On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems

Authors:
Felipe Cucker, Huaian Diao and Yimin Wei

Journal:
Math. Comp. **76** (2007), 947-963

MSC (2000):
Primary 15A09, 15A12; Secondary 65F35

DOI:
https://doi.org/10.1090/S0025-5718-06-01913-2

Published electronically:
November 2, 2006

MathSciNet review:
2291844

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Abstract: Classical condition numbers are normwise: they measure the size of both input perturbations and output errors using some norms. To take into account the relative of each data component, and, in particular, a possible data sparseness, componentwise condition numbers have been increasingly considered. These are mostly of two kinds: mixed and componentwise. In this paper, we give explicit expressions, computable from the data, for the mixed and componentwise condition numbers for the computation of the Moore-Penrose inverse as well as for the computation of solutions and residues of linear least squares problems. In both cases the data matrices have full column (row) rank.

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Additional Information

**Felipe Cucker**

Affiliation:
Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, P.R. of China

Email:
macucker@math.cityu.edu.hk

**Huaian Diao**

Affiliation:
Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, P.R. of China

Email:
50007445@student.cityu.edu.hk

**Yimin Wei**

Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai 200433 and Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, P.R. of China

Email:
ymwei@fudan.edu.cn

DOI:
https://doi.org/10.1090/S0025-5718-06-01913-2

Keywords:
Condition numbers,
componentwise analysis,
least squares

Received by editor(s):
July 21, 2005

Received by editor(s) in revised form:
November 23, 2005

Published electronically:
November 2, 2006

Additional Notes:
The first author was partially supported by City University SRG grant 7001860.

The third author was partially supported by the National Natural Science Foundation of China under grant 10471027 and Shanghai Education Committee.

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.