對於線性系統 Ax=b，傳統的條件數針對所有的向量b，因此是最糟糕的情形，並在許多問題常會被高估。對某一特定的向量b，有效條件數給予了x 的相對誤差更好的上界，但是仍有可能發生有效條件數高估的情形。在這篇論文中，我們研究x 的相對誤差與b 的相對擾動的真實比值，稱做真實條件數。我們得到真實條件數的數個新上界及估計，並且探究藉由平移b 使線性系統轉變成一等價系統後，能將有效條件數最小化。最後我們把這些結果應用到函數近似的問題上。

For linear system Ax = b, the traditional condition number is the worst case for all
b’s and often overestimated in many problems. For a specific b, the effective condition
number is a better upper bound for the relative error of x. But, it is also possible
that this effective condition number is overestimated. In this thesis, we study the true
ratio of the relative error of x to the relative perturbation of b, called the true condition
number. We obtain several new upper bounds and estimates for true condition
number. We also explore to change the system to an equivalent one by shifting b to
minimize its effective condition number. Finally we apply all our results to functional
approximation.

1 Introduction 1
1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 True Condition Number for Linear System 8
2.1 True condition number . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 New condition numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Approximate ∥x∥ by ∥˜x∥ . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Equivalent linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 Shifted vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 True Condition Number for Functional Approximation 29
3.1 Approximation methods . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Orthogonal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Polynomial function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Bibliography 48

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