Modern Computer Arithmetic

221 Pages·2009·1.72 MB·English

Modern Computer Arithmetic
Richard P. Brent and Paul Zimmermann
Version 0.3
Copyright c 2003-2009 Richard P. Brent and Paul Zimmermann
(cid:13)
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Preface
This is a book about algorithms for performing arithmetic, and their imple-
mentation on modern computers. We are concerned with software more than
hardware —we do notcover computer architecture orthe design ofcomputer
hardware since good books are already available on these topics. 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 special functions.
Thealgorithmsthatwepresentaremainlyintendedforarbitrary-precision
arithmetic. That is, they are not limited by the computer wordsize of 32 or
64 bits, only by the memory and time available for the computation. We
consider both integer and real (floating-point) computations.
The bookis divided into four mainchapters, plus anappendix. Chapter 1
covers integer arithmetic. This has, of course, been considered in many other
books and papers. However, there has been much recent progress, inspired in
part by the application to public key cryptography, so most of the published
books are now partly out of date or incomplete. Our aim has been to present
the latest developments in a concise manner.
Chapter 2 is concerned with the FFT and modular arithmetic, and their
applications to computer arithmetic. We consider different number represen-
tations, fast algorithms for multiplication, division and exponentiation, and
the use of the Chinese Remainder Theorem (CRT).
Chapter 3 covers floating-point arithmetic. Our concern is with high-
precision floating-point arithmetic, implemented in software if the precision
provided by the hardware (typically IEEE standard 64-bit arithmetic) is in-
adequate. Thealgorithmsdescribedinthischapterfocusoncorrect rounding,
extending the IEEE standard to arbitrary precision.
Chapter 4 deals with the computation, to arbitraryprecision, of functions
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4 Modern Computer Arithmetic, version 0.3 of June 11, 2009
such as sqrt, exp, ln, sin, cos, and more generally functions defined by power
series or continued fractions. We also consider the computation of certain
constants, such as π and (Euler’s constant) γ. Of course, the computation of
special functions is a huge topic so we have had to be selective. In particular,
wehave concentrated onmethodsthat areefficient andsuitable forarbitrary-
precision computations.
For details that are omitted we give pointers in the Notes and References
sections of each chapter, and in the bibliography. Finally, the Appendix
contains pointers to implementations, useful web sites, mailing lists, and so
on.
The book is intended for anyone interested in the design and implemen-
tation of efficient algorithms for computer arithmetic, and more generally
efficient numerical algorithms. We did our best to present algorithms that
are ready to implement in your favorite language, while keeping a high-level
description.
Although the book is not specifically intended as a textbook, it could be
used in a graduate course in mathematics or computer science, and for this
reason, as well as to cover topics that could not be discussed at length in the
text, we have included exercises at the end of each chapter. For solutions to
the exercises, please contact the authors.
WethanktheFrench NationalInstituteforResearch inComputer Science
and Control (INRIA), the Australian National University (ANU), and the
Australian Research Council (ARC), for their support. The book could not
have been written without the contributions of many friends and colleagues,
toonumerous to mention here, but acknowledged inthe text andintheNotes
and References sections at the end of each chapter.
Finally, we acknowledge Erin Brent, who first suggested writing the book;
and thank our wives, Judy-anne and Marie, for their patience and encour-
agement.
This is a preliminary version — there are still a few exercises to be added.
We welcome comments and corrections. Please send them to either of the
authors.
Richard Brent and Paul Zimmermann
[email protected]
[email protected]
Canberra and Nancy, June 2009
Contents
Preface 3
Contents 5
Notation 11
1 Integer Arithmetic 15
1.1 Representation and Notations . . . . . . . . . . . . . . . . . . 15
1.2 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . 16
1.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.1 Naive Multiplication . . . . . . . . . . . . . . . . . . . 18
1.3.2 Karatsuba’s Algorithm . . . . . . . . . . . . . . . . . . 19
1.3.3 Toom-Cook Multiplication . . . . . . . . . . . . . . . . 20
1.3.4 Fast Fourier Transform (FFT) . . . . . . . . . . . . . . 22
1.3.5 Unbalanced Multiplication . . . . . . . . . . . . . . . . 22
1.3.6 Squaring . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.7 Multiplication by a Constant . . . . . . . . . . . . . . 24
1.4 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.1 Naive Division . . . . . . . . . . . . . . . . . . . . . . . 26
1.4.2 Divisor Preconditioning . . . . . . . . . . . . . . . . . 28
1.4.3 Divide and Conquer Division . . . . . . . . . . . . . . 29
1.4.4 Newton’s Method . . . . . . . . . . . . . . . . . . . . . 32
1.4.5 Exact Division . . . . . . . . . . . . . . . . . . . . . . 32
1.4.6 Only Quotient or Remainder Wanted . . . . . . . . . . 33
1.4.7 Division by a Constant . . . . . . . . . . . . . . . . . . 34
1.4.8 Hensel’s Division . . . . . . . . . . . . . . . . . . . . . 35
1.5 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.5.1 Square Root . . . . . . . . . . . . . . . . . . . . . . . . 36
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1.5.2 k-th Root . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.5.3 Exact Root . . . . . . . . . . . . . . . . . . . . . . . . 39
1.6 Greatest Common Divisor . . . . . . . . . . . . . . . . . . . . 40
1.6.1 Naive GCD . . . . . . . . . . . . . . . . . . . . . . . . 41
1.6.2 Extended GCD . . . . . . . . . . . . . . . . . . . . . . 43
1.6.3 Half GCD, Divide and Conquer GCD . . . . . . . . . . 44
1.7 Base Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.7.1 Quadratic Algorithms . . . . . . . . . . . . . . . . . . 47
1.7.2 Subquadratic Algorithms . . . . . . . . . . . . . . . . . 47
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.9 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 53
2 The FFT and Modular Arithmetic 55
2.1 Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.1.1 Classical Representation . . . . . . . . . . . . . . . . . 55
2.1.2 Montgomery’s Form . . . . . . . . . . . . . . . . . . . 56
2.1.3 Residue Number Systems . . . . . . . . . . . . . . . . 56
2.1.4 MSB vs LSB Algorithms . . . . . . . . . . . . . . . . . 57
2.1.5 Link with Polynomials . . . . . . . . . . . . . . . . . . 57
2.2 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . 58
2.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.3.1 Barrett’s Algorithm . . . . . . . . . . . . . . . . . . . . 59
2.3.2 Montgomery’s Multiplication . . . . . . . . . . . . . . 60
2.3.3 McLaughlin’s Algorithm . . . . . . . . . . . . . . . . . 64
2.3.4 Special Moduli . . . . . . . . . . . . . . . . . . . . . . 65
2.3.5 Fast Multiplication Over GF(2)[x] . . . . . . . . . . . . 66
2.4 Division and Inversion . . . . . . . . . . . . . . . . . . . . . . 72
2.4.1 Several Inversions at Once . . . . . . . . . . . . . . . . 74
2.5 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.5.1 Binary Exponentiation . . . . . . . . . . . . . . . . . . 77
2.5.2 Base 2k Exponentiation . . . . . . . . . . . . . . . . . 78
2.5.3 Sliding Window and Redundant Representation . . . . 79
2.6 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . 80
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.8 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 82
Modern Computer Arithmetic, 0.0 7
§
3 Floating-Point Arithmetic 85
3.1 Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.1.1 Radix Choice . . . . . . . . . . . . . . . . . . . . . . . 86
3.1.2 Exponent Range . . . . . . . . . . . . . . . . . . . . . 87
3.1.3 Special Values . . . . . . . . . . . . . . . . . . . . . . . 88
3.1.4 Subnormal Numbers . . . . . . . . . . . . . . . . . . . 88
3.1.5 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.1.6 Precision: Local, Global, Operation, Operand . . . . . 91
3.1.7 Link to Integers . . . . . . . . . . . . . . . . . . . . . . 92
3.1.8 Ziv’s Algorithm and Error Analysis . . . . . . . . . . . 92
3.1.9 Rounding . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.1.10 Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.2 Addition, Subtraction, Comparison . . . . . . . . . . . . . . . 98
3.2.1 Floating-Point Addition . . . . . . . . . . . . . . . . . 99
3.2.2 Floating-Point Subtraction . . . . . . . . . . . . . . . . 100
3.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.3.1 Integer Multiplication via Complex FFT . . . . . . . . 105
3.3.2 The Middle Product . . . . . . . . . . . . . . . . . . . 107
3.4 Reciprocal and Division . . . . . . . . . . . . . . . . . . . . . 109
3.4.1 Reciprocal . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.4.2 Division . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.5 Square Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.5.1 Reciprocal Square Root . . . . . . . . . . . . . . . . . 119
3.6 Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.6.1 Floating-Point Output . . . . . . . . . . . . . . . . . . 123
3.6.2 Floating-Point Input . . . . . . . . . . . . . . . . . . . 125
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.8 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 128
4 Newton’s Method and Function Evaluation 131
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.2.1 Newton’s Method for Inverse Roots . . . . . . . . . . . 134
4.2.2 Newton’s Method for Reciprocals . . . . . . . . . . . . 134
4.2.3 Newton’s Method for (Reciprocal) Square Roots . . . . 135
4.2.4 Newton’s Method for Formal Power Series . . . . . . . 136
4.2.5 Newton’s Method for Functional Inverses . . . . . . . . 137
4.2.6 Higher Order Newton-like Methods . . . . . . . . . . . 138
8 Modern Computer Arithmetic, version 0.3 of June 11, 2009
4.3 Argument Reduction . . . . . . . . . . . . . . . . . . . . . . . 139
4.3.1 Repeated Use of a Doubling Formula . . . . . . . . . . 141
4.3.2 Loss of Precision . . . . . . . . . . . . . . . . . . . . . 141
4.3.3 Guard Digits . . . . . . . . . . . . . . . . . . . . . . . 142
4.3.4 Doubling versus Tripling . . . . . . . . . . . . . . . . . 143
4.4 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.4.1 Direct Power Series Evaluation . . . . . . . . . . . . . 147
4.4.2 Power Series With Argument Reduction . . . . . . . . 148
4.4.3 Rectangular Series Splitting . . . . . . . . . . . . . . . 149
4.5 Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . 152
4.6 Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . 158
4.7 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . 161
4.7.1 Evaluation of Bessel Functions . . . . . . . . . . . . . . 162
4.7.2 Evaluation of Bernoulli and Tangent numbers . . . . . 163
4.8 Arithmetic-Geometric Mean . . . . . . . . . . . . . . . . . . . 167
4.8.1 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . 167
4.8.2 First AGM Algorithm for the Logarithm . . . . . . . . 168
4.8.3 Theta Functions . . . . . . . . . . . . . . . . . . . . . . 169
4.8.4 Second AGM Algorithm for the Logarithm . . . . . . . 171
4.8.5 The Complex AGM . . . . . . . . . . . . . . . . . . . . 172
4.9 Binary Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.9.1 A Binary Splitting Algorithm for sin,cos . . . . . . . . 176
4.9.2 The Bit-Burst Algorithm . . . . . . . . . . . . . . . . . 176
4.10 Contour Integration . . . . . . . . . . . . . . . . . . . . . . . . 179
4.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.12 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 188
5 Appendix: Implementations and Pointers 193
5.1 Software Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.1.1 CLN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.1.2 GNU MP (GMP) . . . . . . . . . . . . . . . . . . . . . 193
5.1.3 MPFQ . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.1.4 MPFR . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.2 Mailing Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.2.1 The BNIS Mailing List . . . . . . . . . . . . . . . . . . 195
5.2.2 The GMP Lists . . . . . . . . . . . . . . . . . . . . . . 195
5.3 On-Line Documents . . . . . . . . . . . . . . . . . . . . . . . . 195
Modern Computer Arithmetic, 0.0 9
§
Bibliography 197
Index 213
Summary of Complexities 221
10 Modern Computer Arithmetic, version 0.3 of June 11, 2009