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Structure Preserving Algorithms for Computing the Symplectic Singular Value Decom Position

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Structure Preserving Algorithms for Computing the Symplectic Singular Value Decom Position Western Michigan University ScholarWorks at WMU Dissertations Graduate College 4-2005 Structure Preserving Algorithms for Computing the Symplectic Singular Value Decom Position Archara Chaiyakarn Western Michigan University Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Applied Mathematics Commons, and the Mathematics Commons Recommended Citation Chaiyakarn, Archara, "Structure Preserving Algorithms for Computing the Symplectic Singular Value Decom Position" (2005). Dissertations. 1022. https://scholarworks.wmich.edu/dissertations/1022 This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. STRUCTURE PRESERVING ALGORITHMS FOR COMPUTING THE SYMPLECTIC SINGULAR VALUE DECOMPOSITION by Archara Chaiyakarn A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Department of Mathematics Western Michigan University Kalamazoo, Michigan April 2005 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. STRUCTURE PRESERVING ALGORITHMS FOR COMPUTING THE SYMPLECTIC SINGULAR VALUE DECOMPOSITION Archara Chaiyakarn, Ph.D. Western Michigan University, 2005 In this thesis we develop two types of structure preserving Jacobi algo­ rithms for computing the symplectic singular value decomposition of real symplec­ tic matrices and complex symplectic matrices. Unlike general purpose algorithms, these algorithms produce symplectic structure in all factors of the singular value decomposition. Our first algorithm uses the relation between the singular value decomposi­ tion and the polar decomposition to reduce the problem of finding the symplectic singular value decomposition to that of calculating the structured spectral de­ composition of a doubly structured matrix. A Jacobi-like method is developed to compute this doubly structured spectral decomposition. The second algorithm is a one-sided Jacobi method that directly com­ putes the structured singular value decomposition of real or complex symyplectic matrices. Numerical experiments show that our algorithms converge quadratically. Furthermore, the number of sweeps needed for convergence is favorable when compared to Jacobi-like algorithms for other structured matrices. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3164162 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3164162 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. © 2005 Archara Chaiyakarn Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I am greatly indebted to my advisor, Dr. Niloufer Mackey for her valuable time, for her continuous support, for her guidance throughout the process of the completion of this thesis. Her wide knowledge and her logical way of thinking have been a great value for me. Besides being an excellent advisor, she is as close as a relative and a good friend to me. I would like to thank all the members of my committee — Dr. Niloufer Mackey, Dr. Dennis D. Pence, Dr. Clifton E. Ealy Jr., and Dr. Hongguo Xu for providing me with valuable comments and suggestions on early versions of this thesis. I owe thanks to Dr. Niloufer Mackey, Dr. Annegret Paul, and Dr. Clement Burns for letting me use their computers to run numerical experiments. I also want to thank systems analyst Mr. John D. Tucker for his valuable assistance in run­ ning unattended jobs on the Sun network. Thanks also go to all my friends in the Department of Mathematics, par­ ticularly Dr. Kirsty Eisenhart, Dr. Theresa Grant, and Daniela Hernandez whose boundless support and humor have helped me through many of the most difficult days. Thanks to the faculty and staff at the Department of Mathematics for their devotion to their students. I also want to thank my parents, who taught me the value of hard work by their own example. I would like to share this moment of happiness with my ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments—Continued father, mother, sisters and brother. They gave me enormous support during my education. Finally, I would like to give my very special thanks to Pariwat Pacheenbu- rawana, for his encouragement and understanding. Archara Chaiyakarn 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Acknowledgments ii List of Tables vi List of Figures viii N o ta tio n x 1 Introduction 1 1.1 Symplectic G roups .................................................................................... 3 1.1.1 Symplectic Unitary Groups ...................................................... 5 1.2 Symplectic Singular Value Decomposition ........................................... 6 2 Structured Tools 9 2.1 The Classical Jacobi Method ................................................................... 9 2.2 Real Symplectic Plane Rotations ......................................................... 14 2.2.1 Givens-like Action ....................................................................... 16 2.2.2 Jacobi-like Action ........................................................................... 19 2.3 Real Symplectic Double Rotations ......................................................... 21 2.3.1 Direct Sum E m bedding ................................................................. 21 2.3.2 Concentric Embedding ................................................... 23 2.4 Real 4 x 4 Symplectic Orthogonals ..................................................... 27 2.5 Real Symplectic Householders ................................................................ 29 3 Real Symplectic Bird Form 32 3.1 Via Symplectic Householders ................................................................... 33 3.2 Via 4 x 4 Symplectic G iv e n s ................................................................... 37 Computing the Real Symplectic SVD: Using the Polar Decom­ position 41 4.1 4 x 4 C a s e .................................................................................................... 42 4.2 Sweep Designs for 2 n x 2 n C a s e ................................................... 50 4.2.1 Strategy 2>i .................................................................................... 51 IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2.2 Strategy S2 ...................................................................................... 55 4.2.3 Strategy S3 ...................................................................................... 57 4.2.4 Strategy <S4 ...................................................................................... 59 4.2.5 Strategy <S5 .......................................................... 61 4.2.6 Strategy <S6 ...................................................................................... 63 4.3 Description of the Algorithms ................................ 66 4.4 Numerical Experiments ............................................................................. 71 4.4.1 Structured Spectral Decomposition ........................................ 72 4.4.2 Structured SVD ............................................................................. 85 5 Computing the Real Symplectic SVD: One-sided Jacobi 94 5.1 General One-sided Jacobi M ethod .......................................................... 94 5.2 Adaptation to Real Symplectic M atrices ........................................... 96 5.3 Numerical Experiments ............................................................................. 102 6 Computing the Complex Symplectic SVD 110 6.1 Using the Polar Decomposition ...................... 110 6.1.1 Structured Spectral Decomposition of Complex Symplectic Hermitian M atrices ....................................................................... I l l 6.1.2 Numerical Experiments ................................................................ 124 6.2 One-sided Jacobi Method ...................................................................... 132 6.2.1 Numerical Experiments ................................................................ 140 7 Concluding Remarks 146 Bibliography 149 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables 4.1 Average number of sweeps needed for 30
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