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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 劉進賢(Chein-Shan Liu) | |
| dc.contributor.author | Chia-Jou Hsu | en |
| dc.contributor.author | 徐佳柔 | zh_TW |
| dc.date.accessioned | 2021-06-16T23:37:13Z | - |
| dc.date.available | 2013-08-01 | |
| dc.date.copyright | 2012-08-01 | |
| dc.date.issued | 2012 | |
| dc.date.submitted | 2012-07-26 | |
| dc.identifier.citation | [1] Liu, C.-S.; S. N. Atluri (2011): 'An Iterative Method Using an Optimal Descent Vector, for Solving an Ill-Conditioned System Bx=b, Better and Faster than the Conjugate Gradient Method.' CMES: Computer Modeling in Engineering & Sciences 80(4): 275-298.
[2] Newton, I.; J. Colson (1736): The method of fluxions and infinite series; : with its application to the geometry of curve-lines. London, Printed by Henry Woodfall; and sold by John Nourse [3] Newton, I.; Raphson J.; Cunn.; Edmond, H. (1728): Universal arithmetick: or, A treatise of arithmetical composition and resolution. London,, Printed for J. Senex etc. [4] Raphson, J. (1702): Analysis aquationum universalis. Londini,, typis T. B. prostant venales apud A. & I. Churchill etc. [5] Davidenko, D. (1953): 'On a new method of numerical solution of systems of nonlinear equations' Doklady Akad. Nauk SSSR 88: 601-604. [6] Hirsch, M. W.; Smale, S. (1974): Differential equations, dynamical systems, and linear algebra. New York,, Academic Press. [7] Liu, C.-S.; Atluri. S. N. (2008): 'A Novel Time Integration Method for Solving A Large System of Non-Linear Algebraic Equations.' CMES: Computer Modeling in Engineering & Sciences 31: 71-84. [8] Ku, C.-Y.; Yeih, W.-C.; Liu, C.-S.; Chi, C.-C. (2009): 'Applications of the Fictitious Time Integration Method Using a New Time-Like Function.' CMES: Computer Modeling in Engineering & Sciences 43(2): 173-190. [9] Liu, C.-S.; Yeih, W.-C.; Kuo, C.-L.; Atluri. S. N. (2009): 'A Scalar Homotopy Method for Solving an Over/Under-Determined System of Non-Linear Algebraic Equations.' CMES: Computer Modeling in Engineering & Sciences 53: 47-71. [10] Ku, C.-Y., Yeih, W.-C.; Liu, C.-S. (2010): 'Solving Non-Linear Algebraic Equations by a Scalar Newton-homotopy Continuation Method.' International Journal of Nonlinear Sciences & Numerical Simulation 11: 435-450. [11] Liu, C.-S.; Atluri, S. N. (2011): 'An Iterative Algorithm for Solving a System of Nonlinear Algebraic Equations, F(x)=0, Using the System of ODEs with an Optimum a in ; ' CMES: Computer Modeling in Engineering & Sciences 73(4): 395-432. [12] Liu, C.-S.; Kuo, C.-L. (2011): 'A Dynamical Tikhonov Regularization Method for Solving Nonlinear Ill-Posed Problems.' 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(2009): 'A Fictitious Time Integration Method for the Numerical Solution of the Fredholm Integral Equation and for Numerical Differentiation of Noisy Data, and Its Relation to the Filter Theory.' CMES: Computer Modeling in Engineering & Sciences 41(3): 243-261. [17] Aubry, M.; Baues, H. J.; Stephan, H.; Lemaire, J. M. (1995): Homotopy theory and models : based on lectures held at a DMV Seminar in Blaubeuren by H.J. Baues, S. Halperin, and J.-M. Lemaire. Basel ; Boston, Birkhauser. [18] Park, C.-H.; Shim, H.-T. (2005): 'What Is The Homotopy Method For A System Of Nonlinear Equations (Survey) ?' Appl. Math. & Computing 17: 689-700. [19] Wayburns, T. L.; Seader, J. D. (1987): 'Homotopy Continuation Methods For Computer-Aided Process Design.' compur. &em. &gag 11(1): 7-25. [20] Wu, T.-M. (2005): 'A study of convergence on the Newton-homotopy continuation method.' Applied Mathematics and Computation 168: 1169-1174. [21] Wu, T.-M. (2006): 'Solving the nonlinear equations by the Newton-homotopy continuation method with adjustable auxiliary homotopy function.' Applied Mathematics and Computation 173: 383-388. [22] Lahaye, E. (1934): 'Une m'ethod de resolution d'une categorie d'equations transcendantes.' C. R. Acad. Sci. Paris 198(1840-1842) [23] Liu, C.-S. (2001): 'Cone of non-linear dynamical system and group preserving schemes.' International Journal of Non-Linear Mechanics 36(7): 1047-1068. [24] Liu, C.-S. (2004): 'Group preserving scheme for backward heat conduction problems.' International Journal of Heat and Mass Transfer 47(12): 2567-2576. [25] Rossmann, W. (2002): Lie groups: an introduction through linear groups. Oxford; New York, Oxford University Press. [26] Varadarajan, V. S. (1974): Lie groups, Lie algebras, and their representations. Englewood Cliffs, N.J., Prentice-Hall. [27] Ruhl, W. (1970): The Lorentz group and harmonic analysis. New York,, W. A. Benjamin. [28] Salvadori, M. G. (1961): Numerical methods in engineering. Englewood Cliffs, N.J., Prentice-Hall. [29] Carmeli, M. (2000): Group theory and general relativity: representations of the Lorentz group and their applications to the gravitational field. Singapore; River Edge, NJ, World Scientific. [30] Liu, C.-S. (2011): 'A Revision of Relaxed Steepest Descent Method from the Dynamics on an Invariant Manifold.' CMES: Computer Modeling in Engineering & Sciences 80(1): 57-86. [31] Hirsch, M. W.; Smale, S. (1979): 'On Algorithms for Solving f(x)=0.' Communications on Pure and Applied Mathematics 32(3): 281-312. [32] Roose, A.; Kulla, V.; Lomp, M.; Mressoo, T. (1990): 'Test examples of systems of non-linear equations.' Estonian Software Comput. Service Company (Tallin). [33] Spedicato, E.; Huang, Z.; Bergamo. (1997): 'Numerical Experience with Newton-like Methods for Nonlinear Algebraic Systems.' Computing 58: 69-89. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/65336 | - |
| dc.description.abstract | 本文結合了最速下降法、純量同倫及光錐的結構發展出一全局最佳演算法(Globally Optimal Iterative Algorithm)，由預先設定之參數(ϒ、ac) 確定其解的存在並考慮臨界情形(判別式為零)推導出一重要參數 後，分別使用臨界向量u=R+αcr及u=BTF+αcF為下降方向求解線性及非線性代數方程式問題。
本演算法於求解線性代數方程式時展現了良好的效益，其準確性甚至高於共軛梯度法(Conjugate Gradient method)及OIA/ODV; 此外，此方法用於求解非線性代數方程式時也展現了良好了成果，除了良好的效益之外，於求解達芬方程式(Duffing equation)時也有極佳的準確性。 其演算過程除了有效避免了雅可比矩陣(Jacobian matrix)之反矩陣計算，不同於牛頓法，其迭代過程不需擔心出現發散之情形，對於初始猜值的敏感度也明顯低於牛頓法。 | zh_TW |
| dc.description.abstract | It has always been of interest to solve algebraic equations used for describing physical and engineering issues. By using the concepts of the Steepest Descent method, the scalar homotopy method and the structure of light cone, we have developed a novel algorithm with preset parameters ϒ(0≤ϒ<1),ac (ac>1) and the critical parameter αc in the driving vector u=R+αcr and u=BTF+αcF as a descent direction. Due to the criticality of αc, we believe that by using this algorithm, the globally optimal solution can be obtained. It is so call the Globally Optimal Iterative Algorithm (GOIA).
The GOIA has performed both great efficiency and accuracy when it is used for solving algebraic equations. Moreover, by using the GOIA, one can successfully avoid the calculation of the inverse Jacobian matrix which is required when use the Newton’s method instead. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-16T23:37:13Z (GMT). No. of bitstreams: 1 ntu-101-R98521220-1.pdf: 9920221 bytes, checksum: 064fbd17750a0fde04e283cb1e5ccf24 (MD5) Previous issue date: 2012 | en |
| dc.description.tableofcontents | 口試委員會審定書 #
誌謝 i 中文摘要 ii ABSTRACT iii CONTENTS iv LIST OF FIGURES vi LIST OF TABLES ix Chapter 1 Introduction 1 1.1 Motivation and purpose 3 1.2 Literature review 5 1.2.1 Newton’s Method 8 1.2.2 The Fictitious Time Integrating Method 10 1.2.3 The Optimal Iterative Algorithm with an Optimal Descent Vector 14 1.3 Thesis statement 18 Chapter 2 Theory 19 2.1 Steepest Descent Method 19 2.2 Conjugate Gradient method 22 2.2.1 Gram-Schmidt process 24 2.2.2 Krylov subspace 25 2.2.3 Algorithm of the CGM 27 2.3 Homotopy Theory 28 2.3.1 Homotopy Theory 28 2.3.2 Homotopy Method 30 2.3.3 Newton-Homotopy Continuation Method 32 2.3.4 Scalar Homotopy Method 34 2.4 Light Cone 37 2.4.1 Minkowski Spacetime 37 2.4.2 Augmented Dynamic System and the Null Cone 39 2.5 Relaxed Steepest Descent Method 42 Chapter 3 The GOIA for Linear Algebraic equations 46 3.1 Derivation 46 3.2 Examples 52 3.2.1 Hilbert linear problems with n=5 54 3.2.2 Hilbert linear problems with n=50 56 3.2.3 Laplace equation 57 3.2.4 Poisson equation 59 Chapter 4 The GOIA for Nonlinear Algebraic equations 72 4.1 Derivation 72 4.2 Examples 76 4.2.1 Two variables nonlinear equations 77 4.2.2 Three variables nonlinear equations 79 4.2.3 System of nonlinear equations with n=6 81 4.2.4 Duffing oscillator. 82 Chapter 5 Conclusions and Future Work 93 5.1 Conclusions 93 5.2 Future Work 96 REFERENCE 97 | |
| dc.language.iso | en | |
| dc.title | 以臨界向量為下降方向解代數方程式之全局最佳演算法 | zh_TW |
| dc.title | The Globally Optimal Iterative Algorithm with Critical Vector as a Descent Direction to Solve Algebraic Equations | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 100-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 顧承宇(Cheng-Yu Ku),范佳銘(Chia-Ming Fan) | |
| dc.subject.keyword | 最速下降法,純量同倫,光錐,代數方程式,全局最佳演算法,牛頓法,共軛梯度法,OIA/ODV, | zh_TW |
| dc.subject.keyword | algebraic equations,Steepest Descent method,scalar homotopy method,light cone,Globally Optimal Iterative Algorithm (GOIA),Newton’s method, | en |
| dc.relation.page | 100 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2012-07-26 | |
| dc.contributor.author-college | 工學院 | zh_TW |
| dc.contributor.author-dept | 土木工程學研究所 | zh_TW |
| 顯示於系所單位： | 土木工程學系 | |
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