MULTIPOINT METHODS FOR SOLVING NONLINEAR EQUATIONS
By- Miodrag Petkovic, University of Niš, Serbia
- Beny Neta, Naval Postgraduate School, Monterey, CA, USA
This book is the first on the topic and explains the most cutting-edge methods needed for precise calculations and explores the development of powerful algorithms to solve research problems. Multipoint methods have an extensive range of practical applications significant in research areas such as signal processing, analysis of convergence rate, fluid mechanics, solid state physics, and many others. The book takes an introductory approach in making qualitative comparisons of different multipoint methods from various viewpoints to help the reader understand applications of more complex methods. Evaluations are made to determine and predict efficiency and accuracy of presented models useful to wide a range of research areas along with many numerical examples for a deep understanding of the usefulness of each method. This book will make it possible for the researchers to tackle difficult problems and deepen their understanding of problem solving using numerical methods.
Multipoint methods are of great practical importance, as they determine sequences of successive approximations for evaluative purposes. This is especially helpful in achieving the highest computational efficiency. The rapid development of digital computers and advanced computer arithmetic have provided a need for new methods useful to solving practical problems in a multitude of disciplines such as applied mathematics, computer science, engineering, physics, financial mathematics, and biology.
Audience
Research Professionals, Scientists, Engineers, Mathematicans, Graduate Students
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Published: December 2012
Imprint: Academic Press
ISBN: 978-0-12-397013-8
Contents
1 Basic concepts
1.1 Classification of iterative methods
1.2 Order of convergence
1.3 Computational efficiency of iterative methods
1.4 Initial approximations
1.5 One-point iterative methods for simple zeros
1.6 Methods for determining multiple zeros
1.7 Stopping criterion2 Two-Point methods
2.1 Cubically convergent two-point methods
2.2 Ostrowskis fourth-order method and its generalizations
2.3 Family of optimal two-point methods
2.4 Optimal derivative free two-point methods
2.5 Kung-Traubs multipoint methods
2.6 Optimal two-point methods of Jarratts type
2.7 Two-point methods for multiple roots3 Three-Point non-optimal methods
4 Three-Point optimal methods
3.1 Some historical notes
3.2 Methods for constructing sixth-order root-finders
3.3 Ostrowski-like methods of sixth order
3.4 Jarratt-like methods of sixth order
3.5 Other non-optimal three-point methods
4.1 Optimal three-point methods of Bi, Wu, and Ren
4.2 Interpolatory iterative three-point methods
4.3 Optimal methods based on weight functions
4.4 Eighth-order Ostrowski-like methods
4.5 Derivative free family of optimal three-point methods5 Higher-order optimal methods
6 Multipoint methods with memory
5.1 Some comments on higher-order multipoint methods
5.2 Geum-Kims family of four-point methods
5.3 Kung-Traubs families of arbitrary order of convergence
5.4 Methods of higher-order based on inverse interpolation
5.5 Multipoint methods based on Hermites interpolation
5.6 Generalized derivative free family based on Newtonian interpolation
6.1 Early works
6.2 Multipoint methods with memory constructed by inverse interpolation
6.3 Efficient family of two-point self-accelerating methods
6.4 Family of three-point methods with memory
6.5 Generalized multipoint root-solvers with memory
6.6 Computational aspects7 Simultaneous methods for polynomial zeros
7.1 Simultaneous methods for simple zeros
7.2 Simultaneous method for multiple zeros
7.3 Simultaneous inclusion of simple zeros
7.4 Simultaneous inclusion of multiple zeros
7.5 Halley-like inclusion methods of high efficiency
