不精確迭代法求解特徵值問題｜國立清華大學博碩士論文庫

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研究生：	劉青松 Liu, Ching Sung
論文名稱：	不精確迭代法求解特徵值問題 Inexact Iterative Methods For Solving Eigenvalue Problems
指導教授：	林文偉 Lin, Wen-Wei 何南國 Ho, Nan-Kuo
口試委員:	黃聰明吳宗芳郭岳承吳金典
學位類別：	博士 Doctor
系所名稱：	理學院 - 數學系 Department of Mathematics
論文出版年：	2012
畢業學年度：	100
語文別：	英文
論文頁數：	103
中文關鍵詞：	阿洛迪、不精確迭代法、保結構、克雷洛夫子空間、擾動、非負矩陣
外文關鍵詞：	Arnoldi, Inexact iterations, Structure preserving, Krylov subspace, Perturbation, Nonnegative matrix
相關次數：	點閱：2 下載：0
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第一章針對求解大型稀疏矩陣的特徵值問題所使用的不精確Arnoldi法跟Residual Arnoldi法，提出擾動及誤差分析，並說明它們的不同處。在數值實驗上，發現即便擾動很大時，估計的特徵值與真正的特徵值的誤差卻很小，這是跟傳統的擾動分析最大的不同。

第二章當矩陣為Hermitian或者skew-Hermitian時，提出一個保結構的不精確Arnoldi法來求解大型稀疏的特徵值問題；並結合上個章節所提出的擾動分析，提出一個保結構的向後誤差分析。數值結果發現，保結構的不精確Arnoldi法較不保結構的Arnoldi法更為精確。

第三章提出一個不精確的迭代法求解M矩陣的最小特徵值及相對應的特徵向量，然後證明不精確迭代法為全局線性或超線性收斂。在數值實驗上發現不精確迭代法較原本的迭代法提升了大約3倍的效率。

Recent studies on the numerical methods for solving large eigenvalue problems have shown the Arnoldi-like methods can tolerate various types of errors during computation. One of them is the inexact Arnoldi method and the other one is
the residual Arnoldi method. Both methods allow large errors with opposite allowable error patterns. Classical perturbation theorems that use first order approximation are not suitable in the analysis of those methods. In Chapter 1, we develop a perturbation
theorem for eigensystems, which makes no assumption on the error size, and use it to analyze the perturbations of both methods.

In Chapter 2, we study the Inexact Structure-Preserving Arnoldi Methods (ISPAM) for solving Hermitian and skew-Hermitian eigenvalue problems, by which the solutions can preserve the desirable numerical properties as those by the exact methods. The difference between ISPAM and IAM is in the approximation extraction stage, where ISPAM uses the structured Rayleigh quotients that preserve the structures of the original matrices. We provide the formulation for their backward errors, which are also Hermitian and skew-Hermitian, and analyze their allowable inexactness based on the residual gap hypothesis. The solutions obtained by ISPAM can be as accurate as those computed by IAM, under the same allowable error condition.

In Chapter 3, we present an inexact inverse iteration method to find the minimal eigenvalue and the associated eigenvector of an irreducible $M$-matrix. We propose two different relaxation strategies for solving the
linear system of inner iterations. For the convergence of these two iterations, we show they are globally linear and superlinear, respectively.

Abstract ii

Acknowledgements iv

List of Tables ix

List of Figures x

Chapter 1 On the perturbation of inexact Arnoldi-like methods for solving large eigenvalue problems 1
1 Introduction 1
2 Backgrounds 5
2.1 Inexact Arnoldi method 6
2.2 Residual Arnoldi method 8
3 Perturbation theory of eigenvalue problem 12
3.1 Bounds for eigenvectors 13
3.2 Bounds for eigenvalues 14
4 Perturbation analysis of the inexact Arnoldi method 16
4.1 Perturbation of the target eigenpair 17
4.2 Perturbation of non-targeted eigenpairs 19
5 Perturbation analysis of the residual Arnoldi method 20
6 Numerical experiments 22
6.1 Inexact Arnoldi method 23
6.1.1 Results of Example 1 24
6.1.2 Results of Example 2 26
6.2 Residual Arnoldi method 27
6.3 Effect of larger errors 30
6.3.1 Inexact Arnoldi method 30
6.3.2 Residual Arnoldi method 32
7 Conclusion 32

Chapter 2 On the Inexact Structure-Preserving Arnoldi Methods for Hermitian and Skew-Hermitian Eigenvalue Problems 34
1 Introduction 34
2 Inexact structure-preserving Arnoldi methods 37
2.1 Hermitian matrix 37
2.2 Skew-Hermitian matrix 39
2.3 Eigenpair approximations 41
3 Backward error analysis 42
3.1 Hermitian matrix 43
3.2 Skew-Hermitian matrix 47
4 Residual gap of ISPAM 48
4.1 Residual gap of IAM 48
4.2 Residual gap of ISPAM 51
4.2.1 Spectral properties of Rayleigh quotients for Hermitian and skew-Hermitian matrices 51
4.2.2 Residual gap of the type 1 ISPAM for Hermitian matrices 53
4.2.3 Tighter bound on the residual gap 57
4.2.4 Residual gap of type 2, type 3 ISPAM for Hermitian
matrices 61
4.2.5 Residual gap of ISPAM for skew-Hermitian matrices 62
5 Numerical experiments 63
5.1 Synthetic matrices 64
5.2 Problems from DIMACS10 66
6 Conclusion and future work 69

Chapter 3 An inexact inverse iteration for computing the smallest eigenvalue of an irreducible M-matrix 70
1 Introduction 70
2 Preliminaries and Notation 73
2.1 Bounds for eigenvectors 74
2.2 The Noda iteration 75
3 The inexact Noda iteration and convergence theory 76
3.1 The inexact Noda iteration 77
3.2 Convergence Analysis 82
3.3 Convergence Rates 86
4 Computing the smallest eigenpair of an M-matrix 87
5 Computing a good initial vector for IRQI 89
6 Numerical experiments 92
6.1 INI for Nonnegative matrix 92
6.2 INI for M-matrix 97
7 Conclusions 98

Bibliography 100
                                

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