Advanced Arithmetic for the Digital Computer - Design of Arithmetic Units

Advanced computer arithmetic requires that all computer approximations of arithmetic operations - in particular those in the usual vector and matrix spaces - differ from the correct result by at most one rounding. The implementation of advanced computer arithmetic by fast hardware is examined in the book. The new expanded computational capability is gained at modest cost. It increases both the speed of a computation and the accuracy of the computed result. With it fast multiple precision arithmetic can be easily provided. All this strongly supports the case for implementing advanced computer arithmetic on every CPU.

The book also shows that on superscalar processors interval operations can be made as fast as simple floating-point operations with only very modest additional hardware costs.

Contents

• 1.1 Introduction
• 1.2 Implementation Principles
• 1.3 High-Performance Scalar Product Units (SPU)
• 1.4 Comments on the Scalar Product Units
• 1.5 Scalar Product Units for Top-Performance Computers
• 1.6 Hardware Accumulation Window
• 1.7 Theoretical Foundation of Advanced Computer

• 2.1 The one dimensional case
• 2.2 Rounding in product spaces

• 3.1 Introduction and Historical Remarks
• 3.2 Interval Arithmetic, a Powerful Calculus to Deal with Inequalities
• 3.3 Interval Arithmetic as Executable Set Operations
• 3.4 Enclosing the Range of Function Values
• 3.5 The Interval Newton Method
• 3.6 Extended Interval Arithmetic
• 3.7 The Extended Interval Newton Method
• 3.8 Differentiation Arithmetic, Enclosures of Derivatives
• 3.9 Interval Arithmetic on the Computer
• 3.10 Hardware Support for Interval Arithmetic