Mathematical Software

Mathematical Software

1st Edition - January 1, 1971

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  • Editor: John R. Rice
  • eBook ISBN: 9781483267005

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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.

Table of Contents

  • List of Contributors



    Part One Prologue

    Chapter 1. Historical Notes

    I. Introduction

    II. Chronological Record


    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


    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


    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


    5.4 The Estimation of Significance

    I. Introduction

    II. Discussion of Rules

    III. Implementation

    IV. Elementary Functions

    V. Numerical Experiments


    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


    5.6 Making Special Arithmetics Available


    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


    V. Portability

    VI. Testing


    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



    5.11 Software for the Elementary Functions

    I. Introduction

    II. Preliminaries

    III. Primary Routines

    IV. Secondary Routines

    V. Management Routines

    VI. Summary



    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


    5.13 On Writing an Automatic Integration Algorithm


    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


    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


    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


    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


    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


    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


    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


    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


    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


    5.23 A New Algorithm for Nonlinear Least-Squares Curve Fitting

    I. Introduction and Description of the Method

    II. Convergence Results

    III. Numerical Results


    Part Three Selected Mathematical Software

    Chapter 6. Self-Contained Power Routines

    I. Introduction

    II. A Fortran Program

    III. An Assembler Language Program


    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


    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


    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




Product details

  • No. of pages: 538
  • Language: English
  • Copyright: © Academic Press 1971
  • Published: January 1, 1971
  • Imprint: Academic Press
  • eBook ISBN: 9781483267005

About the Editor

John R. Rice

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