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CAMBRIDGE2011Modern Computer Arithmetic
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Preface page ix Acknowledgements xi Notation xiii 1 Integer arithmetic 1 1.1 Representation and notations 1 1.2 Addition and subtraction 2 1.3 Multiplication 3 1.3.1 Naive multiplication 4 1.3.2 Karatsuba’s algorithm 5 1.3.3 Toom–Cook multiplication 6 1.3.4 Use of the fast Fourier transform (FFT) 8 1.3.5 Unbalanced multiplication 8 1.3.6 Squaring 11 1.3.7 Multiplication by a constant 13 1.4 Division 14 1.4.1 Naive division 14 1.4.2 Divisor preconditioning 16 1.4.3 Divide and conquer division 18 1.4.4 Newton’s method 21 1.4.5 Exact division 21 1.4.6 Only quotient or remainder wanted 22 1.4.7 Division by a single word 23 1.4.8 Hensel’s division 24 1.5 Roots 25 1.5.1 Square root 25 1.5.2 kth root 27 1.5.3 Exact root 28 vi Contents 1.6 Greatest common divisor 29 1.6.1 Naive GCD 29 1.6.2 Extended GCD 32 1.6.3 Half binary GCD, divide and conquer GCD 33 1.7 Base conversion 37 1.7.1 Quadratic algorithms 37 1.7.2 Subquadratic algorithms 38 1.8 Exercises 39 1.9 Notes and references 44 2 Modular arithmetic and the FFT 47 2.1 Representation 47 2.1.1 Classical representation 47 2.1.2 Montgomery’s form 48 2.1.3 Residue number systems 48 2.1.4 MSB vs LSB algorithms 49 2.1.5 Link with polynomials 49 2.2 Modular addition and subtraction 50 2.3 The Fourier transform 50 2.3.1 Theoretical setting 50 2.3.2 The fast Fourier transform 51 2.3.3 The Sch¨onhage–Strassen algorithm 55 2.4 Modular multiplication 58 2.4.1 Barrett’s algorithm 58 2.4.2 Montgomery’s multiplication 60 2.4.3 McLaughlin’s algorithm 63 2.4.4 Special moduli 65 2.5 Modular division and inversion 65 2.5.1 Several inversions at once 67 2.6 Modular exponentiation 68 2.6.1 Binary exponentiation 70 2.6.2 Exponentiation with a larger base 70 2.6.3 Sliding window and redundant representation 72 2.7 Chinese remainder theorem 73 2.8 Exercises 75 2.9 Notes and references 77 3 Floating-point arithmetic 79 3.1 Representation 79 3.1.1 Radix choice 80 3.1.2 Exponent range 81 Contents vii 3.1.3 Special values 82 3.1.4 Subnormal numbers 82 3.1.5 Encoding 83 3.1.6 Precision: local, global, operation, operand 84 3.1.7 Link to integers 86 3.1.8 Ziv’s algorithm and error analysis 86 3.1.9 Rounding 87 3.1.10 Strategies 90 3.2 Addition, subtraction, comparison 91 3.2.1 Floating-point addition 92 3.2.2 Floating-point subtraction 93 3.3 Multiplication 95 3.3.1 Integer multiplication via complex FFT 98 3.3.2 The middle product 99 3.4 Reciprocal and division 101 3.4.1 Reciprocal 102 3.4.2 Division 106 3.5 Square root 111 3.5.1 Reciprocal square root 112 3.6 Conversion 114 3.6.1 Floating-point output 115 3.6.2 Floating-point input 117 3.7 Exercises 118 3.8 Notes and references 120 4 Elementary and special function evaluation 125 4.1 Introduction 125 4.2 Newton’s method 126 4.2.1 Newton’s method for inverse roots 127 4.2.2 Newton’s method for reciprocals 128 4.2.3 Newton’s method for (reciprocal) square roots 129 4.2.4 Newton’s method for formal power series 129 4.2.5 Newton’s method for functional inverses 130 4.2.6 Higher-order Newton-like methods 131 4.3 Argument reduction 132 4.3.1 Repeated use of a doubling formula 134 4.3.2 Loss of precision 134 4.3.3 Guard digits 135 4.3.4 Doubling versus tripling 136 4.4 Power series 136 viii Contents 4.4.1 Direct power series evaluation 140 4.4.2 Power series with argument reduction 140 4.4.3 Rectangular series splitting 141 4.5 Asymptotic expansions 144 4.6 Continued fractions 150 4.7 Recurrence relations 152 4.7.1 Evaluation of Bessel functions 153 4.7.2 Evaluation of Bernoulli and tangent numbers 154 4.8 Arithmetic-geometric mean 158 4.8.1 Elliptic integrals 158 4.8.2 First AGM algorithm for the logarithm 159 4.8.3 Theta functions 160 4.8.4 Second AGM algorithm for the logarithm 162 4.8.5 The complex AGM 163 4.9 Binary splitting 163 4.9.1 A binary splitting algorithm for sin, cos 166 4.9.2 The bit-burst algorithm 167 4.10 Contour integration 169 4.11 Exercises 171 4.12 Notes and references 179 5 Implementations and pointers 185 5.1 Software tools 185 5.1.1 CLN 185 5.1.2 GNU MP (GMP) 185 5.1.3 MPFQ 186 5.1.4 GNU MPFR 187 5.1.5 Other multiple-precision packages 187 5.1.6 Computational algebra packages 188 5.2 Mailing lists 189 5.2.1 The GMP lists 189 5.2.2 The MPFR list 190 5.3 On-line documents 190 References 191 Index 207 |
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