# A New Approach to Scientific Computation

## 1st Edition

**Editors:**Ulrich W. Kulisch Willard L. Miranker

**eBook ISBN:**9781483272047

**Imprint:**Academic Press

**Published Date:**28th October 1983

**Page Count:**400

## Description

A New Approach to Scientific Computation is a collection of papers delivered at a symposium held at the IBM Thomas J. Watson Research Center on August 3, 1982. The symposium provided a forum for reviewing various aspects of an approach to scientific computation based on a systematic theory of computer arithmetic. Computer demonstration packages for standard problems of numerical mathematics are considered.

Comprised of 12 chapters, this volume begins by summarizing an extensive research activity in scientific computation as well as the experience gained through various implementations of a new approach to arithmetic on diverse processors, including even microprocessors. A complete listing of the spaces that occur in numerical computations is presented, followed by a discussion of aspects of traditional computer arithmetic and a new definition of computer arithmetic. The properties of semimorphisms are also considered. Subsequent chapters focus on potential applications of programming packages to standard problems in numerical analysis implemented on a Z80 based minicomputer, with a PASCAL extension called PASCAL-SC as the programming language; methods for solving algebraic problems with high accuracy; and the use of a computer with floating-point arithmetic to obtain guaranteed sharp bounds for the value of an arithmetic expression. An extension of FORTRAN which satisfies contemporary requirements of numerical computation is also described.

This book will be helpful to students and practitioners in the fields of computer science and applied mathematics.

## Table of Contents

Contributors

Preface

Acknowledgments

A New Arithmetic for Scientific Computation

1. Introduction

2. The Spaces of Numerical Computations

3. Traditional Definition of Computer Arithmetic: The Vertical Method

4. The New Definition of Computer Arithmetic: The Horizontal Method

5. Computer Arithmetic and Programming Languages

6. Realization and Applications

References

Computer Demonstration Packages for Standard Problems of Numerical Mathematics

Language Extension PASCAL-SC

Computing Inclusions—Old and New

Precise Dot Product

Linear Systems

Inversion of a Matrix

Eigenproblems

Rounding Error and Cancellation

Evaluation of a Polynomial

Zero of a Polynomial

Polynomial Package

Arithmetic Expressions

Systems of Non-linear Equations

Differential Equations

Solving Algebraic Problems with High Accuracy

Introduction

1. Computer Arithmetic

2. Linear Systems

3. Over- and Underdetermined Linear Systems

4. Linear Systems with Band Matrices

5. Sparse Linear Systems

6. Matrix Inversion

7. Non-linear Systems

8. The Algebraic Eigenvalue Problem

9. Real and Complex Zeros of Polynomials

10. Linear, Quadratic, and Convex Programming

11. Arithmetic Expressions

Conclusions

References

Evaluation of Arithmetic Expressions with Maximum Accuracy

Introduction

1. Evaluation of Polynomials

2. Evaluation of Arbitrary Arithmetic Expressions

3. Numerical Results

References

Solving Function Space Problems with Guaranteed Close Bounds

1. Introduction

2. Mathematical Preliminaries

3. Practical Use of the Fixed Point Theorems

4. Functional Arithmetic and Roundings

5. Algorithmic Execution of Iterations

6. Applications to Differential and Integral Equations

7. Some Examples

References

Ultra-Arithmetic: The Digital Computer Set in Function Space

1. Introduction

2. A Review of Ultra-Arithmetic

3. Applications of Ultra-Arithmetic

4. The Arithmetic of Intervals of Polynomials

References

A FORTRAN Extension for Scientific Computation

1. Motivation

2. Notation of the Language Extension

3. Syntax and Semantics of the Extension

References

An Introduction to MATRIX PASCAL: A PASCAL Extension for Scientific Computation

A. Data Types

B. Expressions

C. Procedures, Functions, Operators

D. Universal Operator Concept

E. Expressions with Maximum Accuracy

F. Standard Functions

References

Realization of an Optimal Computer Arithmetic

1. Introduction—Mathematical Foundations

2. Organization of the Arithmetic

3. Implementation of the Elementary Operations

4. Operations in the Higher Spaces

5. Realization on a Micro Computer

References

Features of a Hardware Implementation of an Optimal Arithmetic

1. Introduction

2. Implementation of Scalar Products

3. Algorithmic and Flowchart Description of a Hardware Unit

4. Parallelism in Scalar Products

5. Pipelining of the Arithmetic Operations

Appendix

References

Differentiation and Generation of Taylor Coefficients in PASCAL-SC

1. Automation of Evaluation and Differentiation of Functions

2. Derivative Data Types

3. Derivative Operators

4. Examples of Multiplication Operators in PASCAL-SC

5. Standard Derivative Functions

6. Applications of Derivative Types in Scientific Computation

References

Matrix Pascal

A. Introduction

B. Spaces, Operations, and Data Types

C. Informal Description of the Language Extension

D. Formal Description of MATRIX PASCAL

E. Comments on the Implementation of the Arithmetic

References

## Details

- No. of pages:
- 400

- Language:
- English

- Copyright:
- © Academic Press 1983

- Published:
- 28th October 1983

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483272047