Rice Mathematical Software

Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin-Madison, March 28-30, 1977
1. Auflage 2014
ISBN: 978-1-4832-6714-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 398 Seiten, Web PDF

Mathematical Software III contains the proceedings of the Symposium on Mathematical Software held in Madison, Wisconsin, on March 28-30, 1977, under the auspices of the Mathematics Research Center at the University of Wisconsin-Madison. The papers focus on software designed for mathematical applications such as LINPACK for the solution of linear systems and least squares problems and ELLPACK for elliptic partial differential equations. Comprised of 14 chapters, this volume begins with an overview of LINPACK, a software package designed to solve linear systems and least squares problems. The reader is then introduced to an extension to the exchange algorithm for solving overdetermined linear equations; infallible calculation of polynomial zeros to specified precision; and representation and approximation of surfaces. Subsequent chapters discuss the ways in which mathematical software and exploratory data analysis should interact to satisfy their respective needs; production of mathematical software; computational aspects of the finite element method; and multi-level adaptive techniques for partial differential equations. The book also describes a realistic model of floating-point computation before concluding with an evaluation of the Block Lanczos method for computing a few of the least or greatest eigenvalues of a sparse symmetric matrix. This monograph should be of considerable interest to students and specialists in the fields of mathematics and computer science.

Autoren/Hrsg.

Rice, John R.

Inhaltsverzeichnis

1;Front Cover;1
2;Mathematical Software III;4
3;Copyright Page;5
4;Table of Contents;6
5;Contributors;8
6;Preface;10
7;Chapter 1. Research, Development, and LINPACK;12
7.1;ABSTRACT;12
7.2;1. INTRODUCTION;12
7.3;2. ESTIMATION OF CONDITION NUMBERS;13
7.4;3. STABILITY OF LEAST SQUARES SOLUTIONS;16
7.5;4. SCALING AND COLUMN ELIMINATION;18
7.6;5. DOWNDATING;20
7.7;6. TIMING THE BLAS;22
7.8;REFERENCES;25
8;Chapter 2. A Technique that Gains Speed and Accuracy in the Minimax Solution of Overdetermined Linear Equations;26
8.1;ABSTRACT;26
8.2;1. INTRODUCTION;26
8.3;2. NUMERICAL INSTABILITY IN THE EXCHANGE ALGORITHM;29
8.4;3. THE NEW TECHNIQUE;31
8.5;4. THEORY;36
8.6;5. DISCUSSION;39
8.7;REFERENCES;43
9;Chapter 3. Infallible Calculation of Polynomial Zeros to Specified Precision;46
9.1;ABSTRACT;46
9.2;1. INTRODUCTION;47
9.3;2. PRELIMINARIES;50
9.4;3. STURM SEQUENCES FOR REAL ZEROS;56
9.5;4. STURM SEQUENCES FOR COMPLEX ZEROS;60
9.6;5. ROLLE'S THEOREM FOR REAL ZEROS;63
9.7;6. DESCARTES' THEOREM FOR REAL ZEROS;66
9.8;7. APPLICATION OF INTERVAL ARITHMETIC;70
9.9;8. COMPLEX ZEROS WITHOUT STURM SEQUENCES;74
9.10;REFERENCES;77
10;Chapter 4. Representation and Approximation of Surfaces;80
10.1;ABSTRACT;80
10.2;1. INTRODUCTION;81
10.3;2. INTERPOLATION METHODS DEFINED OVER RECTANGLES;83
10.4;3. INTERPOLATION SCHEMES DEFINED OVER TRIANGLES;93
10.5;4. INTERPOLATION METHODS FOR ARBITRARILY PLACED DATA;121
10.6;5. CONCLUSIONS;129
10.7;REFERENCES;129
10.8;ACKNOWLEDGEMENTS;131
11;Chapter 5. Simulation: Conflicts between Real-Time and Software;132
11.1;ABSTRACT;132
11.2;1. INTRODUCTION;133
11.3;2. SIMULATION;134
11.4;3. REAL-TIME OPERATION;136
11.5;4. NUMERICAL INTEGRATION IN REAL-TIME;137
11.6;5. ERRORS IN NUMERICAL INTEGRATION;140
11.7;6. METHODS FOR REDUCING DELAY AND IMPROVING STABILITY;142
11.8;7. CONCLUSIONS;148
11.9;REFERENCES;149
12;Chapter 6. Mathematical Software and Exploratory Data Analysis;150
12.1;ABSTRACT;150
12.2;1. INTRODUCTION;150
12.3;2. OVERVIEW OF EXPLORATORY DATA ANALYSIS;151
12.4;3. AN EXAMPLE: TIMING DATA;153
12.5;4. SOFTWARE NEEDS OF EXPLORATORY DATA ANALYSIS;165
12.6;5. SUMMARY;168
12.7;REFERENCES;168
13;Chapter 7.
Software for C1 Surface Interpolation;172
13.1;1. INTRODUCTION;172
13.2;2. PROBLEM STATEMENT;173
13.3;3. EXPECTED APPLICATIONS;173
13.4;4. PUBLISHED WORK ON SURFACE INTERPOLATION TO IRREGULARLY LOCATED DATA;174
13.5;5. OUTLINE OF THE ALGORITHMIC APPROACH SELECTED;175
13.6;6. CONSTRUCTING A TRIANGULAR GRID;176
13.7;7. ESTIMATING PARTIAL DERIVATIVES AT THE GRID NODES;181
13.8;8. LOOKUP IN THE TRIANGULAR GRID;182
13.9;9. INTERPOLATION IN A TRIANGLE;182
13.10;10. EXAMPLES;183
13.11;11. THREE CRITERIA FOR TRIANGULATION OF A STRICTLY CONVEX QUADRILATERAL;187
13.12;12. GLOBAL CONSEQUENCES OF THE LOCAL OPTIMIZATION PROCEDURE;193
13.13;13. McLAIN'S TRIANGULATION METHOD;198
13.14;14. LIMITS ON GRID CHANGES WHEN ADDING A NEW POINT;199
13.15;15. CONCLUSIONS;201
13.16;REFERENCES;202
14;Chapter 8. Mathematical Software Production;206
14.1;ABSTRACT;206
14.2;I. Introduction;207
14.3;II. The Evolution of Mathematical Software Production;209
14.4;III. Intellectual Challenges;214
14.5;IV. Projects to Produce Mathematical Software;217
14.6;V. Trends in Mathematical Software Production;228
14.7;REFERENCES;231
15;Chapter 9. Computational Aspects of the Finite Element Method;236
15.1;1. INTRODUCTION;236
15.2;2. GOALS OF THE COMPUTATIONAL ANALYSIS;238
15.3;3. THE PRINCIPAL STAGES OF THE COMPUTATIONAL ANALYSIS;240
15.4;4. SOME SOFTWARE ASPECTS;256
15.5;5. SOME COMPUTATIONAL RESULTS;259
15.6;REFERENCES;264
16;Chapter 10. The Art of Writing a Runge-Kutta Code, Part I;268
16.1;1. INTRODUCTION;268
16.2;2. RUNGE-KUTTA METHODS;269
16.3;3. MEASURES OF QUALITY;273
16.4;4. SUMMARY;284
16.5;REFERENCES;284
17;Chapter 11. Multi-Level Adaptive Techniques (MLAT) for Partial Differential Equations: Ideas and Software;288
17.1;ABSTRACT;288
17.2;INTRODUCTION;288
17.3;1. SURVEY OF MULTI-GRID PROCESSES ON UNIFORM GRIDS;292
17.4;2. NON-UNIFORM GRIDS: ORGANIZATION AND MULTI-GRID PROCESSING;304
17.5;3. SURVEY OF ADAPTATION TECHNIQUES;313
17.6;4. DATA STRUCTURE AND SOFTWARE;318
17.7;REFERENCES;329
18;Chapter 12. ELLPACK: A Research Tool for Elliptic Partial Differential Equations Software;330
18.1;ABSTRACT;330
18.2;1. BACKGROUND;330
18.3;2. THE STRUCTURE OF ELLPACK;334
18.4;3. ELLPACK 77 USER INPUT;338
18.5;4. ELLPACK 77 ORGANIZATION AND INFORMATION;340
18.6;5. ELLPACK 78;346
18.7;6. ELLPACK IMPLEMENTATION AT PURDUE;349
18.8;REFERENCES;350
19;Chapter 13. A Realistic Model of Floating-Point Computation;354
19.1;Abstract;354
19.2;1. Introduction;354
19.3;2. Environment Parameters;355
19.4;3. Properties of Arithmetic;356
19.5;4. Machine Anomalies;359
19.6;5. Error Analysis;360
19.7;6. Arithmetic Comparisons;366
19.8;7. Overflow and Underflow (the Bêtes Noires of Portability);368
19.9;8. Acknowledgments;371
19.10;References;371
20;Chapter 14. The Block Lanczos Method for Computing Eigenvalues;372
20.1;ABSTRACT;372
20.2;1. INTRODUCTION;372
20.3;2. A BLOCK LANCZOS ALGORITHM;373
20.4;3. IMPLEMENTATION;382
20.5;4. EXAMPLES;384
20.6;5. EXTENSIONS;386
20.7;REFERENCES;387
21;Index;390