Computation with recurrence relations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston u.a.
Pitman Advanced Publ. Program
1984
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Applicable mathematics series
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 310 S. |
| ISBN: | 0273085085 |
Internformat
MARC
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| 245 | 1 | 0 | |a Computation with recurrence relations |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Boston u.a. |b Pitman Advanced Publ. Program |c 1984 | |
| 300 | |a XII, 310 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Applicable mathematics series | |
| 650 | 7 | |a Recursieve functies |2 gtt | |
| 650 | 4 | |a Approximation theory | |
| 650 | 4 | |a Functional differential equations | |
| 650 | 4 | |a Point mappings (Mathematics) | |
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Datensatz im Suchindex
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| adam_text | Contents
Preface xi
List of symbols xiii
1 Introduction 1
2 General results on the forward stability of recursion relations 4
2.1 Background 4
2.2 Homogeneous systems 6
2.3 Nonhomogeneous systems 11
2.4 The first order scalar case; forward vs. backward recursion 12
2.5 The computation of successive derivatives 14
2.6 Scalar equations of higher order: minimal and dominant
solutions 17
3 First order equations used in the backward direction: the Miller
algorithm 23
3.1 Introduction: the algorithm 23
3.2 Convergence and error analysis 25
4 Second order homogeneous equations: the Miller algorithm 29
4.1 The algorithm 29
4.2 Reduction of the error by the use of asymptotic
information 37
4.3 Error analysis, the simplified algorithm: case of negative
coefficients 38
4.4 Error analysis, the general algorithm 45
4.5 Algorithms based on continued fractions 46
4.6 Minimal solutions and orthogonal polynomials 53
4.7 The Clenshaw averaging process 57
5 Applications of the Miller algorithm to the computation of special
functions 61
5.1 The confluent hypergeometric function 4 (a, c; x) 61
5.2 The confluent hypergeometric function ^(a,c;x) 63
viii Contents
5.3 The Gaussian hypergeometric function 70
5.4 Associated Legendre functions 73
5.5 The Legendre function Q*(z) 75
5.6 The Jacobi polynomials P^iTl)( i u) 75
5.7 Bessel functions 78
5.8 Zeros of Bessel functions 82
5.9 Eigenvalues of Mathieu s equation 82
6 Second order nonhomogeneous equations: the Olver algorithm 86
6.1 Introduction: the algorithm 86
6.2 Solution by forward elimination (Method A) 90
6.3 The method of averaging (Method B) 94
6.4 The LU decomposition (Method C) 96
6.5 Adaptation to general normalizing conditions 98
6.6 Conclusions 103
7 Higher order systems: homogeneous equations 104
7.1 Observations on higher order equations 104
7.2 The Miller algorithm 105
7.3 The matrix formulation: stability and weak stability 114
7.4 The Clenshaw averaging process 122
!, 7.5 Topics for future research: infinite systems 128
f 7.5.1 Basic series for functions satisfying functional
] equations 128
7.5.2 Stieltjes moment integrals 131
8 Higher order systems (continued): the nonhomogeneous case 135
8.1 Introduction 135
8.2 The Wimp Luke method 137
8.3 The Lozier algorithm 139
9 The computation of 3F2(1) 152
9.1 The recursion 152
9.2 The algorithm; truncation error 154
9.3 Computing the Beta function 157
9.4 Another 3F2(1) 158
10 Computations based on orthogonal polynomials 161
10.1 Preliminaries: properties of some orthogonal poly¬
nomials 161
10.1.1 Chebyshev polynomials 161
10.1.2 Jacobi polynomials 163
10.2 Evaluation of finite sums of functions which satisfy a linear
homogeneous recurrence 165
Contents ix
10.2.1 The algorithm 165
10.2.2 Error analysis; three term recurrence 167
10.2.3 Converting one expansion into another 172
11 Series solutions to ordinary differential equations 177
11.1 Taylor series solutions 178
11.2 The construction of general recurrence relations for the coeffi¬
cients of Gegenbauer series 186
11.2.1 Introduction and basic formulas 186
11.2.2 The algorithms of Clenshaw and Elliott 188
11.2.3 The Lewanowicz construction 196
11.3 Chebyshev series solutions for nonlinear differential
equations 202
12 Multidimensional recursion algorithms; general theory 209
12.1 Background: convergence properties of complex
sequences 209
12.2 Introduction; invariants 211
12.3 Divergence; strange attractors 219
11 12.4 Mean values 223
13 Two dimensional algorithms 229
13.1 General remarks: invariant curves 229
13.2 Evaluation of certain infinite products 231
13.3 Carlson s results 237
13.4 An algorithm of Gatteschi 245
13.5 Tricomi s algorithm 248
13.6 Solutions of linear functional equations 249
14 Higher dimensional algorithms 254
14.1 The computation of a class of trigonometric integrals 254
14.2 Incomplete elliptic integrals 260
Appendix A The general theory of linear difference equations 265
Appendix B The asymptotic theory of linear difference
equations 270
B.I General theory 270
B.2 The construction of formal series solutions 272
B.3 The Olver growth theorems 282
Appendix C Recursion formulas for hypergeometric functions 289
Index of higher mathematical functions discussed 293
Bibliography 295
Index 307
|
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| discipline | Mathematik |
| edition | 1. publ. |
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| id | DE-604.BV000317575 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:12:11Z |
| institution | BVB |
| isbn | 0273085085 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000195042 |
| oclc_num | 9757202 |
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| owner_facet | DE-12 DE-384 DE-29T DE-91G DE-BY-TUM DE-739 DE-19 DE-BY-UBM DE-188 |
| physical | XII, 310 S. |
| publishDate | 1984 |
| publishDateSearch | 1984 |
| publishDateSort | 1984 |
| publisher | Pitman Advanced Publ. Program |
| record_format | marc |
| series2 | Applicable mathematics series |
| spelling | Wimp, Jet Verfasser aut Computation with recurrence relations 1. publ. Boston u.a. Pitman Advanced Publ. Program 1984 XII, 310 S. txt rdacontent n rdamedia nc rdacarrier Applicable mathematics series Recursieve functies gtt Approximation theory Functional differential equations Point mappings (Mathematics) Rekursive Funktion (DE-588)4138367-9 gnd rswk-swf Rekursionsformel (DE-588)4177668-9 gnd rswk-swf Algorithmus (DE-588)4001183-5 gnd rswk-swf Rekursive Funktion (DE-588)4138367-9 s Algorithmus (DE-588)4001183-5 s DE-604 Rekursionsformel (DE-588)4177668-9 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000195042&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wimp, Jet Computation with recurrence relations Recursieve functies gtt Approximation theory Functional differential equations Point mappings (Mathematics) Rekursive Funktion (DE-588)4138367-9 gnd Rekursionsformel (DE-588)4177668-9 gnd Algorithmus (DE-588)4001183-5 gnd |
| subject_GND | (DE-588)4138367-9 (DE-588)4177668-9 (DE-588)4001183-5 |
| title | Computation with recurrence relations |
| title_auth | Computation with recurrence relations |
| title_exact_search | Computation with recurrence relations |
| title_full | Computation with recurrence relations |
| title_fullStr | Computation with recurrence relations |
| title_full_unstemmed | Computation with recurrence relations |
| title_short | Computation with recurrence relations |
| title_sort | computation with recurrence relations |
| topic | Recursieve functies gtt Approximation theory Functional differential equations Point mappings (Mathematics) Rekursive Funktion (DE-588)4138367-9 gnd Rekursionsformel (DE-588)4177668-9 gnd Algorithmus (DE-588)4001183-5 gnd |
| topic_facet | Recursieve functies Approximation theory Functional differential equations Point mappings (Mathematics) Rekursive Funktion Rekursionsformel Algorithmus |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000195042&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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