor. The perturbation analysis for the QR factorization has been considered by several authors. The first norm-based result for R was presented by Stewart =-=[5]-=-. That was further modified and improved by Sun [10]. Using different approaches Sun [10] and Stewart [6] gave first order normwise perturbation analyses for R. First order and strict componentwise pe... |

22 |
Perturbation bounds for the Cholesky and QR factorizations
- Sun
- 1991
(Show Context)
Citation Context ...he perturbation analysis for the QR factorization has been considered by several authors. The first norm-based result for R was presented by Stewart [5]. That was further modified and improved by Sun =-=[10]-=-. Using different approaches Sun [10] and Stewart [6] gave first order normwise perturbation analyses for R. First order and strict componentwise perturbation analyses for R have been given by Zha [13... |

22 |
Component-wise perturbation bounds for some matrix de- compositions
- Sun
- 1992
(Show Context)
Citation Context ...fferent approaches Sun [10] and Stewart [6] gave first order normwise perturbation analyses for R. First order and strict componentwise perturbation analyses for R have been given by Zha [13] and Sun =-=[11]-=-, respectively. A derivation of this first order normwise perturbation result for R follows. Theorem 1.1. [10]. Let A 2 R m\Thetan be of full column rank, with the QR factorizationsA = Q 1 R, and let ... |

18 |
On the perturbation of LU, Cholesky, and QR factorizations
- STEWART
- 1993
(Show Context)
Citation Context ...been considered by several authors. The first norm-based result for R was presented by Stewart [5]. That was further modified and improved by Sun [10]. Using different approaches Sun [10] and Stewart =-=[6]-=- gave first order normwise perturbation analyses for R. First order and strict componentwise perturbation analyses for R have been given by Zha [13] and Sun [11], respectively. A derivation of this fi... |

15 |
On the perturbation of LU and Cholesky factors
- Stewart
- 1997
(Show Context)
Citation Context ... and the size is bounded for fix n when the standard column pivoting is used, whilesS (A) can be arbitrarily large. All these results are being rewritten as a paper using the "up"-notation o=-=f Stewart [8]-=-, see also Chang, Paige and Stewart [1]. Appendix: A. A bound for kW \Gamma1 R ZR k F when the standard column pivoting is used. If we use the standard pivoting strategy in computing the QR factorizat... |

14 |
A componentwise perturbation analysis of the QR decomposition
- Zha
- 1993
(Show Context)
Citation Context ...10]. Using different approaches Sun [10] and Stewart [6] gave first order normwise perturbation analyses for R. First order and strict componentwise perturbation analyses for R have been given by Zha =-=[13]-=- and Sun [11], respectively. A derivation of this first order normwise perturbation result for R follows. Theorem 1.1. [10]. Let A 2 R m\Thetan be of full column rank, with the QR factorizationsA = Q ... |

10 | The triangular matrices of gaussian elimination and related decompositions
- Stewart
- 1995
(Show Context)
Citation Context ...tween the lower and upper bounds on R (A)) and the QR factorization is computed with the standard pivoting, the ill-conditioning of A will usually reveal itself in the diagonal elements of R. Stewart =-=[7]-=- has shown that such upper triangular matrices are artificially illconditioned in the sense that they can be made well-conditioned by scaling the rows via D. Observing (3.4) and (3.5), we can expect r... |

9 | New perturbation analyses for the Cholesky factorization
- Chang, Paige, et al.
- 1996
(Show Context)
Citation Context ...und. We can support this mathematically. In fact, if we take D = diag(r ii ), then d 1sd 2s. . .sd n when the standard column pivoting is used, so that r(T ); c(T )sn+1 2 . It is known that (see e.g. =-=[1]-=-)s2 (sR)sp n(n + 1)(4 n + 6n \Gamma 1)=18. Thus we have the following bound ~ R (A; D)s(n + 1) q n(n + 1)(4 n + 6n \Gamma 1)=6: The practical outcome of this analysis is that we now have an O(n 2 ) co... |