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

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

Published online: 2007-10

Preview Full PDF 138 1823
Export citation

Cited by

• 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

15A12 65F20.

@Article{JCM-25-561, author = {}, 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 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 KW - Condition number KW - 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.