Numerical Methods for Roots of Polynomials - Part II: Chapter 9. Methods Involving Second or Higher Derivatives

Numerical Methods for Roots of Polynomials - Part II: Chapter 9. Methods Involving Second or Higher Derivatives
ISBN-10
0128076992
ISBN-13
9780128076996
Series
Numerical Methods for Roots of Polynomials - Part II
Category
Mathematics
Pages
728
Language
English
Published
2013-07-19
Publisher
Elsevier Inc. Chapters
Authors
J.M. McNamee, V.Y. Pan

Description

Whereas Newton’s method involves only the first derivative, methods discussed in this chapter involve the second or higher. The “classical” methods of this type (such as Halley’s, Euler’s, Hansen and Patrick’s, Ostrowski’s, Cauchy’s and Chebyshev’s) are all third order with three evaluations, so are slightly more efficient than Newton’s method. Convergence of some of these methods is discussed, as well as composite variations (some of which have fairly high efficiency). We describe special methods for multiple roots, simultaneous or interval methods, and acceleration techniques. We treat Laguerre’s method, which is known to be globally convergent for all-real-roots. The Cluster-Adapted Method is useful for multiple or near-multiple roots. Several composite methods are discussed, as well as methods using determinants or various types of interpolation, and Schroeder’s method.

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