Quantifier elimination and cylindrical algebraic decomposition
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| Format: | Buch |
|---|---|
| Sprache: | German |
| Veröffentlicht: |
Wien [u.a.]
Springer
1998
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| Schriftenreihe: | Texts and monographs in symbolic computation
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
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| 245 | 1 | 0 | |a Quantifier elimination and cylindrical algebraic decomposition |c B. F. Caviness ... (eds.) |
| 264 | 1 | |a Wien [u.a.] |b Springer |c 1998 | |
| 300 | |a XIX, 431 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Texts and monographs in symbolic computation | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Algebra |x Data processing |v Congresses | |
| 650 | 4 | |a Algorithms |v Congresses | |
| 650 | 4 | |a Decomposition method |x Data processing |v Congresses | |
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| 700 | 1 | |a Caviness, Bob F. |e Sonstige |4 oth | |
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Datensatz im Suchindex
| DE-19_call_number | 1601/ST 600 C382 |
|---|---|
| DE-19_location | 95 |
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| DE-BY-TUM_katkey | 932486 |
| DE-BY-TUM_location | 01 |
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| DE-BY-UBM_katkey | 2273010 |
| DE-BY-UBM_media_number | 41900097310014 |
| _version_ | 1823051941071028225 |
| adam_text | Contents
List of Contributors xviii
Introduction 1
1 Introduction to the Method 1
2 Importance of QE and CAD Algorithms 2
3 Alternative Approaches 6
4 Practical Issues 7
Acknowledgments 7
Quantifier Elimination by Cylindrical Algebraic Decomposition
Twenty Years of Progress 8
George E. Collins
1 Introduction 8
2 Original Method 8
3 Adjacency and Clustering 10
4 Improved Projection 11
5 Partial CADs 12
6 Interactive Implementation 14
7 Solution Formula Construction 15
8 Equational Constraints 17
9 Subalgorithms 19
10 Future Improvements 21
A Decision Method for Elementary Algebra and Geometry 24
Alfred Tarski
1 Introduction 25
2 The System of Elementary Algebra 31
3 Decision Method for Elementary Algebra 38
4 Extensions to Related Systems 66
5 Notes 69
6 Supplementary Notes 81
xiv Contents
Quantifier Elimination for Real Closed Fields by Cylindrical
Algebraic Decomposition 85
George E. Collins
1 Introduction 85
2 Algebraic Foundations 88
3 The Main Algorithm 96
4 Algorithm Analysis 105
5 Observations 116
Super Exponential Complexity of Presburger Arithmetic 122
Michael J. Fischer and Michael O. Rabin
1 Introduction and Main Theorems 122
2 Algorithms 124
3 Method for Complexity Proofs 125
4 Proof of Theorem 3 (Real Addition) 129
5 Proof of Theorem 4 (Lengths of Proofs for Real Addition) ... 132
6 Proof of Theorems 1 and 2 (Presburger Arithmetic) 133
7 Other Results 134
Cylindrical Algebraic Decomposition I: The Basic Algorithm 136
Dennis S. Arnon, George E. Collins, and Scott McCallum
1 Introduction 136
2 Definition of Cylindrical Algebraic Decomposition 139
3 The Cylindrical Algebraic Decomposition Algorithm: Projection
Phase 140
4 The Cylindrical Algebraic Decomposition Algorithm: Base Phase 146
5 The Cylindrical Algebraic Decomposition Algorithm: Extension
Phase 147
6 An Example 148
Cylindrical Algebraic Decomposition II: An Adjacency Algorithm
for the Plane 152
Dennis S. Arnon, George E. Collins, and Scott McCallum
1 Introduction 152
2 Adjacencies in Proper Cylindrical Algebraic Decompositions . . 153
3 Determination of Section Section Adjacencies 157
4 Construction of Proper Cylindrical Algebraic Decompositions . 161
5 An Example 163
An Improvement of the Projection Operator in Cylindrical
Algebraic Decomposition 166
Hoon Hong
1 Introduction 166
2 Idea 167
3 Analysis 170
Contents xv
4 Empirical Results 170
Partial Cylindrical Algebraic Decomposition for Quantifier
Elimination 174
George E. Collins and Hoon Hong
1 Introduction 174
2 Main Idea 176
3 Partial CAD Construction Algorithm 183
4 Strategy for Cell Choice 189
5 Illustration . .• 191
6 Empirical Results 197
7 Conclusion 200
Simple Solution Formula Construction in Cylindrical Algebraic
Decomposition Based Quantifier Elimination 201
Hoon Hong
1 Introduction 201
2 Problem Statement 202
3 (Complex) Solution Formula Construction 203
4 Simplification of Solution Formulas 205
5 Experiments 210
Recent Progress on the Complexity of the Decision Problem for
the Reals 220
James Renegar
1 Some Terminology 220
2 Some Complexity Highlights 221
3 Discussion of Ideas Behind the Algorithms 224
An Improved Projection Operation for Cylindrical Algebraic
Decomposition 242
Scott McCallum
1 Introduction 242
2 Background Material 243
3 Statements of Theorems about Improved Projection Map . . . 244
4 Proof of Theorem 3 (and Lemmas) 247
5 Proof of Theorem 4 (and Lemmas) 252
6 CAD Construction Using Improved Projection 257
7 Examples 262
8 Appendix 267
Algorithms for Polynomial Real Root Isolation 269
J. R. Johnson
1 Introduction 269
2 Preliminary Mathematics 269
xvj Contents
3 Algorithms 276
4 Computing Time Analysis 284
5 Empirical Computing Times 287
Sturm Habicht Sequences, Determinants and Real Roots of
Univariate Polynomials 300
L. Gonzalez Vega, T. Redo, H. Lombardi, and M. F. Roy
1 Introduction 300
2 Algebraic Properties of Sturm Habicht Sequences 301
3 Sturm Habicht Sequences and Real Roots of Polynomials . . . 304
4 Sturm Habicht Sequences and Hankel Forms 309
5 Applications and Examples 314
Characterizations of the Macaulay Matrix and Their Algorithmic
Impact 317
Georg Hagel
1 Introduction 317
2 Notation 318
3 Definitions of the Macaulay Matrix 318
4 Extraneous Factor and First Properties of the Macaulay
Determinant 320
5 Characterization of the Macaulay Matrix 320
6 Characterization of the Macaulay Matrix, if It Is Used to
Calculate the u Resultant 322
7 Two Sorts of Homogenization 323
8 Characterization of the Matrix of the Extraneous
Factor 324
9 Conclusion 326
Computation of Variant Resultants 327
Hoon Hong and J. Rafael Sendra
1 Introduction 327
2 Problem Statement 328
3 Review of Determinant Based Method 329
4 Quotient Based Method 330
5 Modular Methods 333
6 Theoretical Computing Time Analysis 335
7 Experiments 337
A New Algorithm to Find a Point in Every Cell Defined by a
Family of Polynomials 341
Saugata Basu, Richard Pollack, and Marie Frangoise Roy
1 Introduction 341
2 Proof of the Theorem 346
Contents xvii
Local Theories and Cylindrical Decomposition 351
Daniel Richardson
1 Introduction 351
2 Infinitesimal Sectors at the Origin 352
3 Neighborhoods of Infinity 360
4 Exponential Polynomials in Two Variables 361
A Combinatorial Algorithm Solving Some Quantifier Elimination
Problems 365
Laureano Gonzalez Vega
1 Introduction 365
2 Sturm Habicht Sequence 366
3 The Algorithms 369
4 Conclusions 374
A New Approach to Quantifier Elimination for Real Algebra 376
V. Weispfenning
1 Introduction 376
2 The Quantifier Elimination Problem for the
Elementary Theory of the Reals 378
3 Counting Real Zeros Using Quadratic Forms 378
4 Comprehensive Grobner Bases 381
5 Steps of the Quantifier Elimination Method 382
6 Examples 386
References 393
Index 420
|
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| genre_facet | Konferenzschrift 1993 Linz |
| id | DE-604.BV010951189 |
| illustrated | Illustrated |
| indexdate | 2025-02-03T16:44:36Z |
| institution | BVB |
| isbn | 3211827943 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007324905 |
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| physical | XIX, 431 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
| record_format | marc |
| series2 | Texts and monographs in symbolic computation |
| spellingShingle | Quantifier elimination and cylindrical algebraic decomposition Datenverarbeitung Algebra Data processing Congresses Algorithms Congresses Decomposition method Data processing Congresses Quantorenelimination (DE-588)4308709-7 gnd Reell-abgeschlossener Körper (DE-588)4225673-2 gnd |
| subject_GND | (DE-588)4308709-7 (DE-588)4225673-2 (DE-588)1071861417 |
| title | Quantifier elimination and cylindrical algebraic decomposition |
| title_auth | Quantifier elimination and cylindrical algebraic decomposition |
| title_exact_search | Quantifier elimination and cylindrical algebraic decomposition |
| title_full | Quantifier elimination and cylindrical algebraic decomposition B. F. Caviness ... (eds.) |
| title_fullStr | Quantifier elimination and cylindrical algebraic decomposition B. F. Caviness ... (eds.) |
| title_full_unstemmed | Quantifier elimination and cylindrical algebraic decomposition B. F. Caviness ... (eds.) |
| title_short | Quantifier elimination and cylindrical algebraic decomposition |
| title_sort | quantifier elimination and cylindrical algebraic decomposition |
| topic | Datenverarbeitung Algebra Data processing Congresses Algorithms Congresses Decomposition method Data processing Congresses Quantorenelimination (DE-588)4308709-7 gnd Reell-abgeschlossener Körper (DE-588)4225673-2 gnd |
| topic_facet | Datenverarbeitung Algebra Data processing Congresses Algorithms Congresses Decomposition method Data processing Congresses Quantorenelimination Reell-abgeschlossener Körper Konferenzschrift 1993 Linz |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007324905&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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