Mathematical Software

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.

List of Contributors

Preface

Acknowledgments

Part One Prologue

Chapter 1. Historical Notes

I. Introduction

II. Chronological Record

References

Chapter 2. The Distribution and Sources of Mathematical Software

I. Introduction

II. Local Distribution Methods

III. Assessment of General Sources

IV. Summary

Chapter 3. The Challenge for Mathematical Software

I. Introduction

II. Algorithm Construction

III. Evaluation—Charting the Unknown

IV. Dissemination—Some Alternatives

V. Two Recommendations

References

Chapter 4. Discussion of the Papers

I. The User's Voice

II. Arithmetic

III. Libraries

IV. The Automation of Numerical Analysis

V. Comparative Evaluation

VI. Systems for Mathematical Software

VII. Nonnumerical Software

VIII. Mathematical Procedures

Part Two Proceedings of the Symposium

Chapter 5. The Papers

5.1 A User's Experience with Sophisticated Least-Squares Software in the Discovery of the Lunar Mass Concentrations (Mascons)

I. Nature of the Data Reduction

II. Implication for Program Development and Distribution

III. Summary of Conclusions

Reference

5.2 User Modifiable Software

I. The Argument for Easy-to-Modify Software

II. Writing Easy-to-Modify Software

5.3 Number Representation and Significance Monitoring

I. Number Representation

II. Error Classification

III. Significance Analysis

IV. Significance Monitoring

V. Mathematical Software

References

5.4 The Estimation of Significance

I. Introduction

II. Discussion of Rules

III. Implementation

IV. Elementary Functions

V. Numerical Experiments

References

5.5 Nonstandard Arithmetic

I. Reliability

II. Subroutine Library

III. Efficiency in Execution

IV. Ease of Use

V. Implementation of Nonstandard Arithmetic

VI. Use of Precompiler

VII. Type Other

VIII. Conclusion

References

5.6 Making Special Arithmetics Available

References

5.7 The Production of Mathematical Software for a Mass Audience

I. Introduction

II. Discussion Assumptions

III. Problems in Mathematical Software Production

IV. Environmental Conditions Affecting Mathematical Software Production

V. Production of Mathematical Software

VI. User Attitudes

VII. Summary

5.8 High Quality Portable Numerical Mathematics Software

I. Introduction

II. The Bell Laboratories Numerical Mathematics Program Library One

III. Status of Library One

IV. ZERBND

V. Portability

VI. Testing

References

5.9 The Development and Maintenance of a Technical Subprogram Library

I. Introduction

II. Coding Standards

III. Documentation Format

IV. Review Procedures

V. Maintenance Procedures

VI. Multiple Precision in Fortran

VII. Support and Maintenance Requirements

VIII. Access Procedures

IX. Summary and Conclusions

X. Current Category Index

XI. Sample Documentation

5.10 The Boeing Library and Handbook of Mathematical Routines

Reference

Appendix

5.11 Software for the Elementary Functions

I. Introduction

II. Preliminaries

III. Primary Routines

IV. Secondary Routines

V. Management Routines

VI. Summary

Appendix

References

5.12 Mathematical Function Subprograms for Basic System Libraries—Objectives, Constraints, and Trade-Off

I. Objectives

II. Choice of Programming Language

III. Systems Specifications

IV. Standard Reference for Accuracy

V. Effect of an Argument Error

VI. Errors Due to Straight Coding

VII. Techniques of Reducing Generated Errors

VIII. Two Levels of Accuracy Objectives

References

5.13 On Writing an Automatic Integration Algorithm

References

5.14 Experience and Problems with the Software for the Automatic Solution of Ordinary Differential Equations

I. Introduction

II. The Implementation and Use of Automatic Packages

III. Problems and Some Solutions

IV. Future Plans for the Package

References

5.15 Comparison of Numerical Quadrature Formulas

I. Introduction

II. Quadrature Codes

III. Functions

IV. Tests

V. Summary

VI. Conclusion

VII. Multiple Integrals

VIII. Appendixes

References

5.16 Evaluation of NAPSS Expressions Involving Polyalgorithms, Functions, Recursion, and Untyped Variables

I. Introduction

II. Types of Expressions

III. Basic Construction of the Interpreter

IV. Normal Arithmetic Expressions with Nonrecursive Operands

V. Evaluation Arithmetic Expression with Recursive Operands

VI. Evaluation of Arithmetic Expressions Involving Symbolic Functions

VII. Evaluation of Arithmetic Expressions with Polyalgorithm Calls

References

5.17 Toward Computer-Aided Production of Software for Mathematical Programming

I. Introduction and General Problem

II. Matrix Calculi for Mathematical Programming

III. Ranges and Range Manipulation

IV. Some Language Design and Implementation Problems

References

5.18 Software for Nonnumerical Mathematics

I. Introduction

II. Formula Manipulation

III. Theorem Proving

IV. Pure Mathematics

V. Tools for Developing Nonnumerical Mathematics Software

VI. Future Developments—A Scientific Assistant

References

5.19 Continuous Distribution Sampling: Accuracy and Speed

I. Introduction

II. Conditional Bit Sampling

III. Computation of Conditional Probabilities

IV. Pseudonormal Number Generator

V. Discussion

References

5.20 Applications of Singular Value Analysis

I. Introduction

II. Definition of the Singular Value Decomposition

III. Singular Value Analysis of Systems of Linear Equations

IV. An Example of Singular Value Analysis

V. Algorithms and Subroutines

VI. Experience in Using Singular Value Analysis

VII. Conclusions

References

5.21 Numerical Implementation of Variational Methods for Eigenvalue Problems

I. Introduction

II. A Sequence of Related Problems

III. Problems in Writing the Matrices Generation Routine

IV. Perturbations in (Ax-λBx)

V. A Lower Bound on the Smallest Eigenvalue of B

VI. The Compatible Quadrature Order

VII. Invariance and Optimality of the Compatible Order

VIII. Observations on the Sharpness of the Perturbation Bounds

IX. Quadrature Schemes with Basis Elements as Weights

X. Methods which Utilize the Structure of the Algebraic Problem

XI. The Selection of Program Arguments Governing Error

XII. Cost and Accuracy

References

5.22 Taylor Series Methods for Ordinary Differential Equations—An Evaluation

I. Introduction

II. The Taylor Series Method

III. Numerical Techniques

IV. Implementation

V. Results

VI. Conclusions

References

5.23 A New Algorithm for Nonlinear Least-Squares Curve Fitting

I. Introduction and Description of the Method

II. Convergence Results

III. Numerical Results

References

Part Three Selected Mathematical Software

Chapter 6. Self-Contained Power Routines

I. Introduction

II. A Fortran Program

III. An Assembler Language Program

Reference

Chapter 7. CADRE: An Algorithm for Numerical Quadrature

I. Introduction

II. Mathematical Analysis

III. Numerical Procedures

IV. Fortran Listing of CADRE

V. Testing and Examples

References

Chapter 8. SQUARS: An Algorithm for Least-Squares Approximation

I. Introduction

II. Mathematical Analysis

III. Numerical Procedures

IV. The Algorithm SQUARS

V. Example Program, Testing, and Evaluation

References

Chapter 9. DESUB: Integration of a First-Order System of Ordinary Differential Equations

I. Program Purpose and Use

II. Method

III. History

IV. Adaptation of the Program

V. Testing and Results

VI. Example Problems

VII. Organization of the Program

VIII. DESUB

References

Index

Life Sciences

Physical Sciences & Engineering

Social Sciences & Humanities

Health

Mathematical Software

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