Linear Algebra, Rational Approximation and Orthogonal Polynomials (eBook)

Cover 1
Contents 12
Preface 6
List of symbols 16
Chapter 1. Euclidean fugues 20
1.1 The algorithm of Euclid 20
1.2 Euclidean ring and g.c.l.d 22
1.3 Extended Euclidean algorithm 28
1.4 Continued fraction expansions 34
1.5 Approximating formal series 40
1.6 Atomic Euclidean algorithm 51
1.7 Viscovatoff algorithm 56
1.8 Layer peeling vs. layer adjoining methods 71
1.9 Left-Riight duality 73
Chapter 2. Linear algebra of Hankels 80
2.1 Conventions and notations 80
2.2 Hankel matrices 82
2.3 Tridiagonal matrices 93
2.4 Structured Hankel information 100
2.5 Block Gram-Schmidt algorithm 103
2.6 The Schur algorithm 104
2.7 The Viscovatoff algorithm 114
Chapter 3. Lanczos algorithm 118
3.1 Krylov spaces 119
3.2 Biorthogonality 120
3.3 The generic algorithm 123
3.4 The Euclidean Lanczos algorithm 124
3.5 Breakdown 134
3.6 Note of warning 151
Chapter 4. Orthogonal polynomials 154
4.1 Generalities 154
4.2 Orthogonal polynomials 157
4.3 Properties 168
4.4 Hessenberg matrices 170
4.5 Schur algorithm 174
4.6 Rational approximation 178
4.7 Generalization of Lanczos algorithm 185
4.8 The Hankel case 188
4.9 Toeplitz case 197
4.10 Formal orthogonality on an algebraic curve 245
Chapter 5. Pade approximation 250
5.1 Definitions and terminology 251
5.2 Computation of diagonal PAs 254
5.3 Computation of antidiagonal PAs 260
5.4 Computation of staircase PAs 267
5.5 Minimal indices 269
5.6 Minimal Padé approximation 273
5.7 The Massey algorithm 284
Chapter 6. Linear systems 290
6.1 Definitions 291
6.2 More definitions and properties 306
6.3 The minimal partial realization problem 311
6.4 Interpretation of the Padé results 317
6.5 The mixed problem 319
6.6 Interpretation of the Toeplitz results 323
6.7 Stability checks 325
Chapter 7. General rational interpolation 370
7.1 General framework 370
7.2 Elementary updating and downdating steps 379
7.3 A general recurrence step 384
7.4 Padé approximation 385
7.5 Other applications 399
Chapter 8. Wavelets 404
8.1 Interpolating subdivisions 404
8.2 Multiresolution 409
8.3 Wavelet transforms 414
8.4 The lifting scheme 417
8.5 Polynomial formulation 421
8.6 Euclidean domain of Laurent polynomials 426
8.7 Factorization algorithm 428
Bibliography 432
List of Algorithms 454
Index 455