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The numerical solution of systems of polynomials arising in engineering and science /

Main Author: Sommese, Andrew John.
Other Authors: Wampler, Charles Weldon, II.
Format: Book
Language:English
Published: Hackensack, NJ : World Scientific, 2005.
Subjects:
Polynomials.
Polynômes.
MATHEMATICS > Algebra > Elementary.
Online Access:http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=148565
Table of Contents:
  • Cover
  • Preface
  • Contents
  • Conventions
  • PART I Background
  • Chapter 1 Polynomial Systems
  • 1.1 Polynomials in One Variable
  • 1.2 Multivariate Polynomial Systems
  • 1.3 Trigonometric Equations as Polynomials
  • 1.4 Solution Sets
  • 1.5 Solution by Continuation
  • 1.6 Overview
  • 1.7 Exercises
  • Chapter 2 Homotopy Continuation
  • 2.1 Continuation for Polynomials in One Variable
  • 2.2 Complex Versus Real Solutions
  • 2.3 Path Tracking
  • 2.4 Exercises
  • Chapter 3 Projective Spaces
  • 3.1 Motivation: Quadratic Equations
  • 3.2 Definition of Projective Space
  • 3.3 The Projective Line P1
  • 3.4 The Projective Plane P2
  • 3.5 Projective Algebraic Sets
  • 3.6 Multiprojective Space
  • 3.7 Tracking Solutions to Infinity
  • 3.8 Exercises
  • Chapter 4 Genericity and Probability One
  • 4.1 Generic Points
  • 4.2 Example: Generic Lines
  • 4.3 Probabilistic Null Test
  • 4.4 Algebraic Probability One
  • 4.5 Numerical Certainty
  • 4.6 Other Approaches to Genericity
  • 4.7 Final Remarks
  • 4.8 Exercises
  • Chapter 5 Polynomials of One Variable
  • 5.1 Some Algebraic Facts about Polynomials of One Complex Variable
  • 5.2 Some Analytic Facts about Polynomials of One Complex Variable (Optional)
  • 5.3 Some Numerical Aspects of Polynomials of One Variable
  • 5.4 Exercises
  • Chapter 6 Other Methods
  • 6.1 Exclusion Methods
  • 6.2 Elimination Methods
  • 6.3 Gr246;bner Methods
  • 6.4 More Methods
  • 6.5 Floating Point vs. Exact Arithmetic
  • 6.6 Discussion
  • 6.7 Exercises
  • PART II Isolated Solutions
  • Chapter 7 Coefficient-Parameter Homotopy
  • 7.1 Coefficient-Parameter Theory
  • 7.2 Parameter Homotopy in Application
  • 7.3 An Illustrative Example: Triangles
  • 7.4 Nested Parameter Homotopies
  • 7.5 Side Conditions
  • 7.6 Homotopies that Respect Symmetry Groups
  • 7.7 Case Study: Stewart-Gough Platforms
  • 7.8 Historical Note: The Cheaters Homotopy
  • 7.9 Exercises
  • Chapter 8 Polynomial Structures
  • 8.1 A Hierarchy of Structures
  • 8.2 Notation
  • 8.3 Homotopy Paths for Linearly Parameterized Families
  • 8.4 Product Homotopies
  • 8.5 Polytope Structures
  • 8.6 A Summarizing Example
  • 8.7 Exercises
  • Chapter 9 Case Studies
  • 9.1 Nash Equilibria
  • 9.2 Chemical Equilibrium
  • 9.3 Stewart-Gough Forward Kinematics
  • 9.4 Six-Revolute Serial-Link Robots
  • 9.5 Planar Seven-Bar Structures
  • 9.6 Four-Bar Linkage Design
  • 9.7 Exercises
  • Chapter 10 Endpoint Estimation
  • 10.1 Nonsingular Endpoints
  • 10.2 Singular Endpoints
  • 10.3 Singular Endgames
  • 10.4 Losing the Endgame
  • 10.5 Deflation of Isolated Singularities
  • 10.6 Exercises
  • Chapter 11 Checking Results and Other Implementation Tips
  • 11.1 Checks
  • 11.2 Corrective Actions
  • 11.3 Exercises
  • PART III Positive Dimensional Solutions
  • Chapter 12 Basic Algebraic Geometry
  • 12.1 Affine Algebraic Sets
  • 12.2 The Irreducible Decomposition for Affine Algebraic Sets
  • 12.3 Further Remarks on Projective Algebraic Sets
  • 12.4 Quasiprojective Algebraic Sets
  • 12.5 Constructible Algebraic Sets
  • 12.6 Multiplicity
  • 12.7 Exercises
  • Chapter 13 Basic Numerical Algebraic Geometry
  • 13.1 Introduction to Witness Sets
  • 13.2 Linear Slicing.

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