Extrapolation Methods (eBook)

Front Cover 1
Extrapolation Methods: Theory and Practice 4
Copyright Page 5
Table of Contents 8
Preface 6
Chapter 1. INTRODUCTION TO THE THEORY 12
1.1 First steps 12
1.2 What is an extrapolation method? 16
1.3 What is an extrapolation algorithm? 19
1.4 Quasi-linear sequence transformations 22
1.5 Sequence transformations as ratios of determinants 29
1.6 Triangular recursive schemes 32
1.7 Normal forms of the algorithms 37
1.8 Progressive forms of the algorithms 39
1.9 Particular rules of the algorithms 45
1.10 Accelerability and non-accelerability 50
1.11 Optimality 53
1.12 Asymptotic behaviour of sequences 58
Chapter 2. SCALAR EXTRAPOLATION ALGORITHMS 66
2.1 The E-algorithm 66
2.2 Richardson extrapolation process 83
2.3 The e-algorithm 89
2.4 The G-transformation 106
2.5 Rational extrapolation 112
2.6 Generalizations of the e-algorithm 119
2.7 Levin's transforms 124
2.8 Overholt's process 130
2.9 T-type algorithms 132
2.10 The iterated .2 process 139
2.11 Miscellaneous algorithms 142
Chapter 3. SPECIAL DEVICES 156
3.1 Error estimates and acceleration 156
3.2 Convergence tests and acceleration 162
3.3 Construction of asymptotic expansions 170
3.4 Construction of extrapolation processes 176
3.5 Extraction procedures 185
3.6 Automatic selection 189
3.7 Composite sequence transformations 196
3.8 Error control 204
3.9 Contractive sequence transformations 212
3.10 Least squares extrapolation 221
Chapter 4. VECTOR EXTRAPOLATION ALGORITHMS 224
4.1 The vector e-algorithm 227
4.2 The topological e-algorithm 231
4.3 The vector E-algorithm 239
4.4 The recursive projection algorithm 244
4.5 The H-algorithm 249
4.6 The Ford-Sidi algorithms 255
4.7 Miscellaneous algorithms 258
Chapter 5. CONTINUOUS PREDICTION ALGORITHMS 264
5.1 The Taylor expansion 265
5.2 Confluent Overholt's process 266
5.3 Confluent e-algorithms 267
5.4 Confluent .-algorithm 273
5.5 Confluent G-transform 276
5.6 Confluent E-algorithm 277
5.7 T-type confluent algorithms 278
Chapter 6. APPLICATIONS 280
6.1 Sequences and series 281
6.2 Systems of equations 313
6.3 Eigenelements 343
6.4 Integral and differential equations 349
6.5 Interpolation and approximation 365
6.6 Statistics 368
6.7 Integration and differentiation 376
6.8 Prediction 400
Chapter 7. SOFTWARE 408
7.1 Programming the algorithms 408
7.2 Computer arithmetic 411
7.3 Programs 414
Bibliography 424
Index 466