Testing eigenvalue software. Item Type text; Dissertation-Reproduction (electronic) Authors Henderson, Lehman Edwin, Jr. Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 03/10/2021 22:22:26 Link to Item http://hdl.handle.net/10150/185744 INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. 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 UMI 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. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each o'riginal is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. U·M·I University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor. M148106-1346 USA 313/761-4700 800/521-0600 Order Number 9213693 Testing eigenvalue· software Henderson, Lehman Edwin, Jr., Ph.D. The University of Arizona, 1991 Copyright ®1991 by Henderson, Lehman Edwin, Jr. All rights reserved. U·M·I 300 N. Zeeb Rd. Ann Arbor, MI48106 Testing Eigenvalue Software by Lehman Edwin Henderson, Jr. Copyright © 1991, Lehman Edwin Henderson, Jr. A Dissertation Submitted to the Faculty of the Department of Computer Science in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate College of THE UNIVERSITY OF ARIZONA 199 1 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Final Examination Committee, we certify that we have read the dissertation prepared by Lehman Edwin Henderson, Jr. entitled _________T~E~S~T~I~N~G~EI~G~E~NV~AL~UE~S~O~F~TW~AR~E~ ______________________ and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy ID/31/'11 Date lolso/cr.1 Date Date' I Date Date Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. rd D. Schlichting Date Webb Miller 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under the rules of the Library. Brief quotations from this dissertation are allowed without special permission, provided that accurate acknowledgement of source is made. Requests for permis­ sion for extended quotation from, or reproduction of this manuscript in whole or in part may be granted by the copyright holder. I wish to acknowledge the help of my parents, Lehman and Vivian Henderson, for their early and continued support and encouragement of my interest in the sciences. ·J TABLE OF CONTENTS LIST OF FIGURES . .. 10 LIST OF TABLES. .. 11 ABSTRACT ...... .. 13 CHAPTER 1: Introduction 1-1 1.1 What I Have Done 1-1 1.2 What It Is Based Upon . 1.5 1.3 Other Related Work Ii 1.4 My Contribution 19 1.5 Conclusions ... 21 1.6 Organization of This Dissertation 2:3 CHAPTER 2: Definitions and Relevant Theorems. 2.5 2.1 Reduction to Tridiagonal Form :32 2.2 Plane Rotations . :33 6 CHAPTER 3: Testing Methods Reported in the Literature. 3·5 3.1 The Handbook ..... :3i 3.2 EISPACK test matrices. 60 3.3 Other Recent Papers .. 61 CHAPTER 4: The QR Algorithm. 69 4.1 Rotations ... iO 4.2 Why QL Instead of QR? 80 4.3 Generic QL Algorithm, Non-optimized 81 4.4 Choice of Shift ......... 8.5 4.5 Explicitly Shifted QL Algorithm. 8i 4.6 Optimized QL Algorithm. 88 4.7 Convergence Criteria 9.5 4.8 Error of Result ... 96 4.9 Programs That Were Tested 96 CHAPTER 5: Measuring How Well a Program Performs 102 5.1 Nearness: Vector and Matrix Norms. 10:3 5.2 Instability score . 106 5.3 An Estimate of w lOi CHAPTER 6: Random Testing 11-1 6.1 Introduction...... 115 7 11.j 6.2 Statement of the Problem . " 6.3 Experimental Method. 116 6.4 Results......... 118 CHAPTER 7: Hill-Climbing Software 129 7.1 The Compiler ........ 1:31 7.2 Straight-Line Code Generator 138 7.3 Maximizer.......... 1:39 7.4 Operation of the Maximizer 140 CHAPTER 8: Results from Hill Climbing . 142 8.1 Comparing Hill Climbing to Random Testing. 143 8.2 Will a Few Matrices Do? 14.j 8.3 Finding the Instability 148 8.4 Conclusion. 16:3 CHAPTER 9: Conclusions and Recommendations. 16.j 9.1 Conclusions ............. 165 9.2 Recommendations for future research 166 APPENDIX A: Data from Random Testing 169 A.1 IMTQL2. 170 A.2 IMTQLV. 170 8 A.3 OKRATV . ............... 173 AA OKWRAV ................................ Ii.j A.5 PWK ................................... 178 A.6 PWKU. 179 A.7 SKRAT ................................. 179 A.S SKRATN ............ ................... 181 A.9 SKRATW . 182 A.10 TQL1 .................................. 182 A.ll TQL1MB .............. 18.5 A.12 TQLIMA . 0 0 0 • 0 0 • 0 0 0 0 0 0 0 00 ••• 0 0 0 0 • 0 0 0 • 0 0 • 186 A.13 TQLIMB ................................ 187 A.14 TQL1NB ................................. 187 A.15 TQL1 U ................................. 188 A.16 TQL2 .................................. 190 A.17 TQLRAT ................................ 191 APPENDIX B: Data from Hill Climbing .................... 19.j Bol OKRATV 0 • 0 • 0 0 0 0 •••••• 0 • 0 ••• 0 •••• 0 ••• 0 •• 19i B.2 OKWRAV .. 0 ••••••••••••••••••••••••••••• 203 B.3 PWK ................................... 210 B.4 PWKU. 0 • • • • • • • • • • • • • • • • • 217 B.5 SKRAT ........ ............. 22-1 9 B.6 SKRATN ... 2:31 B.7 TQL1 .................................. 238 B.S TQL1MA . 245 B.9 TQL1MB ....................... ........ 2.j1 B.lO TQL1NA ................................ 258 B.11 TQL1NB ................................. 26.5 B.12 TQL1S .................................. 272 B.13 TQL1 U ........................... ...... 279 B.14 TQLRAT ................................ 286 REFERENCES .................................. 294 10 LIST OF FIGURES 2.1 Elimination of Off-Diagonal Element by Rotation . .. :34 4.1 Elements of the Matrix A ..... 84 4.2 Elements of the Matrix A during iteration 84 4.3 Two Rows During Rotation 84 4.4 Two Rows After Rotation . 85 4.5 Inner Loop of EISPACK TQLl Code 89 5.1 Typical relation of w to w 108 5.2 Linear Transform of Outputs With Respect to Inputs 110 5.3 Linear Transform of Outputs \Vith Respect to Rounding Errors 111 5.4 Geometric Interpretation of w . 112 7.1 Block Diagram of the Hill-climbing Software ............. 130 11 LIST OF TABLES 1.1 Results of testing 4 x 4 Linear Equation Solvers 17 1.2 Results of testing 4 x 4 Eigenvalue Solvers ... 20 5.1 Instabilities in Linear Equation Software Found By Hill-Climbing . 109 6.1 Maximum instability score, unmodified software 119 6.2 Comparing the Effects of Re-Orthogonalization 120 6.3 Maximum w Using Different Constants for Machine Precision. 121 6.4 Comparing the Effects of Scoring Precision . : . 123 6.5 Comparing the Effects of Scoring Precision of TQLRAT versus the Size of the Test Matrix . 12-1 6.6 Comparing the Effects of the Magnitude of the Input Data 12.5 6.7 Maximum Instability Score Varying the Size of the Test Matrix 126 6.8 Distribution of Maximum Instability Scores for OKWRAV 127 8.1 Summary of Instabilities Found by Random Testing . 1-1:3 8.2 Instabilities Found By Hill-climbing in Eigenvalue Software . 1-1-1 12 8.3 Using Selected Matrices on Unstable Routines 1-16 8.4 Using Selected Matrices on Stable Routines . 1-18 ABSTRACT This dissertation describes a significant advance in automated testing of eigenvalue software. Several programs are described that assist the re­ searcher in verifying that a new program is stable. Using backwards error techniques popularized by Wilkinson, a maximizer or "hill climber" system­ atically searches for instabilities in the program being tested. This work builds on software first reported by Miller and removes the restriction of not being able to work on iterative methods. Testing eigenvalue solver programs with sets of small random input data can often find instabilities, but the described hill climbing technique is more efficient. Using only ten sets of starting points, the maximizer will often find the instability, if it exists, in only a few tries. 1-1 CHAPTER 1 Introduction "The fundamental principle of science, the definition almost, is this: the sole test of the validity of any idea is experiment." R. P. Feynman 1.1 What I Have Done This dissertation is about testing eigenvalue software. Its high point is the de­ scription of a software package that will systematically search for input data that causes an unstable eigenvalue routine to fail.