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Chapter 5 Floating-Point Arithmetic

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Computer Arithmetic and Validity
This chapter is in the book Computer Arithmetic and Validity
Chapter 5Floating-Point ArithmeticThus far our considerations have proceededunder the general assumptionthat the setRin Figure 1 is a linearly ordered ringoid. We are now goingto be more specific and assume thatRis the linearly orderedfield of realnumbers and that the subsetsDandSare special screens, which are calledfloating-point systems. In Section 5.1 we brieflyreviewthedefinition ofthe real numbers and their representation byb-adic expansions. Section 5.2deals withfloating-point numbers and systems. We also discuss severalbasic roundings of the real numbers into afloating-point system, and wederive error bounds for these roundings.In Section 5.3 we considerfloating-point operations defined by semi-morphisms, and we derive error bounds for these operations. Then weconsider the operations defined in the product sets listed underSandDin Figure 1, and we derive error formulas and bounds for these operationsalso. All this is done under the assumption that the operations are definedby semimorphisms.For scalar product and matrix multiplication, we also derive error for-mulas and bounds for the conventional definition of the operations. Theerror formulas are simpler if the operations are defined by semimorphisms,and the error bounds are smaller by a factor of at leastncompared to thebounds obtained by the conventional definition of the operations. These twoproperties are reproduced in the error analysis of many algorithms.Finally we review the so-called IEEEfloating-point arithmetic standard.5.1 Definition and Properties of the Real NumbersTwo methods of defining the real numbers are most commonly used. These may becalled the constructive method and the axiomatic method. We have mentioned theseconcepts already in the introduction of this book. For clarity we briefly consider theaxiomatic method here.The real numbers{R,+,·,≤}can be defined as a conditionally complete linearlyorderedfield, i.e.,I{R,+,·}is afield.II{R,≤}is a conditionally complete, linearly ordered set.

Chapter 5Floating-Point ArithmeticThus far our considerations have proceededunder the general assumptionthat the setRin Figure 1 is a linearly ordered ringoid. We are now goingto be more specific and assume thatRis the linearly orderedfield of realnumbers and that the subsetsDandSare special screens, which are calledfloating-point systems. In Section 5.1 we brieflyreviewthedefinition ofthe real numbers and their representation byb-adic expansions. Section 5.2deals withfloating-point numbers and systems. We also discuss severalbasic roundings of the real numbers into afloating-point system, and wederive error bounds for these roundings.In Section 5.3 we considerfloating-point operations defined by semi-morphisms, and we derive error bounds for these operations. Then weconsider the operations defined in the product sets listed underSandDin Figure 1, and we derive error formulas and bounds for these operationsalso. All this is done under the assumption that the operations are definedby semimorphisms.For scalar product and matrix multiplication, we also derive error for-mulas and bounds for the conventional definition of the operations. Theerror formulas are simpler if the operations are defined by semimorphisms,and the error bounds are smaller by a factor of at leastncompared to thebounds obtained by the conventional definition of the operations. These twoproperties are reproduced in the error analysis of many algorithms.Finally we review the so-called IEEEfloating-point arithmetic standard.5.1 Definition and Properties of the Real NumbersTwo methods of defining the real numbers are most commonly used. These may becalled the constructive method and the axiomatic method. We have mentioned theseconcepts already in the introduction of this book. For clarity we briefly consider theaxiomatic method here.The real numbers{R,+,·,≤}can be defined as a conditionally complete linearlyorderedfield, i.e.,I{R,+,·}is afield.II{R,≤}is a conditionally complete, linearly ordered set.

Chapters in this book

  1. Frontmatter i
  2. Contents vii
  3. Introduction 1
  4. I Theory of Computer Arithmetic
  5. Chapter 1 First Concepts 12
  6. Chapter 2 Ringoids and Vectoids 42
  7. Chapter 3 Definition of Computer Arithmetic 60
  8. Chapter 4 Interval Arithmetic 84
  9. II Implementation of Arithmetic on Computers
  10. Chapter 5 Floating-Point Arithmetic 154
  11. Chapter 6 Implementation of Floating-Point Arithmetic on a Computer 187
  12. Chapter 7 Hardware Support for Interval Arithmetic 233
  13. Chapter 8 Scalar Products and Complete Arithmetic 245
  14. III Principles of Verified Computing
  15. Chapter 9 Sample Applications 302
  16. Backmatter 356
Readers are also interested in:
https://doi.org/10.1515/9783110203196.2.154

Chapters in this book

  1. Frontmatter i
  2. Contents vii
  3. Introduction 1
  4. I Theory of Computer Arithmetic
  5. Chapter 1 First Concepts 12
  6. Chapter 2 Ringoids and Vectoids 42
  7. Chapter 3 Definition of Computer Arithmetic 60
  8. Chapter 4 Interval Arithmetic 84
  9. II Implementation of Arithmetic on Computers
  10. Chapter 5 Floating-Point Arithmetic 154
  11. Chapter 6 Implementation of Floating-Point Arithmetic on a Computer 187
  12. Chapter 7 Hardware Support for Interval Arithmetic 233
  13. Chapter 8 Scalar Products and Complete Arithmetic 245
  14. III Principles of Verified Computing
  15. Chapter 9 Sample Applications 302
  16. Backmatter 356
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