Handbook of computational methods for integration:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
Chapman & Hall/CRC
2005
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 598 S. graph. Darst. 1 CD-ROM (12 cm) |
| ISBN: | 1584884282 |
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MARC
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| 100 | 1 | |a Kythe, Prem K. |d 1930- |e Verfasser |0 (DE-588)12072362X |4 aut | |
| 245 | 1 | 0 | |a Handbook of computational methods for integration |c Prem K. Kythe ; Michael R. Schäferkotter |
| 246 | 1 | 3 | |a Computational methods for integration |
| 264 | 1 | |a Boca Raton [u.a.] |b Chapman & Hall/CRC |c 2005 | |
| 300 | |a XXII, 598 S. |b graph. Darst. |e 1 CD-ROM (12 cm) | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Numerieke methoden |2 gtt | |
| 650 | 4 | |a Numerical integration | |
| 650 | 4 | |a Integrals | |
| 650 | 4 | |a Orthogonal polynomials | |
| 650 | 0 | 7 | |a Numerische Integration |0 (DE-588)4172168-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Numerische Integration |0 (DE-588)4172168-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Schäferkotter, Michael R. |e Verfasser |4 aut | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013147938&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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| _version_ | 1804133325944651776 |
|---|---|
| adam_text | Contents
Preface ............................................................. xiii
Notation .............................................................xv
1 Preliminaries ...................................................... 1
1.1 Notation and Definitions........................................... 1
1.2 Orthogonal Polynomials...........................................3
1.2.1 Chebyshev Polynomials of the First Kind......................5
1.2.2 Chebyshev Polynomials of the Second Kind...................7
1.2.3 Gegenbauer (or Ultraspherical) Polynomials...................7
1.2.4 Hermite Polynomials........................................8
1.2.5 Jacobi Polynomials ......................................... 8
1.2.6 Laguerre Polynomials.......................................9
1.2.7 Generalized Laguerre Polynomials........................... 9
1.2.8 Legendre Polynomials ..................................... 10
1.2.9 Shifted Legendre Polynomials .............................. 10
1.2.10 Shifted Chebyshev Polynomials............................11
1.2.11 Shifted Jacobi Polynomials................................ 12
1.3 Finite and Divided Differences.................................... 13
1.3.1 Finite Differences ......................................... 13
1.3.2 Divided Differences ....................................... 15
1.3.3 Newton-Gregory Formula.................................. 16
1.4 Interpolation .................................................... 18
1.4.1 Lagrangian Polynomial .................................... 18
1.4.2 Neville s Method..........................................20
1.4.3 Osculating Polynomials ....................................21
1.4.4 Richardson Extrapolation...................................22
1.4.5 Error Bounds in Polynomial Interpolation....................23
1.4.6 Newton-Gregory Forward Polynomial ....................... 25
vi CONTENTS
1.5 Semi-Infinite Interval.............................................26
1.5.1 Linear Interpolation........................................26
1.5.2 Quadratic Interpolation.....................................26
1.5.3 Cubic Interpolation........................................27
1.5.3(a) Cubic Interpolation with Four Single Points ............ 27
1.5.3(b) Cubic Interpolation with Two Double Points............27
1.5.3(c) Cubic Interpolation with Three Double Points .......... 28
1.5.4 Transformation to the interval [0,1]..........................28
1.6 Convergence Accelerators ........................................ 31
1.6.1 Fixed-Point Iteration.......................................31
1.6.2 Aitkin s A2 Process........................................32
1.6.3 Shanks Process............................................34
1.6.4 e-Algorthm ............................................... 36
1.6.5 Levin s Transformation.................................... 37
1.6.6 i-Transformation .......................................... 37
1.6.7 D- and ^-Transformations.................................38
1.7 Polynomial Splines .............................................. 39
2 Interpolatory Quadrature ....................................... 44
2.1 Riemann Integration ............................................. 44
2.1.1 Approximate Integration ................................... 45
2.2 Euler-Maclaurin Expansion.......................................49
2.2.1 Darboux s Formula........................................49
2.2.2 Bernoulli Numbers and Bernoulli Polynomials................50
2.2.3 Euler-Maclaurin Expansion Formula.........................50
2.3 Interpolatory Quadrature Rules....................................52
2.3.1 General Integration Rules .................................. 52
2.3.2 Lagrange Formula.........................................54
2.4 Newton-Cotes Formulas..........................................54
2.4.1 Closed-Type Newton-Cotes Rules........................... 58
2.4.2 Open-Type Newton-Cotes Rules ............................ 60
2.5 Basic Quadrature Rules...........................................63
2.6 Repeated Quadrature Rules ....................................... 69
2.6.1 An Initial Value Problem................................... 72
2.7 Romberg s Scheme .............................................. 75
2.8 Gregory s Correction Scheme..................................... 82
2.9 Interpolatory Product Integration .................................. 83
2.9.1 Application to Integral Equations............................ 88
2.10 Iterative and Adaptive Schemes .................................. 89
2.10.1 An Iterative Method for Simpson s Rule .................... 90
2.10.2 An Adaptive Quadrature with Trapezoidal Rule..............91
2.10.3 An Adaptive Method Based on Simpson s Rule..............94
2.10.4 Convergence.............................................95
2.11 Test Integrals..................................................101
CONTENTS vii
3 Gaussian Quadrature .......................................... 108
3.1 Gaussian Rules................................................. 108
3.2 Extended Gaussian Rules........................................ 112
3.2.1 Gauss-Jacobi Rule........................................113
3.2.2 Gauss-Legendre Rule..................................... 115
3.2.3 Gauss-Laguerre Rule ..................................... 118
3.2.4 Gauss-Hermite Rule...................................... 118
3.2.5 Gauss-Radau Rule........................................119
3.2.6 Gauss-Lobatto Rule ...................................... 120
3.2.7 Gauss-Chebyshev Rule....................................120
3.2.8 Gauss-Log Rule.......................................... 124
3.3 Other Extended Rules........................................... 125
3.3.1 Clenshaw-Curtis Rules.................................... 125
3.3.2 Gauss-Kronrod Rule...................................... 131
3.3.3 Patterson s Rules......................................... 133
3.3.4 Basu Rule ............................................... 134
3.4 Analytic Functions.............................................. 136
3.4.1 Symmetric Weight Functions .............................. 137
3.4.2 Optimal Quadrature Rules.................................139
3.5 Bessel s Rule...................................................140
3.5.1 Extrapolation ............................................ 143
3.5.2 Nodes inside the Integration Interval........................145
3.6 Gaussian Rules for the Moments ................................. 148
3.6.1 Gauss-Jacobi Moments ................................... 148
3.6.2 Gauss-Legendre Moments................................. 148
3.6.3 Gauss-Legendre Moments on an Arbitrary Interval........... 149
3.6.4 Gauss-Chebyshev Moments ............................... 149
3.6.5 Gauss-Chebyshev Moments on an Arbitrary Interval ......... 150
3.6.6 Weights and Modified Moments............................ 150
3.6.7 Lozier s Algorithm ....................................... 160
3.6.8 Values of the Moments.................................... 160
3.7 Finite Oscillatory Integrals....................................... 161
3.7.1 Filon s Formulas for Trigonometric Functions ............... 161
3.7.2 Use of Chebyshev Polynomials and Bessel Functions.........164
3.7.3 Nontrigonometric Oscillatory Functions .................... 169
3.8 Noninterpolatory Product Integration ............................. 170
3.8.1 Approximation by Fourier Series...........................175
3.8.2 Asymptotic Error Estimates............................... 176
3.9 Test Integrals...................................................180
4 Improper Integrals ............................................. 187
4.1 Infinite Range Integrals..........................................187
4.1.1 Truncation Method....................................... 187
4.1.2 Gaussian Rules...........................................188
viii CONTENTS
4.1.3 Mapped Finite Range Rules ............................... 192
4.2 Improper Integrals .............................................. 193
4.2.1 Method of Variable Transformations........................197
4.2.2 Types of Inherent Errors...................................200
4.3 Slowly Convergent Integrals .....................................203
4.4 Oscillatory Integrals ............................................ 207
4.4.1 Summation, then Integration...............................207
4.4.2 Pantis Method...........................................209
4.4.3 Piessens Method.........................................211
4.4.4 Price s Method...........................................213
4.4.5 Use of Sine Transform....................................214
4.4.6 Test Integrals............................................ 214
4.5 Product Integration............................................. 216
5 Singular Integrals...............................................218
5.1 Quadrature Rules...............................................218
5.1.1 Gauss-Christoffel Rules...................................220
5.1.2 Logarithmic Weight Function..............................225
5.1.3 Singular Integrals ........................................ 226
5.1.4 Singularities near and on the Real Axis ..................... 227
5.1.5 Branch Singularities ......................................229
5.1.6 Gautschi s Computation Codes.............................230
5.1.7 Rational Gaussian Formulas...............................230
5.2 Product Integration ............................................. 232
5.2.1 Endpoint Singularities .................................... 234
5.2.2 Interior Singularities......................................235
5.3 Acceleration Methods...........................................237
5.3.1 Endpoint Singularity......................................237
5.3.2 Logarithmic Singularity...................................237
5.3.3 Levin s Transformations .................................. 240
5.3.4 Sidi s Rules..............................................242
5.3.5 Infinite Range Integration..................................245
5.4 Singular and Hypersingular Integrals..............................247
5.4.1 P.V. of a Singular Integral on a Contour.....................251
5.4.2 Application to Cauchy P.V. Integrals........................253
5.4.3 Hadamard s Finite-Part Integrals...........................254
5.4.4 Two-Sided Finite-Part Integrals............................255
5.4.5 One-Sided Finite-Part Integrals........................----256
5.5 Computer-Aided Derivations.....................................258
5.5.1 Gauss-Chebyshev Rule....................................260
5.5.2 Gauss-Legendre Rule.....................................261
5.5.3 Singular Point as a Node ..................................261
6 Fourier Integrals and Transforms ............................. 264
6.1 Fourier Transforms............................................. 264
CONTENTS ix
6.1.1 Use of Simpson s Rule....................................268
6.1.2 Uncertainty Principle ..................................... 269
6.1.3 Short Time Fourier Transform ............................. 270
6.2 Interpolatory Rules for Fourier Integrals...........................271
6.2.1 Spline Interpolatory Rules.................................272
16.2.2 Hurwitz-Zweifel s Method...................................282
6.3 Interpolatory Rules by Rational Functions.........................286
6.3.1 Equally Spaced Interpolation Points........................287
6.3.2 Computation of the Integrals...............................288
6.3.3 High-Precision Formulas.................................. 289
6.4 Trigonometric Integrals..........................................291
6.4.1 Basic Interpolatory Formulas ..............................292
6.4.2 Euler Expansion..........................................293
6.5 Finite Fourier Transforms........................................296
6.6 Discrete Fourier Transforms..................................... 299
6.6.1 Discrete Sine and Cosine Transforms.......................301
6.6.2 Radix 2 FFT.............................................303
6.6.3 FFT Procedures..........................................307
6.6.4 FFT Algorithm...........................................312
6.6.5 Linear Multistep Method..................................312
6.6.6 Computation of Fourier Coefficients........................314
6.6.7 Filon-Luke Rule..........................................315
6.7 Hartley Transform..............................................315
6.7.1 Comparison of the FFT and FHT Algorithms................321
6.7.2 FHT2D Programs ........................................ 322
7 Inversion of Laplace Transforms .............................. 323
7.1 Use of Orthogonal Polynomials .................................. 323
7.2 Interpolatory Methods...........................................328
7.2.1 Use of Legendre Polynomials..............................329
7.2.1(a) Papoulis Method...................................329
7.2.1(b) Lanczos Method...................................330
7.2.2 Delta Function Methods....................................331
7.2.2(a) Cost s Method ..................................... 331
7.2.2(b) Alfrey s Approximation.............................331
7.2.2(c) ter Haar s Formula..................................332
7.2.2(d) Schapery s Approximation .......................... 332
7.2.2(e) Least Squares Method...............................332
7.2.3 Use of Chebyshev Polynomials ............................333
7.2.3(a) Lanczos Method...................................333
7.2.3(b) Papoulis Method .................................. 334
7.2.4 Use of Jacobi Polynomials ................................ 337
7.2.4(a) Piessens Method...................................337
7.2.5 Use of Laguerre Polynomials ..............................340
CONTENTS
7.2.5(a) Lanczos Method...................................340
7.2.5(b) Papoulis Method .................................. 341
7.2.5(c) Weeks Method .................................... 342
7.2.5(d) Piessens Method...................................347
7.2.5(e) Lyness and Giunta s Method.........................349
7.2.5(f) de Hoog s Method..................................350
7.3 Use of Gaussian Quadrature Rules................................351
7.3.1 Bellman-Kalaba-Lockett s Method.........................351
7.3.2 Felts and Cook s Method..................................354
7.3.3 Piessens Method.........................................355
7.3.4 Kajiwara and Tsuji s Method..............................359
7.4 Use of Fourier Series............................................361
7.4.1 Dubner and Abate s Method...............................361
7.4.2 Crump s Improvement....................................364
7.4.3 Durbin s Improvement....................................370
7.4.4 Discretization Errors......................................376
7.4.5 Use of FFT Algorithm....................................379
7.4.6 Improvement of the FFT Algorithm ........................ 382
7.4.7 Use of the FHT Algorithm.................................384
7.5 Use of Bromwich Contours......................................387
7.5.1 Talbot s Method..........................................387
7.5.2 Duffy s Method ..........................................391
7.6 Inversion by the Riemann Sum...................................392
7.7 New Exact Laplace Inverse Transforms ........................... 394
Wavelets ......................................................... 396
8.1 Orthogonal Systems.............................................396
8.2 Trigonometric System...........................................398
8.3 Haar System...................................................401
8.3.1 Extension to the Real Axis................................ 405
8.3.2 Wavelet Transform .......................................408
8.3.3 Ordered Fast Haar Wavelet Transform......................409
8.4 Other Wavelet Systems..........................................412
8.4.1 Schauder System.........................................412
8.4.2 Shannon System......................................... 412
8.4.3 Franklin System..........................................414
8.4.4 Strömberg s System...................................... 415
8.4.5 Lusin s System...........................................416
8.4.6 Grossmann and Mortel s System...........................417
8.4.7 Hat Wavelet System......................................417
8.4.8 Quadratic Battle-Lemarié System .......................... 418
8.4.9 Mexican Hat Wavelet System..............................418
8.4.10 Meyer System .......................................... 418
8.4.11 Wavelet Analysis and Synthesis...........................419
CONTENTS xi
8.5 Daubechies System ............................................ 420
8.5.1 Approximation of Functions...............................425
8.6 Fast Daubechies Transforms .....................................431
9 Integral Equations .............................................. 436
9.1 Nyström System................................................436
9.2 Integral Equations of the First Kind...............................440
9.2.1 Use of Numerical Quadrature..............................440
9.2.2 Regularization Methods...................................443
9.2.2(a) Phillips Method ................................... 443
9.2.2(b) Twomey s Method..................................445
9.2.2(c) Caldwell s Method ................................. 446
9.3 Integral Equations of the Second Kind ............................448
9.3.1 Quadrature Methods......................................449
9.3.2 Modified Quadrature Methods .............................452
9.3.3 Quadrature Formulas ..................................... 453
9.3.4 Error Analysis ...........................................456
9.3.5 Expansion Methods.......................................460
9.3.5(a) Deferred Correction Method.........................461
9.3.5(b) Iterative Deferred Correction Scheme.................463
9.3.6 Product Integration Method................................466
9.4 Singular Integral Equations......................................468
9.4.1 Singularities in Linear Equations...........................468
9.5 Weakly Singular Equations ......................................471
9.5.1 Product Integration Methods...............................472
9.5.2 Asymptotic Expansions...................................479
9.5.3 Modified Quadrature Rule.................................481
9.6 Cauchy Singular Equations of the First Kind.......................484
9.6.1 Use of Trigonometric Polynomials ......................... 485
9.7 Cauchy Singular Equations of the Second Kind ....................488
9.7.1 Gauss-Jacobi Quadrature..................................491
9.7.2 Solution by Jacobi Polynomials............................492
9.8 Canonical Equation.............................................494
9.9 Finite-Part Singular Equations ................................... 498
9.10 Integral Equations over a Contour...............................501
9.10.1 Boundary Element Method...............................502
A Quadrature Tables ............................................. 503
A.I Cotesian Numbers, Tabulated for k n/2, n = 1(1)11 ............ 503
A.2 Weights for a Single Trapezoidal Rule and Repeated Simpson s Rule 504
A.3 Weights for Repeated Simpson s Rule and a Single Trapezoidal Rule 504
A.4 Weights for a Single 3/8-Rule and Repeated Simpson s Rule........504
A.5 Weights for Repeated Simpson s Rule and a Single 3/8-Rule.........505
A.6 Gauss-Legendre Quadrature.....................................506
A.7 Gauss-Laguerre Quadrature ..................................... 507
xii CONTENTS
A.8 Gauss-Hermite Quadrature......................................508
A.9 Gauss-Radau Quadrature........................................510
A.10 Gauss-Lobatto Quadrature .....................................512
A.I 1 Nodes of Equal-Weight Chebyshev Rule.........................513
A.12 Gauss-Log Quadrature.........................................513
A.13 Gauss-Kronrod Quadrature Rule................................515
A.14 Patterson s Quadrature Rule....................................516
A.15 Filon s Quadrature Formula....................................520
A.16 Gauss-Cos Quadrature on [ît/2, tt/2] ............................520
A.17 Gauss-Cos Quadrature on [0, tt/2] .............................. 521
A.18 Coefficients in (5.1.15) with w(x) = ln(l/a;), 0 x 1 ......... 522
A.19 Coefficients in (5.1.15) with w{x) = Ei(x), 0 x oo..........523
A.20 10-point Gauss-Christoffel Rule with w(x) = lr^l/a;), 0 x 1 . 523
A.21 20-point Gauss-Christoffel Rule with w(x) = ln(l/x), 0 x 1 . 524
A.22 10-point Gauss-Christoffel Rule with w(x) = Ei(x), 0 x oo .. 524
A.23 20-point Gauss-Christoffel Rule with w(x) = Ei(x), 0 x oo .. 525
A.24 Sidi s Nodes xk,i for w(x) = (1 - x)ax0 (log xf ............... 526
A.25 Nodes xk,i for Sidi s Rule Sk...................................528
A.26 Nodes xkA for Sidi s Rule Sk...................................530
A.27 Sidi s Quadrature Rules........................................532
A.28 Values of Bj defined in (7.2.5)..................................537
A.29 Values of Cn, n = 0(1)10, defined by (7.2.17) ................... 537
A.30 Coefficients of t [N)(x)........................................538
A.31 Nodes and Weights for the G^-Rule............................539
A.32 Values of p0..................................................539
B Figures .......................................................... 540
C Contents of the CD-R......................................... 547
Bibliography ....................................................... 551
Index ............................................................... 585
|
| any_adam_object | 1 |
| author | Kythe, Prem K. 1930- Schäferkotter, Michael R. |
| author_GND | (DE-588)12072362X |
| author_facet | Kythe, Prem K. 1930- Schäferkotter, Michael R. |
| author_role | aut aut |
| author_sort | Kythe, Prem K. 1930- |
| author_variant | p k k pk pkk m r s mr mrs |
| building | Verbundindex |
| bvnumber | BV019822689 |
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| callnumber-label | QA299 |
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| callnumber-search | QA299.3 |
| callnumber-sort | QA 3299.3 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 910 |
| classification_tum | MAT 655f |
| ctrlnum | (OCoLC)56414086 (DE-599)BVBBV019822689 |
| dewey-full | 518/.54 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518/.54 |
| dewey-search | 518/.54 |
| dewey-sort | 3518 254 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV019822689 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T20:06:56Z |
| institution | BVB |
| isbn | 1584884282 |
| language | English |
| lccn | 2004058208 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013147938 |
| oclc_num | 56414086 |
| open_access_boolean | |
| owner | DE-703 DE-20 DE-91G DE-BY-TUM DE-634 DE-11 |
| owner_facet | DE-703 DE-20 DE-91G DE-BY-TUM DE-634 DE-11 |
| physical | XXII, 598 S. graph. Darst. 1 CD-ROM (12 cm) |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Chapman & Hall/CRC |
| record_format | marc |
| spelling | Kythe, Prem K. 1930- Verfasser (DE-588)12072362X aut Handbook of computational methods for integration Prem K. Kythe ; Michael R. Schäferkotter Computational methods for integration Boca Raton [u.a.] Chapman & Hall/CRC 2005 XXII, 598 S. graph. Darst. 1 CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Numerieke methoden gtt Numerical integration Integrals Orthogonal polynomials Numerische Integration (DE-588)4172168-8 gnd rswk-swf Numerische Integration (DE-588)4172168-8 s DE-604 Schäferkotter, Michael R. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013147938&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kythe, Prem K. 1930- Schäferkotter, Michael R. Handbook of computational methods for integration Numerieke methoden gtt Numerical integration Integrals Orthogonal polynomials Numerische Integration (DE-588)4172168-8 gnd |
| subject_GND | (DE-588)4172168-8 |
| title | Handbook of computational methods for integration |
| title_alt | Computational methods for integration |
| title_auth | Handbook of computational methods for integration |
| title_exact_search | Handbook of computational methods for integration |
| title_full | Handbook of computational methods for integration Prem K. Kythe ; Michael R. Schäferkotter |
| title_fullStr | Handbook of computational methods for integration Prem K. Kythe ; Michael R. Schäferkotter |
| title_full_unstemmed | Handbook of computational methods for integration Prem K. Kythe ; Michael R. Schäferkotter |
| title_short | Handbook of computational methods for integration |
| title_sort | handbook of computational methods for integration |
| topic | Numerieke methoden gtt Numerical integration Integrals Orthogonal polynomials Numerische Integration (DE-588)4172168-8 gnd |
| topic_facet | Numerieke methoden Numerical integration Integrals Orthogonal polynomials Numerische Integration |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013147938&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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