Rice Mathematical Software

1. Auflage 2014
ISBN: 978-1-4832-6700-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 538 Seiten, Web PDF

Mathematical Software deals with software designed for mathematical applications such as Fortran, CADRE, SQUARS, and DESUB. The distribution and sources of mathematical software are discussed, along with number representation and significance monitoring. User-modifiable software and non-standard arithmetic programs are also considered. Comprised of nine chapters, this volume begins with a historical background in the form of a chronological list of events that trace the development of computing in general and mathematical software in particular. The next chapter examines where and how mathematical software is being created and how it is being disseminated to eventual consumers. A number of important shortcomings are identified. The future of mathematical software and the challenges facing mathematical software are then discussed. Subsequent chapters focus on the point of view of people outside the professional community of mathematical software; the monitoring of significance in computation and its relation to number representation; libraries of mathematical software; and the automation of numerical analysis. Eleven algorithms for numerical quadrature are also compared. This book 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;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;16
6;Acknowledgments;18
7;PART ONE: PROLOGUE;22
7.1;Chapter 1. Historical Notes;24
7.1.1;I. Introduction;24
7.1.2;II. Chronological Record;24
7.1.3;References;32
7.2;Chapter 2. The Distribution and Sources of Mathematical Software;34
7.2.1;I. Introduction;34
7.2.2;II. Local Distribution Methods;35
7.2.3;III. Assessment of General Sources;40
7.2.4;IV. Summary;45
7.3;Chapter 3. The Challenge for Mathematical Software;48
7.3.1;I. Introduction;48
7.3.2;II. Algorithm Construction;49
7.3.3;III. Evaluation—Charting the Unknown;54
7.3.4;IV. Dissemination—Some Alternatives;58
7.3.5;V. Two Recommendation;61
7.3.6;References;62
7.4;Chapter 4. Discussion of the Papers;64
7.4.1;I. The User's Voice;64
7.4.2;II. Arithmetic;65
7.4.3;III. Libraries;65
7.4.4;IV. The Automation of Numerical Analysis;66
7.4.5;V. Comparative Evaluation;66
7.4.6;VI. Systems for Mathematical Software;67
7.4.7;VII. Nonnumerical Software;67
7.4.8;VIII. Mathematical Procedures;67
8;PART TWO: PROCEEDINGS OF THE SYMPOSIUM;70
8.1;Chapter 5. The Papers;72
8.1.1;5.1 A USER'S EXPERIENCE WITH SOPHISTICATED LEAST-SQUARES SOFTWARE IN THE DISCOVERY OF THE LUNAR MASS CONCENTRATIONS(MASCONS);72
8.1.1.1;I. Nature of the Data Reduction;72
8.1.1.2;II. Implication for Program Development and Distribution;76
8.1.1.3;III. Summary of Conclusions;78
8.1.1.4;Reference;78
8.1.2;5.2 USER MODIFIABLE SOFTWARE;80
8.1.2.1;I. The Argument for Easy-to-Modify Software;80
8.1.2.2;II. Writing Easy-to-Modify Software;83
8.1.3;5.3 NUMBER REPRESENTATION AND SIGNIFICANCE MONITORING;88
8.1.3.1;I. Number Representation;89
8.1.3.2;II. Error Classification;94
8.1.3.3;III. Significance Analysis;97
8.1.3.4;IV. Significance Monitoring;108
8.1.3.5;V. Mathematical Software;110
8.1.3.6;References;112
8.1.4;5.4 THE ESTIMATION OF SIGNIFICANCE;114
8.1.4.1;I. Introduction;114
8.1.4.2;II. Discussion of Rules;115
8.1.4.3;III. Implementation;117
8.1.4.4;IV. Elementary Functions;121
8.1.4.5;V. Numerical Experiments;123
8.1.4.6;References;125
8.1.5;5.5 NONSTANDARD ARITHMETIC;126
8.1.5.1;I. Reliability;126
8.1.5.2;II. Subroutine Library;126
8.1.5.3;III. Efficiency in Execution;127
8.1.5.4;IV. Ease of Use;127
8.1.5.5;V. Implementation of Nonstandard Arithmetic;128
8.1.5.6;VI. Use of Precompiler;129
8.1.5.7;VII. Type Other;131
8.1.5.8;VIII. Conclusion;131
8.1.5.9;References;132
8.1.6;5.6 MAKING SPECIAL ARITHMETICS AVAILABLE;134
8.1.6.1;References;140
8.1.7;5.7 THE PRODUCTION OF MATHEMATICAL SOFTWARE FOR A MASS AUDIENCE;142
8.1.7.1;I. Introduction;142
8.1.7.2;II. Discussion Assumptions;143
8.1.7.3;III. Problems in Mathematical Software Production;143
8.1.7.4;IV. Environmental Conditions Affecting Mathematical Software Production;148
8.1.7.5;V. Production of Mathematical Software;149
8.1.7.6;VI. User Attitudes;151
8.1.7.7;VII. Summary;151
8.1.8;5.8 HIGH QUALITY PORTABLE NUMERICAL MATHEMATICS SOFTWARE;152
8.1.8.1;I. Introduction;152
8.1.8.2;II. The Bell Laboratories Numerical Mathematics Program Library One;152
8.1.8.3;III. Status of Library One;153
8.1.8.4;IV. ZERBND;156
8.1.8.5;V. Portability;156
8.1.8.6;VI. Testing;157
8.1.8.7;References;159
8.1.9;5.9 THE DEVELOPMENT AND MAINTENANCE OF A TECHNICAL SUBPROGRAM LIBRARY;162
8.1.9.1;I. Introduction;162
8.1.9.2;II. Coding Standards;163
8.1.9.3;III. Documentation Format;164
8.1.9.4;IV. Review Procedures;166
8.1.9.5;V. Maintenance Procedures;167
8.1.9.6;VI. Multiple Precision in Fortran;167
8.1.9.7;VII. Support and Maintenance Requirements;167
8.1.9.8;VIII. Access Procedures;168
8.1.9.9;IX. Summary and Conclusions;168
8.1.9.10;X. Current Category Index;169
8.1.9.11;XI. Sample Documentation;169
8.1.10;5.10 THE BOEING LIBRARY AND HANDBOOK OF MATHEMATICAL ROUTINES;174
8.1.10.1;Reference;176
8.1.10.2;Appendix;177
8.1.11;5.11 SOFTWARE FOR THE ELEMENTARY FUNCTIONS;192
8.1.11.1;1. Introduction;192
8.1.11.2;II. Preliminaries;193
8.1.11.3;III. Primary Routines;195
8.1.11.4;IV. Secondary Routines;199
8.1.11.5;V. Management Routines;203
8.1.11.6;VI. Summary;205
8.1.11.7;Appendix;205
8.1.11.8;References;206
8.1.12;5.12 MATHEMATICAL FUNCTION SUBPROGRAMS FOR BASIC SYSTEM LIBRARIES—OBJECTIVES, CONSTRAINTS, AND TRADE-OFF;208
8.1.12.1;I. Objectives;208
8.1.12.2;II. Choice of Programming Language;210
8.1.12.3;III. Systems Specifications;212
8.1.12.4;IV. Standard Reference for Accuracy;214
8.1.12.5;V. Effect of an Argument Error;216
8.1.12.6;VI. Errors Due to Straight Coding;216
8.1.12.7;VII. Techniques of Reducing Generated Errors;217
8.1.12.8;VIII. Two Levels of Accuracy Objectives;218
8.1.12.9;References;220
8.1.13;5.13 ON WRITING AN AUTOMATIC INTEGRATION ALGORITHM;222
8.1.13.1;References;230
8.1.14;5.14 EXPERIENCE AND PROBLEMS WITH THE SOFTWARE FOR THE AUTOMATIC SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS;232
8.1.14.1;I. Introduction;232
8.1.14.2;II. The Implementation and Use of Automatic Packages;236
8.1.14.3;III. Problems and Some Solutions;240
8.1.14.4;IV. Future Plans for the Package;246
8.1.14.5;References;247
8.1.15;5.15 COMPARISON OF NUMERICAL QUADRATURE FORMULAS;250
8.1.15.1;I. Introduction;250
8.1.15.2;II. Quadrature Codes;251
8.1.15.3;III. Functions;252
8.1.15.4;IV. Tests;252
8.1.15.5;V. Summary;253
8.1.15.6;VI. Conclusion;254
8.1.15.7;VII. Multiple Integrals;254
8.1.15.8;VIII. Appendixes;254
8.1.15.9;References;280
8.1.16;5.16 EVALUATION OF NAPSS EXPRESSIONS INVOLVING POLYALGORITHMS,FUNCTIONS, RECURSION, AND UNTYPED VARIABLES;282
8.1.16.1;I. Introduction;282
8.1.16.2;II. Types of Expressions;282
8.1.16.3;III. Basic Construction of the Interpreter;285
8.1.16.4;IV. Normal Arithmetic Expressions with Nonrecursive Operands;286
8.1.16.5;V. Evaluation Arithmetic Expression with Recursive Operands;289
8.1.16.6;VI. Evaluation of Arithmetic Expressions Involving Symbolic Functions;291
8.1.16.7;VII. Evaluation of Arithmetic Expressions with Polyalgorithm Calls;293
8.1.16.8;References;295
8.1.17;5.17 TOWARD COMPUTER-AIDED PRODUCTION OF SOFTWARE FOR MATHEMATICAL PROGRAMMING;296
8.1.17.1;I. Introduction and General Problem;296
8.1.17.2;II. Matrix Calculi for Mathematical Programming;297
8.1.17.3;III. Ranges and Range Manipulation;300
8.1.17.4;IV. Some Language Design and Implementation Problems;304
8.1.17.5;References;314
8.1.18;5.18 SOFTWARE FOR NONNUMERICAL MATHEMATICS;316
8.1.18.1;I. Introduction;316
8.1.18.2;II. Formula Manipulation;318
8.1.18.3;III. Theorem Proving;329
8.1.18.4;IV. Pure Mathematics;335
8.1.18.5;V. Tools for Developing Nonnumerical Mathematics Software;339
8.1.18.6;VI. Future Developments—A Scientific Assistant;342
8.1.18.7;References;345
8.1.19;5.19 CONTINUOUS DISTRIBUTION SAMPLING: ACCURACY AND SPEED;352
8.1.19.1;I. Introduction;352
8.1.19.2;II. Conditional Bit Sampling;354
8.1.19.3;III. Computation of Conditional Probabilities;356
8.1.19.4;IV. Pseudonormal Number Generator;358
8.1.19.5;V. Discussion;360
8.1.19.6;References;365
8.1.20;5.20 APPLICATIONS OF SINGULAR VALUE ANALYSIS;368
8.1.20.1;I. Introduction;368
8.1.20.2;II. Definition of the Singular Value Decomposition;369
8.1.20.3;III. Singular Value Analysis of Systems of Linear Equations;369
8.1.20.4;IV. An Example of Singular Value Analysis;371
8.1.20.5;V. Algorithms and Subroutines;374
8.1.20.6;VI. Experience in Using Singular Value Analysis;375
8.1.20.7;VII. Conclusions;376
8.1.20.8;References;377
8.1.21;5.21 NUMERICAL IMPLEMENTATION OF VARIATIONAL METHODS FOR EIGENVALUE PROBLEMS;378
8.1.21.1;I. Introduction;378
8.1.21.2;II. A Sequence of Related Problems;379
8.1.21.3;III. Problems in Writing the Matrices Generation Routine;380
8.1.21.4;IV. Perturbations in (Ax-.Bx);380
8.1.21.5;V. A Lower Bound on the Smallest Eigenvalue of B;382
8.1.21.6;VI. The Compatible Quadrature Order;383
8.1.21.7;VII. Invariance and Optimality of the Compatible Order;384
8.1.21.8;VIII. Observations on the Sharpness of the Perturbation Bounds;385
8.1.21.9;IX. Quadrature Schemes with Basis Elements as Weights;385
8.1.21.10;X. Methods which Utilize the Structure of the Algebraic Problem;386
8.1.21.11;XI. The Selection of Program Arguments Governing Error;387
8.1.21.12;XII. Cost and Accuracy;388
8.1.21.13;References;389
8.1.22;5.22 TAYLOR SERIES METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS—AN EVALUATION;390
8.1.22.1;I. Introduction;390
8.1.22.2;II. The Taylor Series Method;391
8.1.22.3;III. Numerical Techniques;394
8.1.22.4;IV. Implementation;397
8.1.22.5;V. Results;399
8.1.22.6;VI. Conclusions;409
8.1.22.7;References;410
8.1.23;5.23 A NEW ALGORITHM FOR NONLINEAR LEAST-SQUARES CURVE FITTING;412
8.1.23.1;I. Introduction and Description of the Method;412
8.1.23.2;II. Convergence Results;414
8.1.23.3;III. Numerical Results;415
8.1.23.4;References;417
9;PART THREE: SELECTED MATHEMATICAL SOFTWARE;418
9.1;Chapter 6. Self-Contained Power Routines;420
9.1.1;I. Introduction;420
9.1.2;II. A Fortran Program;421
9.1.3;III. An Assembler Language Program;427
9.1.4;Reference;436
9.2;Chapter 7. CADRE: An Algorithm for Numerical Quadrature;438
9.2.1;1. Introduction;438
9.2.2;II. Mathematical Analysis;438
9.2.3;III. Numerical Procedures;446
9.2.4;IV. Fortran Listing of CADRE;451
9.2.5;V. Testing and Examples;459
9.2.6;References;470
9.3;Chapter 8. SQUARS: An Algorithm for Least-Squares Approximation;472
9.3.1;I. Introduction;472
9.3.2;II. Mathematical Analysis;474
9.3.3;III. Numerical Procedures;477
9.3.4;IV. The Algorithm SQUARS;480
9.3.5;V. Example Program, Testing, and Evaluation;493
9.3.6;References;497
9.4;Chapter 9. DESUB: Integration of a First-Order System of Ordinary Differential Equations;498
9.4.1;I. Program Purpose and Use;498
9.4.2;II. Method;501
9.4.3;III. History;504
9.4.4;IV. Adaptation of the Program;504
9.4.5;V. Testing and Results;507
9.4.6;VI. Example Problems;507
9.4.7;VII. Organization of the Program;509
9.4.8;VIII. DESUB;514
9.4.9;References;528
10;Index;530