Numerical Methods for Roots of Polynomials - Part II - 1st Edition - ISBN: 9780444527301, 9780080931432

Numerical Methods for Roots of Polynomials - Part II, Volume 16

1st Edition

Authors: J.M. McNamee Victor Pan
eBook ISBN: 9780080931432
Hardcover ISBN: 9780444527301
Imprint: Elsevier Science
Published Date: 22nd July 2013
Page Count: 728
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Table of Contents






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


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


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


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


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


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


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


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


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




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.

Key Features

  • 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


Primary Markets: Academic faculty and libraries

Secondary Market: Industry (engineering)


No. of pages:
© Elsevier Science 2013
Elsevier Science
eBook ISBN:
Hardcover ISBN:


"...a well-written handbook of numerical methods for polynomial root-solving...covers most of the traditional methods for well as a great many invented in the last few decades of the 20th and early 21st centuries."--MathSciNet, Numerical Methods for Roots of Polynomials - Part II

"This book comprehensively covers traditional and latest methods on the calculation of roots of polynomials. The readers will benefit from this book greatly since these numerical methods in this book are accurate practical and have wide applications in control theory, information processing, statistics, etc. This book is well-written and accessible…"--Zentralblatt MATH, 1279.65053
"In this second of two parts, McNamee and Pan describe methods that are mostly numerical, or iterative, though they do devote one chapter to analytic methods for polynomials of degree up to five. Readers only need knowledge of polynomials at the senior high-school level, they say, but should have completed at least undergraduate courses in calculus and linear algebra."--
Reference & Research Book News, October 2013

About the Authors

J.M. McNamee Author

Affiliations and Expertise

York University, Toronto, Canada

Victor Pan Author