NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2007; 14:603–610 Published online 18 June 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nla.540 On Frobenius normwise condition numbers for Moore–Penrose inverse and linear least-squares problems Huaian Diao 1 and Yimin Wei 2, 3, ∗, † 1 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, People’s Republic of China 2 School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China 3 Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, Shanghai, People’s Republic of China SUMMARY Condition numbers play an important role in numerical analysis. Classical condition numbers are norm- wise: they measure the size of both input perturbations and output errors using norms. In this paper, we give explicit, computable expressions depending on the data, for the normwise condition numbers for the computation of the Moore–Penrose inverse as well as for the solutions of linear least-squares problems with full-column rank. Copyright 2007 John Wiley & Sons, Ltd. Received 11 November 2006; Revised 15 May 2007; Accepted 20 May 2007 KEY WORDS: Moore–Penrose inverse; linear least squares; condition number; perturbation 1. INTRODUCTION Condition numbers measure the worst-case sensitivity of an input data to small perturbations. To the best of our knowledge, a general theory of condition numbers was first given by Rice [1]. Let  : R s → R t be a mapping, where R s and R t are the usual real s -dimensional and t -dimensional Euclidean spaces equipped with the norms ‖·‖ D and ‖·‖ S , respectively. If  is continuous and Fr´ echet differentiable in the neighbourhood of a 0 ∈ R s then, according to [1], the relative normwise condition number of a 0 is given by (; a 0 ) := lim →0 sup ‖a‖ D   ‖(a 0 + a) - (a 0 )‖ S ‖(a 0 )‖ S  ‖a‖ D ‖a 0 ‖ D  = ||| ′ (a 0 )||| · ‖a 0 ‖ D ‖(a 0 )‖ S (1) ∗ Correspondence to: Yimin Wei, School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China. † E-mail: ymwei@fudan.edu.cn Contract/grant sponsor: National Natural Science Foundation of China; contract/grant number: 10471027 Contract/grant sponsor: Shanghai Education Committee Copyright 2007 John Wiley & Sons, Ltd.