Mathematical Software
- 1st Edition - May 27, 2014
- Editor: John R. Rice
- Language: English
- eBook ISBN:9 7 8 - 1 - 4 8 3 2 - 6 7 0 0 - 5
Mathematical Software deals with software designed for mathematical applications such as Fortran, CADRE, SQUARS, and DESUB. The distribution and sources of mathematical software… Read more

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Request a sales quoteMathematical 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
- No. of pages: 538
- Language: English
- Edition: 1
- Published: May 27, 2014
- Imprint: Academic Press
- eBook ISBN: 9781483267005
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