Book description
This is the revised and extended second edition of the successful basic book on computer arithmetic. It is consistent with the newest recent standard developments in the field. The book shows how the arithmetic and mathematical capability of the digital computer can be enhanced in a quite natural way. The work is motivated by the desire and the need to improve the accuracy of numerical computing and to control the quality of the computed results (validity). The accuracy requirements for the elementary floatingpoint operations are extended to the customary product spaces of computations including interval spaces. The mathematical properties of these models are extracted into an axiomatic approach which leads to a general theory of computer arithmetic. Detailed methods and circuits for the implementation of this advanced computer arithmetic on digital computers are developed in part two of the book. Part three then illustrates by a number of sample applications how this extended computer arithmetic can be used to compute highly accurate and mathematically verified results. The book can be used as a highlevel undergraduate textbook but also as reference work for research in computer arithmetic and applied mathematics.
Table of contents
 Foreword to the second edition
 Preface (1/3)
 Preface (2/3)
 Preface (3/3)
 Introduction (1/2)
 Introduction (2/2)

I Theory of computer arithmetic

1 First concepts
 1.1 Ordered sets
 1.2 Complete lattices and complete subnets (1/2)
 1.2 Complete lattices and complete subnets (2/2)
 1.3 Screens and roundings (1/3)
 1.3 Screens and roundings (2/3)
 1.3 Screens and roundings (3/3)
 1.4 Arithmetic operations and roundings (1/2)
 1.4 Arithmetic operations and roundings (2/2)
 2 Ringoids and vectoids

3 Definition of computer arithmetic
 3.1 Introduction
 3.2 Preliminaries
 3.3 The traditional definition of computer arithmetic
 3.4 Definition of computer arithmetic by semimorphisms (1/2)
 3.4 Definition of computer arithmetic by semimorphisms (2/2)
 3.5 A remark about roundings
 3.6 Uniqueness of the minus operator
 3.7 Rounding near zero (1/2)
 3.7 Rounding near zero (2/2)

4 Interval arithmetic
 4.1 Interval sets and arithmetic (1/2)
 4.1 Interval sets and arithmetic (2/2)
 4.2 Interval arithmetic over a linearly ordered set
 4.3 Interval matrices (1/2)
 4.3 Interval matrices (2/2)
 4.4 Interval vectors
 4.5 Interval arithmetic on a screen (1/2)
 4.5 Interval arithmetic on a screen (2/2)
 4.6 Interval matrices and interval vectors on a screen (1/2)
 4.6 Interval matrices and interval vectors on a screen (2/2)
 4.7 Complex interval arithmetic (1/2)
 4.7 Complex interval arithmetic (2/2)
 4.8 Complex interval matrices and interval vectors
 4.9 Extended interval arithmetic
 4.10 Exceptionfree arithmetic for extended intervals
 4.11 Extended interval arithmetic on the computer
 4.12 Exceptionfree arithmetic for closed real intervals on the computer
 4.13 Comparison relations and lattice operations
 4.14 Algorithmic implementation of interval multiplication and division

1 First concepts

II Implementation of arithmetic on computers

5 Floatingpoint arithmetic
 5.1 Definition and properties of the real numbers
 5.2 Floatingpoint numbers and roundings (1/2)
 5.2 Floatingpoint numbers and roundings (2/2)
 5.3 Floatingpoint operations (1/2)
 5.3 Floatingpoint operations (2/2)
 5.4 Subnormal floatingpoint numbers
 5.5 On the IEEE floatingpoint arithmetic standard (1/2)
 5.5 On the IEEE floatingpoint arithmetic standard (2/2)

6 Implementation of floatingpoint arithmetic on a computer
 6.1 A brief review of the realization of integer arithmetic (1/2)
 6.1 A brief review of the realization of integer arithmetic (2/2)
 6.2 Introductory remarks about the level 1 operations
 6.3 Addition and subtraction
 6.4 Normalization
 6.5 Multiplication
 6.6 Division
 6.7 Rounding
 6.8 A universal rounding unit
 6.9 Overflow and underflow treatment
 6.10 Algorithms using the short accumulator (1/2)
 6.10 Algorithms using the short accumulator (2/2)
 6.11 The level 2 operations (1/2)
 6.11 The level 2 operations (2/2)
 7 Hardware support for interval arithmetic

8 Scalar products and complete arithmetic
 8.1 Introduction and motivation
 8.2 Historical remarks
 8.3 The ubiquity of the scalar product in numerical analysis
 8.4 Implementation principles (1/2)
 8.4 Implementation principles (2/2)
 8.5 Informal sketch for computing an exact dot product
 8.6 Scalar product computation units (SPUs)
 8.7 Comments
 8.8 The data format complete and complete arithmetic (1/2)
 8.8 The data format complete and complete arithmetic (2/2)
 8.9 Top speed scalar product units (1/3)
 8.9 Top speed scalar product units (2/3)
 8.9 Top speed scalar product units (3/3)
 8.10 Hardware complete register window

5 Floatingpoint arithmetic

III Principles of verified computing

9 Sample applications
 9.1 Basic properties of interval mathematics (1/3)
 9.1 Basic properties of interval mathematics (2/3)
 9.1 Basic properties of interval mathematics (3/3)
 9.2 Differentiation arithmetic, enclosures of derivatives (1/2)
 9.2 Differentiation arithmetic, enclosures of derivatives (2/2)
 9.3 The interval Newton method
 9.4 The extended interval Newton method
 9.5 Verified solution of systems of linear equations (1/2)
 9.5 Verified solution of systems of linear equations (2/2)
 9.6 Accurate evaluation of arithmetic expressions (1/2)
 9.6 Accurate evaluation of arithmetic expressions (2/2)
 9.7 Multiple precision arithmetics (1/3)
 9.7 Multiple precision arithmetics (2/3)
 9.7 Multiple precision arithmetics (3/3)
 9.8 Remarks on Kaucher arithmetic (1/2)
 9.8 Remarks on Kaucher arithmetic (2/2)

9 Sample applications
 A Frequently used symbols
 B On homomorphism
 Bibliography (1/10)
 Bibliography (2/10)
 Bibliography (3/10)
 Bibliography (4/10)
 Bibliography (5/10)
 Bibliography (6/10)
 Bibliography (7/10)
 Bibliography (8/10)
 Bibliography (9/10)
 Bibliography (10/10)
 List of figures
 List of tables
 Index (1/2)
 Index (2/2)
Product information
 Title: Computer Arithmetic and Validity
 Author(s):
 Release date: April 2013
 Publisher(s): De Gruyter
 ISBN: 9783110301793
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