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Effective condition number and its applications

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Abstract

Consider the over-determined system Fx = b where \({{\bf F}\in\mathcal{R}^{m \times n}, m \geq n}\) and rank (F) = r ≤ n, the effective condition number is defined by \({{\rm Cond_{-}eff }= \frac {\|{\bf b}\|}{\sigma_r\|{\bf x}\|}}\), 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+ΔA)(x+Δx) = b+Δb involving both ΔA and Δb, the new error bounds pertinent to Cond_eff are derived. Next, we apply the effective condition number to the solutions of Motz’s problem by the collocation Trefftz methods (CTM). Motz’s problem is the benchmark of singularity problems. We choose the general particular solutions \({v_L = \sum\nolimits_{k=0}^L d_k (\frac {r}{R_p})^{k+\frac 12}}\) \({{\rm cos}(k +\frac 12)\theta}\) with a radius parameter R p . The CTM is used to seek the coefficients d i by satisfying the boundary conditions only. Based on the new effective condition number, the optimal parameter R p = 1 is found. which is completely in accordance with the numerical results. However, if based on the traditional condition number Cond, the optimal choice of R p is misleading. Under the optimal choice R p = 1, the Cond grows exponentially as L increases, but Cond_eff is only linear. The smaller effective condition number explains well the very accurate solutions obtained. The error analysis in [14,15] and the stability analysis in this paper grant the CTM to become the most efficient and competent boundary method.

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Authors and Affiliations

  1. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

    Zi-Cai Li

  2. Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

    Zi-Cai Li

  3. Department of Applied Mathematics, Chung Hua University, Hsin-Chu, Taiwan

    Zi-Cai Li

  4. Department of Applied Mathematics, I-Shou University, Kaohsiung, 840, Taiwan

    Hung-Tsai Huang

  5. Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, 80424, Taiwan

    Jeng-Tzong Chen

  6. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Yimin Wei

  7. Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Ministry of Education, Shanghai, People’s Republic of China

    Yimin Wei

Authors
  1. Zi-Cai Li
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  2. Hung-Tsai Huang
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  3. Jeng-Tzong Chen
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  4. Yimin Wei
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Corresponding author

Correspondence to Yimin Wei.

Additional information

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).

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Li, ZC., Huang, HT., Chen, JT. et al. Effective condition number and its applications. Computing 89, 87–112 (2010). https://doi.org/10.1007/s00607-010-0098-8

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  • DOI: https://doi.org/10.1007/s00607-010-0098-8

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