Quadrature theory: the theory of numerical integration on a compact interval
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, R.I.
American Math. Soc.
2011
|
| Series: | Mathematical surveys and monographs
178 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. 335-356) and index |
| Physical Description: | VII, 363 S. |
| ISBN: | 9780821853610 0821853619 |
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| 245 | 1 | 0 | |a Quadrature theory |b the theory of numerical integration on a compact interval |c Helmut Brass ; Knut Petras |
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Record in the Search Index
| _version_ | 1804149414533529600 |
|---|---|
| adam_text | Titel: Quadrature theory
Autor: Braß, Helmut
Jahr: 2011
Contents
Preface vii
Chapter 1. Introduction 1
Chapter 2. The Abstract Framework 7
2.1. Standard estimation framework 7
2.2. Linear rules that are exact on a subspace 16
2.3. Strong optimality: inner product Spaces 21
2.4. Varying the Observation 26
Chapter 3. Norm and Kernel of the Remainder Functional 29
3.1. Norm of an estimation rule 29
3.2. The interpolation theorem 33
3.3. Quadrature formulas and one-sided approximation 36
3.4. Krein s theory 40
Chapter 4. Co-observations 47
4.1. Survey 47
4.2. The Peano kernel theorem 50
4.3. Bounded derivatives as co-observation 59
4.4. Bounded Variation as co-observation 63
4.5. Error bounds using the modulus of continuity 66
4.6. Derivatives of bounded Variation 72
4.7. Sard s co-observation 73
4.8. Co-observations of Davis type 76
4.9. Bounds in the complex plane as co-observations 82
4.10. Convex functions 93
Chapter 5. Quadrature Rules of Interpolatory Type 99
5.1. Recapitulation 99
5.2. The Newton-Cotes method 104
5.3. A theorem of Sloan and Smith 110
5.4. Error bounds for the Clenshaw-Curtis method 114
5.5. Relatives of the Clenshaw-Curtis method 121
5.6. The distribution of nodes 130
5.7. Bounds for the norms of Peano kerneis of interpolatory rules 135
5.8. Asymptotic behaviour of a class of Peano kerneis 144
Chapter 6. Gaussian Quadrature 149
6.1. Rules of high degree and orthogonal polynomials 149
6.2. Coefficients and nodes for general weights 156
vi CONTENTS
6.3. Nodes and coefficients for w = 1 162
6.4. Peano kerneis for general weights 166
6.5. Peano kerneis for w = 1 173
6.6. Error bounds 180
6.7. Asymptotics of the error 189
6.8. Extremal properties of Gaussian rules 195
6.9. Why Gaussian quadrature? 197
6.10. The Kronrod method 201
6.11. Kronrod rules for w - 1 204
Chapter 7. Quadrature Rules with Equidistant Nodes 211
7.1. The trapezoidal method and the Euler-Maclaurin formula 211
7.2. More on the trapezoidal method 217
7.3. Simpson s method 230
7.4. The Filon method 233
7.5. Gregory methods 236
7.6. Romberg methods 244
7.7. Equidistant nodes and the degree of polynomial exactness 254
7.8. The midpoint method 256
Chapter 8. Periodic Integrands 261
8.1. The special role of the trapezoidal rule for w - 1 261
8.2. Error bounds for the trapezoidal rule 264
8.3. Trigonometrie Interpolation 271
8.4. Universality 273
8.5. Standard rules for Fourier coefficients 276
8.6. Other rules for Fourier coefficients 283
Chapter 9. Variance and Chebyshev-type Rules 291
9.1. Fundamentals 291
9.2. Chebyshev methods 296
9.3. The special case of w = 1 299
9.4. Variance 303
Chapter 10. Problems 307
Appendix A. Orthogonal Polynomials 315
Appendix B. Bernoulli Polynomials 325
Appendix C. Validation of Co-observations 329
C.l. Automatic generation of Taylor coefficients 329
C.2. Real interval arithmetic 331
C.3. Complex interval arithmetic 333
Bibliography 335
Books on quadrature 335
References 336
Symbols 357
Index 361
|
| any_adam_object | 1 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV040368706 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T00:22:39Z |
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| language | English |
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| physical | VII, 363 S. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
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| publisher | American Math. Soc. |
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| series | Mathematical surveys and monographs |
| series2 | Mathematical surveys and monographs |
| spelling | Brass, Helmut Verfasser aut Quadrature theory the theory of numerical integration on a compact interval Helmut Brass ; Knut Petras Providence, R.I. American Math. Soc. 2011 VII, 363 S. txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs 178 Includes bibliographical references (p. 335-356) and index Gaussian quadrature formulas Numerical integration Numerische Integration (DE-588)4172168-8 gnd rswk-swf Gauß-Quadraturformel (DE-588)4156108-9 gnd rswk-swf Numerische Integration (DE-588)4172168-8 s Gauß-Quadraturformel (DE-588)4156108-9 s DE-604 Petras, Knut Verfasser aut Erscheint auch als Online-Ausgabe 978-1-4704-1405-4 Mathematical surveys and monographs 178 (DE-604)BV000018014 178 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025222358&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Brass, Helmut Petras, Knut Quadrature theory the theory of numerical integration on a compact interval Mathematical surveys and monographs Gaussian quadrature formulas Numerical integration Numerische Integration (DE-588)4172168-8 gnd Gauß-Quadraturformel (DE-588)4156108-9 gnd |
| subject_GND | (DE-588)4172168-8 (DE-588)4156108-9 |
| title | Quadrature theory the theory of numerical integration on a compact interval |
| title_auth | Quadrature theory the theory of numerical integration on a compact interval |
| title_exact_search | Quadrature theory the theory of numerical integration on a compact interval |
| title_full | Quadrature theory the theory of numerical integration on a compact interval Helmut Brass ; Knut Petras |
| title_fullStr | Quadrature theory the theory of numerical integration on a compact interval Helmut Brass ; Knut Petras |
| title_full_unstemmed | Quadrature theory the theory of numerical integration on a compact interval Helmut Brass ; Knut Petras |
| title_short | Quadrature theory |
| title_sort | quadrature theory the theory of numerical integration on a compact interval |
| title_sub | the theory of numerical integration on a compact interval |
| topic | Gaussian quadrature formulas Numerical integration Numerische Integration (DE-588)4172168-8 gnd Gauß-Quadraturformel (DE-588)4156108-9 gnd |
| topic_facet | Gaussian quadrature formulas Numerical integration Numerische Integration Gauß-Quadraturformel |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025222358&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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