Modern Computer Arithmetic (version 0.5.1)

TL;DR

This draft book discusses algorithms for efficient arbitrary-precision arithmetic on modern computers, covering integer, modular, floating-point, and special function computations, with a focus on software implementations and recent developments.

Contribution

It provides a comprehensive, up-to-date overview of algorithms for arbitrary-precision arithmetic, connecting classical methods with modern computational techniques.

Findings

Algorithms for arbitrary-precision integer arithmetic

Efficient methods for modular arithmetic and FFT

Techniques for elementary and special function evaluation

Abstract

This is a draft of a book about algorithms for performing arithmetic, and their implementation on modern computers. We are concerned with software more than hardware - we do not cover computer architecture or the design of computer hardware. Instead we focus on algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics such as modular arithmetic, greatest common divisors, the Fast Fourier Transform (FFT), and the computation of elementary and special functions. The algorithms that we present are mainly intended for arbitrary-precision arithmetic. They are not limited by the computer word size, only by the memory and time available for the computation. We consider both integer and real (floating-point) computations. The book is divided into four main chapters, plus an appendix. Our aim is to present the latest developments in a concise manner. At the same time, we provide a self-contained introduction for the reader who is not an expert in the field, and exercises at the end of each chapter. Chapter titles are: 1, Integer Arithmetic; 2, Modular Arithmetic and the FFT; 3, Floating-Point Arithmetic; 4, Elementary and Special Function Evaluation; 5 (Appendix), Implementations and Pointers. The book also contains a bibliography of 236 entries, index, summary of notation, and summary of complexities.