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論文名稱 Title |
自伴隨常對角矩陣的特徵值分布探討 A study of eigenvalue distribution of Hermitian Toeplitz matrix |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
34 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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口試日期 Date of Exam |
2019-06-27 |
繳交日期 Date of Submission |
2019-07-04 |
關鍵字 Keywords |
特徵值分布、Hermitian Toeplitz矩陣、Toeplitz矩陣、收斂速度、共軛梯度法 convergence rate, conjugate gradient method, eigenvalue distribution, Hermitian Toeplitz matrix, Toeplitz matrix |
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統計 Statistics |
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中文摘要 |
Hermitian Toeplitz矩陣源自很多實際上的應用,比如說用有限差分法解微分方 程。在這篇文章,我們利用共軛梯度法來解線性系統Ax = b,其中A是Hermitian Toeplitz矩陣,藉此找出它的收斂速度。基本上,共軛梯度法的收斂速度是被矩 陣A的條件數所決定的。我們會表示出Hermitian Toeplitz矩陣的特徵值的上下限。 在數值結果上,我們注意到當Hermitian Toeplitz矩陣越來越大的時候,它的特徵值 分布會逼近這個矩陣的生成函數。 |
Abstract |
Hermitian Toeplitz matrix arises from many real applications, such as solving differential equations by using finite difference method. In this paper, we consider the convergence rate of solving linear system Ax = b with the Hermitian Toeplitz matrix by using conjugate gradient method. The convergence rate of conjugate gradient method basically depends on the condition number on the matrix. We demonstrate the upper bound and lower bound of eigenvalues of Hermitian Toeplitz matrix. In the mathematical experiments, we notice that when the size of Hermitian Toeplitz matrix becomes larger, the distribution of the eigenvalues seems approach the associated distribution function. |
目次 Table of Contents |
Contents 論文審定書i 摘要iii Abstract iv 1 Introduction 1 2 Preliminaries 3 2.1 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Toeplitz matrix generated by a function f(x) . . . . . . . . . . . . . . . . 5 2.3 Properties of Toeplitz matrix Tn(f) . . . . . . . . . . . . . . . . . . . . . 8 2.4 Conjugate Gradient method . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Eigenvalue distribution of Toeplitz matrix 13 4 Solving linear system Ax = b with Toeplitz matrix 15 4.1 Strang circulant matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Symmetric positive definite matrix . . . . . . . . . . . . . . . . . . . . . 16 4.3 Finding the locally best step length . . . . . . . . . . . . . . . . . . . . . 17 5 Appendix 19 5.1 Two-sided Preconditioning method . . . . . . . . . . . . . . . . . . . . . 19 5.2 Finite-Difference method . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6 Results and Experiments 22 6.1 Example 1 in section 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6.2 Example 2 in section 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.3 The eigenvalue distribution of f(x) = x4 + 1 . . . . . . . . . . . . . . . 24 6.4 The eigenvalue distribution of f(x) = x4 + x2 + 1 . . . . . . . . . . . . 25 6.5 The convergence rate of the example in Appendix . . . . . . . . . . . . . 26 |
參考文獻 References |
[1] G. Strang, A proposal for Toeplitz Matrix Calculations, Studies 1986. [2] Ivan Oseledets, Eugene Tyrtyshnikov, A unifying approach to the construction of circulant preconditioners, Linear Algebra and its Applications Volume 418, Issues 2–3, 15 October 2006, Pages 435-449 [3] Silvia Noschese, Lothar Reichel, Generalized circulant Strang-type preconditioners, Volume 19, Issue 1 Special Issue: Structured Matrices and Tensors, January 2012, Pages 3-17 [4] A. C. R. Newbery, Pei matrix eigenvectors, Magazine Communications of the ACM CACM Homepage archive Volume 6 Issue 9, Sept. 1963 Page 515 [5] P. Tilli, Locally Toeplitz sequences: spectral properties and applications, Linear Algebra and its applications, Volume 278, Issues 1–3, 15 July 1998, Pages 91-120 27 |
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