Extrapolation Methods: Theory and Practice

Cover......Page 1
Preface
......Page 6
Contents......Page 8
1.1 First steps......Page 12
1.2 What is an extrapolation method?......Page 16
1.3 What is an extrapolation algorithm?......Page 19
1.4 Quasi-linear sequence transformations......Page 22
1.5 Sequence transformations as ratios of determinants......Page 29
1.6 Triangular recursive schemes......Page 32
1.7 Normal forms of the algorithms......Page 37
1.8 Progressive forms of the algorithms......Page 39
1.9 Particular rules of the algorithms......Page 45
1.10 Accelerability and non-accelerability......Page 50
1.11 Optimality......Page 53
1.12 Asymptotic behaviour of sequences......Page 58
2.1 The E-algorithm......Page 66
2.2 Richardson extrapolation process......Page 83
2.3 The \epsilon-algorithm......Page 89
2.4 The G-transformation......Page 106
2.5 Rational extrapolation......Page 112
2.6 Generalizations of the \epsilon-algorithm......Page 119
2.7 Levin's transforms......Page 124
2.8 Overholt's process......Page 130
2.9 \Theta-type algorithms......Page 132
2.10 The iterated \Delta^2 process......Page 139
2.11 Miscellaneous algorithms......Page 142
3.1 Error estimates and acceleration......Page 156
3.2 Convergence tests and acceleration......Page 162
3.3 Construction of asymptotic expansions......Page 170
3.4 Construction of extrapolation processes......Page 176
3.5 Extraction procedures......Page 185
3.6 Automatic selection......Page 189
3.7 Composite sequence transformations......Page 196
3.8 Error control......Page 204
3.9 Contractive sequence transformations......Page 212
3.10 Least squares extrapolation......Page 221
4 Vector Extrapolation Algorithms......Page 224
4.1 The vector \epsilon-algorithm......Page 227
4.2 The topological \epsilon-algorithm......Page 231
4.3 The vector E-algorithm......Page 239
4.4 The recursive projection algorithm......Page 244
4.5 The H-algorithm......Page 249
4.6 The Ford-Sidi algorithms......Page 255
4.7 Miscellaneous algorithms......Page 258
5 Continuous Prediction Algorithms......Page 264
5.1 The Taylor expansion......Page 265
5.2 Confluent Overholt's process......Page 266
5.3 Confluent \epsilon-algorithms......Page 267
5.4 Confluent \rho-algorithm......Page 273
5.5 Confluent G-transform......Page 276
5.6 Confluent E-algorithm......Page 277
5.7 \Theta-type confluent algorithms......Page 278
6 Applications......Page 280
6.1.1 Simple sequences......Page 281
6.1.2 Double sequences......Page 289
6.1.3 Chebyshev and Fourier series......Page 293
6.1.4 Continued fractions......Page 295
6.1.5 Vector sequences......Page 309
6.2 Systems of equations......Page 313
6.2.1 Linear systems......Page 314
6.2.2 Projection methods......Page 318
6.2.3 Regularization and penalty techniques......Page 320
6.2.4 Nonlinear equations......Page 326
6.2.5 Continuation methods......Page 341
6.3 Eigenelements......Page 343
6.3.1 Eigenvalues and eigenvectors......Page 344
6.3.2 Derivatives of eigensystems......Page 347
6.4 Integral and differential equations......Page 349
6.4.1 Implicit Runge-Kutta methods......Page 350
6.4.2 Boundary value problems......Page 351
6.4.3 Nonlinear methods......Page 357
6.4.4 Laplace transform inversion......Page 359
6.4.5 Partial differential equations......Page 363
6.5 Interpolation and approximation......Page 365
6.6 Statistics......Page 368
6.6.1 The jackknife......Page 369
6.6.2 ARMA models......Page 370
6.6.3 Monte-Carlo methods......Page 372
6.7 Integration and differentiation......Page 376
6.7.1 Acceleration of quadrature formulae......Page 377
6.7.2 Nonlinear quadrature formulas......Page 383
6.7.3 Cauchy's principal values......Page 384
6.7.4 Infinite integrals......Page 389
6.7.5 Multiple integrals......Page 398
6.8 Prediction......Page 400
7.1 Programming the algorithms......Page 408
7.2 Computer arithmetic......Page 411
7.3 Programs......Page 414
Bibliography......Page 424
Index......Page 466