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Volume 25, Issue 5
Condition Number for Weighted Linear Least Squares Problem

Yimin Wei, Huaian Diao & Sanzheng Qiao

J. Comp. Math., 25 (2007), pp. 561-572.

Published online: 2007-10

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  • Abstract

In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem: $\min_x \|Ax-b\|_2$, where $A$ is an $m$-by-$n$ ($m \ge n$) rank deficient matrix. We first derive an explicit expression for the condition number in the weighted Frobenius norm $\|\left[AT, \beta b\right] \|_F$ of the data $A$ and $b$, where $T$ is a positive diagonal matrix and $\beta$ is a positive scalar. We then discuss the sensitivity of the standard 2-norm condition numbers for the generalized matrix inversion and rank deficient least squares and establish relations between the condition numbers and their condition numbers called level-2 condition numbers.

  • Keywords

Moore-Penrose inverse, Condition number, Linear least squares.

  • AMS Subject Headings

15A12, 65F20.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-25-561, author = {Yimin Wei, Huaian Diao and Sanzheng Qiao}, title = {Condition Number for Weighted Linear Least Squares Problem}, journal = {Journal of Computational Mathematics}, year = {2007}, volume = {25}, number = {5}, pages = {561--572}, abstract = {

In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem: $\min_x \|Ax-b\|_2$, where $A$ is an $m$-by-$n$ ($m \ge n$) rank deficient matrix. We first derive an explicit expression for the condition number in the weighted Frobenius norm $\|\left[AT, \beta b\right] \|_F$ of the data $A$ and $b$, where $T$ is a positive diagonal matrix and $\beta$ is a positive scalar. We then discuss the sensitivity of the standard 2-norm condition numbers for the generalized matrix inversion and rank deficient least squares and establish relations between the condition numbers and their condition numbers called level-2 condition numbers.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8713.html} }
TY - JOUR T1 - Condition Number for Weighted Linear Least Squares Problem AU - Yimin Wei, Huaian Diao & Sanzheng Qiao JO - Journal of Computational Mathematics VL - 5 SP - 561 EP - 572 PY - 2007 DA - 2007/10 SN - 25 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8713.html KW - Moore-Penrose inverse, Condition number, Linear least squares. AB -

In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem: $\min_x \|Ax-b\|_2$, where $A$ is an $m$-by-$n$ ($m \ge n$) rank deficient matrix. We first derive an explicit expression for the condition number in the weighted Frobenius norm $\|\left[AT, \beta b\right] \|_F$ of the data $A$ and $b$, where $T$ is a positive diagonal matrix and $\beta$ is a positive scalar. We then discuss the sensitivity of the standard 2-norm condition numbers for the generalized matrix inversion and rank deficient least squares and establish relations between the condition numbers and their condition numbers called level-2 condition numbers.

Yimin Wei, Huaian Diao and Sanzheng Qiao. (2007). Condition Number for Weighted Linear Least Squares Problem. Journal of Computational Mathematics. 25 (5). 561-572. doi:
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