Numerical Methods for Roots of Polynomials - Part II
- 1st Edition, Volume 16 - August 28, 2013
- Authors: J.M. McNamee, Victor Pan
- Language: English
- Paperback ISBN:9 7 8 - 0 - 4 4 4 - 6 3 8 3 5 - 9
- Hardback ISBN:9 7 8 - 0 - 4 4 4 - 5 2 7 3 0 - 1
- eBook ISBN:9 7 8 - 0 - 0 8 - 0 9 3 1 4 3 - 2
Numerical Methods for Roots of Polynomials - Part II along with Part I (9780444527295) covers most of the traditional methods for polynomial root-finding such as interp… Read more
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Numerical Methods for Roots of Polynomials - Part II along with Part I (9780444527295) covers most of the traditional methods for polynomial root-finding such as interpolation and methods due to Graeffe, Laguerre, and Jenkins and Traub. It includes many other methods and topics as well and has a chapter devoted to certain modern virtually optimal methods. Additionally, there are pointers to robust and efficient programs. This book is invaluable to anyone doing research in polynomial roots, or teaching a graduate course on that topic.
- First comprehensive treatment of Root-Finding in several decades with a description of high-grade software and where it can be downloaded
- Offers a long chapter on matrix methods and includes Parallel methods and errors where appropriate
- Proves invaluable for research or graduate course
Academic faculty and libraries, Industry (engineering)
Dedication
Acknowledgment
Preface
Introduction
References
Chapter 7. Bisection and Interpolation Methods
7.1 Introduction and History
7.2 Secant Method and Variations
7.3 The Bisection Method
7.4 Methods Involving Quadratics
7.5 Methods of Higher Order or Degree
7.6 Rational Approximations
7.7 Hybrid Methods
7.8 Parallel Methods
7.9 Multiple Roots
7.10 Method of Successive Approximation
7.11 Miscellaneous Methods Without Using Derivatives
7.12 Methods Using Interval Arithmetic
7.13 Programs
References
Chapter 8. Graeffe’s Root-Squaring Method
8.1 Introduction and History
8.2 The Basic Graeffe Process
8.3 Complex Roots
8.4 Multiple Modulus Roots
8.5 The Brodetsky–Smeal–Lehmer Method
8.6 Methods for Preventing Overflow
8.7 The Resultant Procedure and Related Methods
8.8 Chebyshev-Like Processes
8.9 Parallel Methods
8.10 Errors in Root Estimates by Graeffe Iteration
8.11 Turan’s Methods
8.12 Algorithm of Sebastião e Silva and Generalizations
8.13 Miscellaneous
8.14 Programs
References
Chapter 9. Methods Involving Second or Higher Derivatives
9.1 Introduction
9.2 Halley’s Method and Modifications
9.3 Laguerre’s Method and Modifications
9.4 Chebyshev’s Method
9.5 Methods Involving Square Roots
9.6 Other Methods Involving Second Derivatives
References
Chapter 10. Bernoulli, Quotient-Difference, and Integral Methods
10.1 Bernoulli’s Method for One Dominant Root
10.2 Bernoulli’s Method for Complex and/or Multiple Roots
10.3 Improvements and Generalizations of Bernoulli’s Method
10.4 The Quotient-Difference Algorithm
10.5 The Lehmer–Schur Method
10.6 Methods Using Integration
10.7 Programs
References
Chapter 11. Jenkins–Traub, Minimization, and Bairstow Methods
11.1 The Jenkins–Traub Method
11.2 Jenkins–Traub Method for Real Polynomials
11.3 Precursors and Generalizations of the Jenkins–Traub Method
11.4 Minimization Methods—The Downhill Technique
11.5 Minimization Methods—Use of Gradient
11.6 Hybrid Minimization and Newton’s Methods
11.7 Lin’s Method
11.8 Generalizations of Lin’s Method
11.9 Bairstow’s Method
11.10 Generalizations of Bairstow’s Method
11.11 Bairstow’s Method for Multiple Factors
11.12 Miscellaneous Methods
11.13 Programs
References
Chapter 12. Low-Degree Polynomials
12.1 Introduction
12.2 History of the Quadratic
12.3 Modern Solutions of the Quadratic
12.4 Errors in the Quadratic Solution
12.5 Early History of the Cubic
12.6 Cardan’s Solution of the Cubic
12.7 More Recent Derivations of the Cubic Solution
12.8 Trigonometric Solution of the Cubic
12.9 Discriminants of the Cubic
12.10 Early Solutions of the Quartic
12.11 More Recent Treatment of the Quartic
12.12 Analytic Solution of the Quintic
References
Chapter 13. Existence and Solution by Radicals
13.1 Introduction and Early History of the Fundamental Theorem of Algebra
13.2 Trigonometric Proof-Gauss’ Fourth Proof
13.3 Proofs Using Integration
13.4 Methods Based on Minimization
13.5 Miscellaneous Proofs
13.6 Solution by Radicals (Including Background on Fields and Groups)
13.7 Solution by Radicals: Galois Theory
References
Chapter 14. Stability Considerations
14.1 Introduction
14.2 History
14.3 Roots in the Left (or Right) Half-Plane; Use of Cauchy Index and Sturm Sequences
14.4 Routh’s Method for the Hurwitz Problem
14.5 Routh Method—the Singular Cases
14.6 Other Methods for the Hurwitz Problem
14.7 Robust Hurwitz Stability
14.8 The Number of Zeros in the Unit Circle, and Schur Stability
14.9 Robust Schur Stability
14.10 Programs on Stability
References
Chapter 15. Nearly Optimal Universal Polynomial Factorization and Root-Finding
15.1 Introduction and Main Results
15.2 Definitions and Preliminaries
15.3 Norm Bounds
15.4 Root Radii: Estimates and Algorithms
15.5 Approximating the Power Sums of Polynomial Zeros
15.6 Initial Approximate Splitting
15.7 Refinement of Approximate Splitting: Algorithms
15.8 Refinement of Splitting: Error Norm Bounds
15.9 Accelerated Refinement of Splitting. An Algorithm and the Error Bound
15.10 Computation of the Initial Basic Polynomial for the Accelerated Refinement
15.11 Updating the Basic Polynomials
15.12 Relaxation of the Initial Isolation Constraint
15.13 The Bitwise Precision and the Complexity of Padé Approximation and Polynomial Splitting
15.14 Perturbation of a Padé Approximation
15.15 Avoiding Degeneration of Padé Approximations
15.16 Splitting into Factors over an Arbitrary Circle
15.17 Recursive Splitting into Factors: Error Norm Bounds
15.18 Balanced Splitting and Massive Clusters of Polynomial Zeros
15.19 Balanced Splitting via Root Radii Approximation
15.20 -Centers of a Polynomial and Zeros of a Higher Order Derivative
15.21 Polynomial Splitting with Precomputed -Centers
15.22 How to Avoid Approximation of the Zeros of Higher Order Derivatives
15.23 NAPF and PFD for Any Number of Fractions
15.24 Summary and Comparison with Alternative Methods (Old and New). Some Directions to Further Progress
15.25 The History of Polynomial Root-Finding and Factorization via Recursive Splitting
15.26 Exercises
References
Index
- Edition: 1
- Volume: 16
- Published: August 28, 2013
- Language: English
JM
J.M. McNamee
Affiliations and expertise
York University, Toronto, CanadaRead Numerical Methods for Roots of Polynomials - Part II on ScienceDirect