## Modern Computer Arithmetic

226. R. P. Brent and P. Zimmermann, Modern Computer Arithmetic, Cambridge Monographs on Computational and Applied Mathematics (No. 18), Cambridge University Press, November 2010, 236 pages. Publisher's web page.

A preliminary (electronic) version of the book is available here: Version 0.5.9 (7 October 2010), xvi+223 pages (2.0 MB).

Versions with larger (12pt) fonts are available from arXiv:1004.4710v1 (version 0.5.1 of 27 April 2010)
or here (version 0.5.2 of 31 July 2010). For other versions, see Paul Zimmermann's page.

For the electronic versions, copying this work is allowed for non-commercial use (see the license on page iii of the pdf file).

## Abstract

This is a book about algorithms for performing arithmetic, and their implementation on modern computers. It collects in the same document all state-of-the-art algorithms for multiple precision arithmetic (integers, integers modulo n, floating-point numbers). The best current reference on this topic is Volume 2 of Knuth's The Art of Computer Programming, which omits some new important algorithms (divide and conquer division, other variants of FFT multiplication, floating-point algorithms, ...). Our aim is to give detailed algorithms:
• for all operations (not just multiplication as in many text books),
• for all size ranges (not just schoolbook methods or FFT-based methods),
• and including all details (for example how to properly deal with carries for integer algorithms, or a rigorous analysis of roundoff errors for floating-point algorithms).
The book should be useful for graduate students in computer science and mathematics (perhaps too specialized for most undergraduates), researchers in discrete mathematics, computer algebra, number theory, cryptography, and developers of multiple-precision libraries.

## Summary

Chapter 1 describes integer arithmetic (representation, addition, subtraction, multiplication, division, roots, gcd, base conversion).
Chapter 2 deals with modular arithmetic (representation, multiplication, division/inversion, exponentiation, conversion, applications of FFT).
Chapter 3 treats with floating-point arithmetic (addition, subtraction, comparison, multiplication, division, algebraic functions, conversion).
Chapter 4 covers Newton's method and function evaluation (Newton's method and its variants, argument reduction, power series, asymptotic expansions, continued fractions, recurrence relations, arithmetic-geometric mean, binary splitting, D-finite functions, contour integration, constants).
Finally, Chapter 5 gives pointers to software tools, mailing-lists, and on-line documents.

## Exercises

The book contains many exercises, which vary considerably in difficulty. For solutions to selected exercises, please contact one of the authors.

## Errata

Errata for the printed (CUP) version of the book and electronic versions 0.5.7 and later may be found here.

Errata for earlier versions of the book are here.

## Software

Many algorithms from the book are implemented in the GNU MP and MPFR libraries. Other relevant packages are:
• fastfunlib is a C library for fast multiprecision evaluation of transcendental functions, using fixed-point arithmetic directly on top of GMP/MPIR. The goal is to provide optimized base implementations of transcendental functions, with minimal overhead at low precision as well as asymptotic speed.
• Zen, from Reynald Lercier and Florent Chabaud, which provides fast arithmetic in polynomial finite rings over Z/nZ, with application to integer factorization and primality testing.
The following programs illustrate algorithms from the book:
• hgcd.c is an implementation of the binary (or 2-adic or LSB or right-to-left) subquadratic gcd of Stehlé and Zimmermann. It is quite efficient compared to the code in GMP 4.3.1 (which implements the classical MSB or left-to-right reduction). For example on a Core 2 at 2.83Ghz we can compute the GCD of two numbers of 107 limbs, i.e, about 193 million decimal digits, in 738.5s with mpz_gcd and 725.1s with mpz_bgcd. For 108 limbs, i.e., about 1.9 billion decimal digits, mpz_gcd takes 11776s and mpz_bgcd takes 11031s, thus a 6.3% speedup.

## Books on Related Topics

The Art of Computer Programming, volume 2: Seminumerical Algorithms, Donald E. Knuth, 3rd edition, 1998.

Modern Computer Algebra , Joachim von zur Gathen and J. Gerhard, Cambridge University Press, 2nd edition, 2003.

The Design and Analysis of Computer Algorithms, A. V. Aho, J. E. Hopcroft and J. D. Ullman, Addison-Wesley, 1974 [chapters 7 and 8].

The Computational Complexity of Algebraic and Numeric Problems, A. Borodin and I. Munro, Elsevier Computer Science Library, 1975.

Handbook of Applied Cryptography, Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, CRC Press, 1997 [chapter 14].

Prime Numbers: A Computational Perspective, Richard E. Crandall and Carl Pomerance, Springer Verlag, 2001 [chapter 9].

Fast Algorithms, A Multitape Turing Machine Implementation, Arnold Schönhage and A. F. W. Grotefeld and E. Vetter, BI-Wissenschaftsverlag, 1994 [out of print].

Handbook of Elliptic and Hyperelliptic Curve Cryptography, Henri Cohen, Gerhard Frey, Roberto Avanzi, Christophe Doche, Tanja Lange, Kim Nguyen, Frederik Vercauteren, Chapman & Hall/CRC, series Discrete Mathematics and its Applications, 2005 [chapters 9-12].

Handbook of Floating-Point Arithmetic, Nicolas Brisebarre, Florent de Dinechin, Claude-Pierre Jeannerod, Vincent Lefèvre, Guillaume Melquiond, Jean-Michel Muller, Nathalie Revol, Damien Stehlé and Serge Torres, Birkhäuser, 2009.

Matters Computational, Jörg Arndt, Springer, 2010, also freely available online.

Finite Precision Number Systems and Arithmetic, Peter Kornerup and David W. Matula, Cambridge University Press, 2010.

## Reviews

For a review of the book by Warren Ferguson, see SIAM Review, 53, 4 (December 2011), 809-810.

## Citations

• 31 December 2009: Fabrice Bellard set a new record for computing digits of Pi, with 2700 billion digits, and referrred to our book when describing the algorithms used to obtain this record.

### The FEE method of E. A. Karatsuba

The reader may wonder why our book does not refer to the FEE method of E. A. Karatsuba. In fact, various drafts of Chapter 4 of the book did refer to this method, but our publisher (CUP) asked us to remove such references to avoid any possibility of legal problems. For brief comments on the FEE method, see for example the draft of the book that is available at arXiv:1004.4710.

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