Modern Computer Arithmetic
R. P. Brent,
Modern Computer Arithmetic,
Cambridge Monographs on Computational and Applied Mathematics (No. 18),
Cambridge University Press,
November 2010, 236 pages.
Publisher's web page.
To cite this document, please use the following:
Modern Computer Arithmetic, Richard Brent and Paul Zimmermann,
Cambridge University Press, 2010.
Preliminary versions of the book are available here:
(October 2010), xvi+223 pages;
this version corresponds quite closely to the version published by Cambridge
University Press (the only differences being that pages i-viii slightly differ,
and the three typos reported by Torbjörn Granlund on version 0.5.7 are fixed
(September 2010), xvi+223 pages,
(September 2010), xvi+223 pages,
(August 2010), xvi+223 pages,
(March 2010), xvi+227 pages,
(February 2010), xvi+245 pages,
(November 2009), 239 pages,
(June 2009), 221 pages,
(March 2009), 215 pages,
(June 2008), 190 pages,
(November 2006), 94 pages,
(October 2006), 91 pages.
For the electronic versions,
copying this work is allowed for non-commercial use (see the license on
page iii of the pdf file).
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 in multiple precision arithmetic (integers, integers modulo n,
floating-point numbers). The best current reference on that topic is
volume 2 from Knuth's The art of computer programming,
which misses some new important algorithms (divide and conquer division,
other variants of FFT multiplication, floating-point algorithms, ...)
Our aim is to give detailed algorithms:
The book is useful for graduate students in computer science and
mathematics (perhaps too specialized for most undergraduates, at least
in its present state), researchers in discrete mathematics, computer
algebra, number theory, cryptography, and developers of multiple-precision
- for all operations (not just multiplication as 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
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, an appendix gives pointers to software tools, mailing-lists, and
The book contains many exercises, which vary considerably in difficulty.
Here are solutions to selected exercises.
Errata for the printed (CUP) version of the book and electronic versions
0.5.7 and later may be found
Errata for earlier versions of the book are
Many algorithms from the book are implemented in the
GNU MP and
Other relevant packages are:
The following programs illustrate algorithms from the book:
- 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.
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.
- hgcd.c is an implementation of the binary (or 2-adic
or LSB or right-to-left)
subquadratic gcd from 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 Jürgen
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
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,
Jörg Arndt, Springer, 2010, also
freely available online.
See also the ``Algorithms'' chapter from the
GMP reference manual.
For a review of the book by Warren Ferguson, see SIAM Review, 53, 4 (December 2011), 809-810.
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.