| Description |
1 online resource (xvii, 446 pages) : illustrations |
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text txt rdacontent |
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computer c rdamedia |
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online resource cr rdacarrier |
| Series |
Studies in computational mathematics ; 6 |
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Studies in computational mathematics ; 6.
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| Summary |
Evolving from an elementary discussion, this book develops the Euclidean algorithm to a very powerful tool to deal with general continued fractions, non-normal Pa 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 Pa approximation, polynomial root location problems in the complex plane, very general rational interpolation problems, and the lifting scheme for wavelet transform computation. The text serves as a supplement to existing books on structured linear algebra problems, rational approximation and orthogonal polynomials. Features of this book: & bull; provides a unifying approach to linear algebra, rational approximation and orthogonal polynomials & bull; requires an elementary knowledge of calculus and linear algebra yet introduces advanced topics. The book will be of interest to applied mathematicians and engineers and to students and researchers. |
| Bibliography |
Includes bibliographical references (pages 413-433) and index. |
| Note |
Print version record. |
| Contents |
Cover -- 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 information -- 2.5 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 approximation -- 5.1 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 steps -- 7.3 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 -- Last Page. |
| Subject |
Euclidean algorithm.
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Algebras, Linear.
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Padé approximant.
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Orthogonal polynomials.
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Algorithme d'Euclide.
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Algèbre linéaire.
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Approximants de Padé.
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Polynômes orthogonaux.
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MATHEMATICS -- Number Theory.
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Algebras, Linear
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Euclidean algorithm
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Orthogonal polynomials
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Padé approximant
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| Added Author |
Barel, Marc van, 1960-
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| Other Form: |
Print version: Bultheel, Adhemar. Linear algebra, rational approximation, and orthogonal polynomials. Amsterdam ; New York : Elsevier, 1997 0444828729 9780444828729 (DLC) 97040610 (OCoLC)37682686 |
| ISBN |
9780444828729 |
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0444828729 |
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9780080535524 (electronic bk.) |
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0080535526 (electronic bk.) |
| Standard No. |
AU@ 000048130700 |
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CHNEW 001005822 |
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DEBBG BV036962261 |
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DEBBG BV042317275 |
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DEBSZ 276841433 |
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DEBSZ 482352760 |
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NZ1 12432924 |
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NZ1 15192838 |
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