NDLI: Effective condition number and its applications

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Effective condition number and its applications

Content Provider	Springer Nature Link
Author	Chen, Jeng Tzong Wei, Yimin Li, Zi Cai Huang, Hung Tsai
Copyright Year	2010
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.
Ending Page	112
Page Count	26
Starting Page	87
File Format	PDF
ISSN	0010485X
e-ISSN	14365057
Journal	Computing
Issue Number	1-2
Volume Number	89
Language	English
Publisher	Springer Vienna
Publisher Date	2010-06-22
Publisher Place	Vienna
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Motz’s problem Collocation Trefftz method Computational Mathematics and Numerical Analysis Computer Science Condition number Effective condition number Stability analysis Particular solutions Singularity problem Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods Radius parameter
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Computational Theory and Mathematics Computational Mathematics Numerical Analysis Computer Science Applications Software

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