Details

Autor(en) / Beteiligte

• Bultheel, Adhemar
• Barel, Marc van,

Titel

Linear algebra, rational approximation, and orthogonal polynomials [electronic resource]

Link zum Volltext

• Elsevier Books AutoHoldings

Beschreibungen/Notizen

• Description based upon print version of record.
• Includes bibliographical references (p. 413-433) and index.
• Front Cover; Linear Algebra, Rational Approximation and Orthogonal Polynomials; Copyright Page; Contents; Preface; List of symbols; Chapter 1. Euclidean fugues; 1.1 The algorithm of Euclid; 1.2 Euclidean ring and g.c.l.d; 1.3 Extended Euclidean algorithm; 1.4 Continued fraction expansions; 1.5 Approximating formal series; 1.6 Atomic Euclidean algorithm; 1.7 Viscovatoff algorithm; 1.8 Layer peeling vs. layer adjoining methods; 1.9 Left-Riight duality; Chapter 2. Linear algebra of Hankels; 2.1 Conventions and notations; 2.2 Hankel matrices; 2.3 Tridiagonal matrices
• 2.4 Structured Hankel information2.5 Block Gram-Schmidt algorithm; 2.6 The Schur algorithm; 2.7 The Viscovatoff algorithm; Chapter 3. Lanczos algorithm; 3.1 Krylov spaces; 3.2 Biorthogonality; 3.3 The generic algorithm; 3.4 The Euclidean Lanczos algorithm; 3.5 Breakdown; 3.6 Note of warning; Chapter 4. Orthogonal polynomials; 4.1 Generalities; 4.2 Orthogonal polynomials; 4.3 Properties; 4.4 Hessenberg matrices; 4.5 Schur algorithm; 4.6 Rational approximation; 4.7 Generalization of Lanczos algorithm; 4.8 The Hankel case; 4.9 Toeplitz case; 4.10 Formal orthogonality on an algebraic curve
• Chapter 5. Pade approximation5.1 Definitions and terminology; 5.2 Computation of diagonal PAs; 5.3 Computation of antidiagonal PAs; 5.4 Computation of staircase PAs; 5.5 Minimal indices; 5.6 Minimal Padé approximation; 5.7 The Massey algorithm; Chapter 6. Linear systems; 6.1 Definitions; 6.2 More definitions and properties; 6.3 The minimal partial realization problem; 6.4 Interpretation of the Padé results; 6.5 The mixed problem; 6.6 Interpretation of the Toeplitz results; 6.7 Stability checks; Chapter 7. General rational interpolation; 7.1 General framework
• 7.2 Elementary updating and downdating steps7.3 A general recurrence step; 7.4 Padé approximation; 7.5 Other applications; Chapter 8. Wavelets; 8.1 Interpolating subdivisions; 8.2 Multiresolution; 8.3 Wavelet transforms; 8.4 The lifting scheme; 8.5 Polynomial formulation; 8.6 Euclidean domain of Laurent polynomials; 8.7 Factorization algorithm; Bibliography; List of Algorithms; Index
• Evolving from an elementary discussion, this book develops the Euclidean algorithm to a very powerful tool to deal with general continued fractions, non-normal Padé tables, look-ahead algorithms for Hankel and Toeplitz matrices, and for Krylov subspace methods. It introduces the basics of fast algorithms for structured problems and shows how they deal with singular situations. Links are made with more applied subjects such as linear system theory and signal processing, and with more advanced topics and recent results such as general bi-orthogonal polynomials, minimal Padé approximation, poly
• English