- ホーム
- > 洋書
- > 英文書
- > Science / Mathematics
Full Description
This book is a guide to concepts and practice in numerical algebraic geometry - the solution of systems of polynomial equations by numerical methods. Through numerous examples, the authors show how to apply the well-received and widely used open-source Bertini software package to compute solutions, including a detailed manual on syntax and usage options. The authors also maintain a complementary web page where readers can find supplementary materials and Bertini input files.
Numerically Solving Polynomial Systems with Bertini approaches numerical algebraic geometry from a user's point of view with numerous examples of how Bertini is applicable to polynomial systems. It treats the fundamental task of solving a given polynomial system and describes the latest advances in the field, including algorithms for intersecting and projecting algebraic sets, methods for treating singular sets, the nascent field of real numerical algebraic geometry, and applications to large polynomial systems arising from differential equations.
Those who wish to solve polynomial systems can start gently by finding isolated solutions to small systems, advance rapidly to using algorithms for finding positive-dimensional solution sets (curves, surfaces, etc.), and learn how to use parallel computers on large problems. These techniques are of interest to engineers and scientists in fields where polynomial equations arise, including robotics, control theory, economics, physics, numerical PDEs, and computational chemistry.
Contents
List of Figures
Conventions
Preface
Part I: Isolated Systems
Chapter 1: Polynomial Systems
Chapter 2: Basic Polynomial Continuation
Chapter 3: Adaptive Precision and Endgames
Chapter 4: Projective Space
Chapter 5: Types of Homotopies
Chapter 6: Parameter Homotopies
Chapter 7: Advanced Topics about Isolated Solutions
Part II: Positive-Dimensional Solution Sets
Chapter 8: Positive-Dimensional Components
Chapter 9: Computing Witness Supersets
Chapter 10: The Numerical Irreducible Decomposition
Chapter 11: Advanced Topics about Positive-Dimensional Solution Sets
Part III: Further Algorithms and Applications
Chapter 12: Intersection
Chapter 13: Singular Sets
Chapter 14: Real Solutions
Chapter 15: Applications to Algebraic Geometry
Chapter 16: Projections of Algebraic Sets
Chapter 17: Big Polynomial Systems Arising from Differential Equations
Part IV: Bertini Users Manual
Appendix A: Bertini Quick Start Guide
Appendix B: Input Format
Appendix C: Calling Options
Appendix D: Output Files
Appendix E: Configuration Settings
Appendix F: Tips and Tricks
Appendix G: Parallel Computing
Appendix H: Related Software
Bibliography
Software Index
Subject Index.
