Iterative Methods for Non-Hermitian Positive Semi-Definite Systems
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Iterative Methods for Non-Hermitian Positive Semi-Definite Systems

Iterative Methods for Non-Hermitian Positive Semi-Definite Systems


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This dissertation, "Iterative Methods for Non-hermitian Positive Semi-definite Systems" by Man-Kiu, Ho, 何文翹, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of thesis entitled ITERATIVE METHODS FOR NON-HERMITIAN POSITIVE SEMI-DEFINITE SYSTEMS submitted by Ho Man-Kiu for the degree of Master of Philosophy at The University of Hong Kong in July 2004 In this thesis, two types of non-Hermitian structured systems, namely two-by- twoblocksystemsandcirculant-plus-diagonalsystems, areconsideredandstudied. These two types of systems can be found in many parts of the world: two-by- two block systems can usually be found in saddle point problems of optimization and partial differential equations, while circulant-plus-diagonal systems appear in solving integral equations. In some special cases, the two systems cannot be solved efficiently, especially when the Hermitian part of the matrix is positive semi-definite, or even indefinite. Hermitian and skew-Hermitian splitting (HSS) method is applied to solve two- by-two block systems. The coefficient matrix can be easily split into its Hermitian and skew-Hermitian parts. Convergence analysis of the HSS method is studied. The main result is that under some full rank conditions, the convergence of the HSS method can be shown by using the singular value decomposition. Precon- ditioning techniques based on Hermitian and skew-Hermitian preconditioners can also be applied to solve such systems. The main result is that the eigenvalues of these preconditioned matrices are mainly clustered around zero and two, when the parameter is chosen to be sufficiently small. Experimental results about the clustering behaviour of the eigenvalues distribution are illustrated using Oseen equations. For circulant-plus-diagonal systems, the normal and skew-Hermitian splitting(NSS) method is studied. By making use of the NSS method, the circulant struc- tureofthecoefficientmatrixiskeptsothatthetwohalf-stepsinthemethodcanbe easily implemented. The convergent condition of the NSS method is investigated. ThemainresultisthatNSSmethodisalwaysconvergentwhenthecirculantmatrix is positive definite, or when it is positive semi-definite but possesses only one zero real part eigenvalue. Block successive over-relaxation (SOR) method is considered to speed up the iteration for solving circulant-plus-diagonal systems. Numerical results are presented to illustrate the effectiveness of the proposed methods. DOI: 10.5353/th_b3028940 Subjects: Iterative methods (Mathematics) Eigenvectors Matrices


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Product Details
  • ISBN-13: 9781374722309
  • Publisher: Open Dissertation Press
  • Publisher Imprint: Open Dissertation Press
  • Height: 279 mm
  • No of Pages: 64
  • Weight: 172 gr
  • ISBN-10: 1374722308
  • Publisher Date: 27 Jan 2017
  • Binding: Paperback
  • Language: English
  • Spine Width: 3 mm
  • Width: 216 mm


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