Computing (2010) 89:87–112 DOI 10.1007/s00607-010-0098-8 Effective condition number and its applications Zi-Cai Li · Hung-Tsai Huang · Jeng-Tzong Chen · Yimin Wei Received: 18 March 2010 / Accepted: 27 May 2010 / Published online: 22 June 2010 © Springer-Verlag 2010 Abstract Consider the over-determined system Fx = b where F ∈ R m×n , m ≥ n and rank (F) = r ≤ n, the effective condition number is defined by Cond_eff = b σ r x , where the singular values of F are given as σ max = σ 1 ≥ σ 2 ≥ ··· ≥ σ r > 0 and σ r +1 =···= σ n = 0. For the general perturbed system (A + A)(x + x) = b + b Communicated by N. Yan. Partial results of this paper were represented at the Mini symposium on Collocation and Trefftz Method for the 7th World Congress on Computational Mechanics, Los Angeles, California, July 16–22, 2006. Y. Wei was supported by the National Natural Science Foundation of China under grant 10871051, Doctoral Program of the Ministry of Education under grant 20090071110003, Shanghai Science and Technology Committee under grant 08511501703, Shanghai Municipal Education Committee (Dawn Project) and 973 Program Project (No. 2010CB327900). Z.-C. Li Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan e-mail: zcli@math.nsysu.edu.tw Z.-C. Li Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan Z.-C. Li Department of Applied Mathematics, Chung Hua University, Hsin-Chu, Taiwan H.-T. Huang Department of Applied Mathematics, I-Shou University, Kaohsiung 840, Taiwan e-mail: huanght@isu.edu.tw J.-T. Chen Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 80424, Taiwan e-mail: jtchen@mail.ntou.edu.tw 123