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NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS HYPERCUBE SOLUTIONS FOR CONJUGATE DIRECTIONS by Jonathan Edward Hartman December, 1991 Thesis Co-Advisor: William B. Gragg Thesis Co-Advisor: Uno R. Kodres Approved for public release; distribution is unlimited. T257844 UNCLASSIFIED juRITY CLASSIFICATION OF THIS_PAGF_ REPORT DOCUMENTATION PAGE REPORT SECURITY CLASSIFICATION lb RESTRICTIVE MARKINGS J UNCLASSIFIED £! SECURITY CLASSIFICATION AUTHORITY 3 DISTRIBUTION/AVAILABILITY OF REPORT DECLASSIFICATION/DOWNGRADING SCHEDULE Approved for public release; distribution is unlimited. ^PERFORMING ORGANIZATION REPORT NUMBER(S) 5 MONITORING ORGANIZATION REPORT NUMBER(S) I NAME OF PERFORMING ORGANIZATION 6b OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION Computer Science Department (ifapplicable) Naval Postgraduate School cs Naval Postgraduate School I ADDRESS (City, State, and ZIP Code) 7b ADDRESS (City, State, and ZIP Code) Monterey, CA 93943 Monterey, CA 93943 I I NAME OF FUNDING/SPONSORING 8b OFFICE SYMBOL 9 PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER ORGANIZATION (ifapplicable) ADDRESS (City, State, and ZIP Code) 10 SOURCE OF FUNDING NUMBERS PROGRAM PROJECT TASK WORK UNIT ELEMENT NO. NO. NO. ACCESSION NO. TITLE (Include Security Classification) HYPERCUBE SOLUTIONS FOR CONJUGATE DIRECTIONS PERSONAL AUTHOR(S) Hartman. Jonathan Edward TYPE OF REPORT 13b TIME COVERED 14 DATE OF REPORT (Year, Month, Day) 15. PAGE COUNT Master's Thesis FROM 06/89 TO 12/91 December 1991 380 SUPPLEMENTARY NOTATION The views expressed in this thesis are those of the author and do not reflect the iTicial policy or position of the Department of Defense or the United States Government. COSATI CODES 18. SUBJECT TERMS (Continue on reverse ifnecessary and identify by block number) Conjugate Gradients, Gaussian Elimination, Gauss Factorization, Hyper- FIELD GROUP SUB-GROUP cube, iPSC/2, Matrices, Multiprocessors, Transputers, Communicating Sequential Processes, Message Passing, Distributed Memory, MIMD. I ABSTRACT (Continue on reverse ifnecessary and identify by block number) As computing machines advance, new fields are explored and old ones are expanded. This thesis considers irallel solutions to several well-known problems from numerical linear algebra, including Gauss Factorization id the method of Conjugate Gradients. The Gauss algorithm was implemented on two parallel machines: an — — itel iPSC/2, and a network of INMOS T-800 transputers. Interprocessor communication in both cases was orne by a hypercube interconnection topology. The results reveal general findings from parallel computing and more specific data and information con- ;rning the systems and algorithms that were employed. Communication is timed and the results are analyzed, lowing typical features of a message passing system. System performance is illustrated by results from the auss codes. The use of two different pivoting strategies shows the potential and the limitations of a parallel iachine. The iPSC/2 and transputer systems both show excellent parallel performance when solving large, snse, unstructured systems. Differences, advantages, and disadvantages of these two systems are examined and cpectations for current and future machines are discussed. D. DISTRIBUTION/AVAILABILITY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATION 3 UNCLASSIFIED/UNLIMITED DsAME AS RPT DTIC USERS UNCLASSIFIED >a. NAME OF RESPONSIBLE INDIVIDUAL 22b. TELEPHONE (Include Area Code) 22c. OFFICE SYMBOL Uno R. Kodres (408) 646-2197 CSKr FORM 1473, 84 MAR 83 APR edition may be used until exhausted SECURITY CLASSIFICATION OF THIS PAGE All other editions are obsolete UNCLASSIFIED Approved for public release; distribution is unlimited. HYPERCUBE SOLUTIONS for CONJUGATE DIRECTIONS by Jonathan Edward Hartman Captain. United States Marine Corps B.S.. United States Naval Academy, 1984 Submitted in partial fulfillment of the requirements for the degrees of MASTER OF SCIENCE IN COMPUTER SCIENCE MASTER OF SCIENCE IN APPLIED MATHEMATICS from tIk- naval POSTGRADUATE SCHOOL December, 1991 ABSTRACT As computing machines advance, new fields are explored and old ones are expanded. This thesis considers parallel solutions to several well-known problems from numerical linear algebra, including Gauss Factorization and the method of Conjugate Gradients. The Gauss algorithm was implemented on two parallel machines: an Intel iPSC/2, and a network of INMOS T-800 transputers. Interprocessor — — communication in both cases was borne by a hypercube interconnection topology. The results reveal general findings from parallel computing and more specific data and information concerning the systems and algorithms that were employed. Communication is timed and the results are analyzed, showing typical features of a message passing system. System performance is illustrated by results from the Gauss codes. The use of two different pivoting strategies shows the potential and the limitations of a parallel machine. The iPSC/2 and transputer systems both show excellent parallel performance when solving large, dense, unstructured systems. Differences, advantages, and disadvantages of these two systems are examined and expectations for current and future machines are discussed. 111 tttftt. d.l THESIS DISCLAIMER The computer programs developed in this research have not been exercised for all cases of interest. Every reasonable effort has been made to eliminate computa- tional and logical errors, but the programs should not be considered fully verified. Any application of these programs without additional verification is at the user's risk. A reasonable effort has been put forth to make the code efficient. Optimization has been suppressed, however, in areas where it would jeopardize the simplicity and clarity of the algorithm without great reward in terms of performance. IMS, inmos, and occam are trademarks of INMOS Limited, a member of the SGS-THOMSON Microelectronics Group. INTEL, intel, and iPSC are trademarks of Intel Corporation. IBM, PC AT, and PC XT are registered trademarks of Inter- national Business Machines Corporation. CIO, LD-ONE, LD-NET, TASM, TCX, TIO, TLIB, and TLNK are trademarks of Logical Systems. MS-DOS is a trademark of Microsoft Corporation. MATLAB is a trademark of The MathWorks, Inc. All other brand and product names are trademarks or registered trademarks of their respective companies. IV