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1 online resource (xiii, 299 pages) |
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text txt rdacontent |
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computer c rdamedia |
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online resource cr rdacarrier |
| Summary |
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. Provides a succinct way of implementing a wide range of useful and important numerical algorithms for solving research problems Illustrates how numerical methods can be used to study problems which have applications in engineering and sciences, including signal processing, and control theory, and financial computation Facilitates a deeper insight into the development of methods, numerical analysis of convergence rate, and very detailed analysis of computational efficiencyProvides a powerful means of learning by systematic experimentation with some of the many fascinating problems in scienceIncludes highly efficient algorithms convenient for the implementation into the most common computer algebra systems such as Mathematica, MatLab, and Maple. |
| Note |
Publisher supplied information; title not viewed. |
| Contents |
1.Basic concepts -- 1.1.Classification of iterative methods -- 1.2.Order of convergence -- 1.2.1.Computational order of convergence (COC) -- 1.2.2.R-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 criterion -- 2.Two-point methods -- 2.1.Cubically convergent two-point methods -- 2.1.1.Composite multipoint methods -- 2.1.2.Traub's two-point methods -- 2.1.3.Two-point methods generated by derivative estimation -- 2.2.Ostrowski's 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-Traub's multipoint methods -- 2.6.Optimal two-point methods of Jarratt's type -- 2.6.1.Jarratt's two-step methods -- 2.6.2.Jarratt-like family of two-point methods -- 2.7.Two-point methods for multiple roots -- 2.7.1.Non-optimal two-point methods for multiple zeros -- 2.7.2.Optimal two-point methods for multiple zeros -- 3.Three-point non-optimal methods -- 3.1.Some historical notes -- 3.2.Methods for constructing sixth-order root-finders -- 3.2.1.Method 1 -- Secant-like method -- 3.2.2.Method 2 -- Rational bilinear interpolation -- 3.2.3.Method 3 -- Hermite's interpolation -- 3.2.4.Method 4 -- Inverse interpolation -- 3.2.5.Method 5 -- Newton's interpolation -- 3.2.6.Method 6 -- Taylor's approximation of derivative -- 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.Three-point optimal methods -- 4.1.Optimal three-point methods of Bi, Wu, and Ren -- 4.2.Interpolatory iterative three-point methods -- 4.2.1.Optimal three-point methods based on Hermite's interpolation -- 4.2.2.Three-point methods based on rational function interpolation -- 4.2.3.Three-point methods constructed by inverse interpolation -- 4.2.4.Numerical examples -- 4.3.Optimal methods based on weight functions -- 4.3.1.Family based on the sum of three weight functions -- 4.3.2.Liu-Wang's family -- 4.3.3.Family based on two weight functions -- 4.3.4.Geum-Kim's families -- 4.4.Eighth-order Ostrowski-like methods -- 4.4.1.First Ostrowski-like family -- 4.4.2.Second Ostrowski-like family -- 4.4.3.Third Ostrowski-like family -- 4.4.4.Family of quasi-Ostrowski's type -- 4.5.Derivative free family of optimal three-point methods -- 5.Higher-order optimal methods -- 5.1.Some comments on higher-order multipoint methods -- 5.2.Geum-Kim's family of four-point methods -- 5.3.Kung-Traub's families of arbitrary order of convergence -- 5.4.Methods of higher-order based on inverse interpolation -- 5.5.Multipoint methods based on Hermite's interpolation -- 5.6.Generalized derivative free family based on Newtonian interpolation -- 6.Multipoint methods with memory -- 6.1.Early works -- 6.1.1.Self-accelerating Steffensen-like method -- 6.1.2.Self-accelerating secant method -- 6.2.Multipoint methods with memory constructed by inverse interpolation -- 6.2.1.Two-step method with memory of Neta's type -- 6.2.2.Three-point method with memory of Neta's type -- 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 -- 6.5.1.Derivative free families with memory -- 6.5.2.Order of convergence of the generalized families with memory -- 6.6.Computational aspects -- 6.6.1.Numerical examples (I) -- two-point methods -- 6.6.2.Numerical examples (II) -- three-point methods -- 6.6.3.Comparison of computational efficiency -- 7.Simultaneous methods for polynomial zeros -- 7.1.Simultaneous methods for simple zeros -- 7.1.1.A third-order simultaneous method -- 7.1.2.Simultaneous methods with corrections -- 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. |
| Subject |
Differential equations, Nonlinear -- Numerical solutions.
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Équations différentielles non linéaires -- Solutions numériques.
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Differential equations, Nonlinear -- Numerical solutions
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| Added Author |
Petkovi, Miodrag, author.
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Neta, Beny, author.
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Petkovi, Ljiljana D., author.
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Duni, Jovana, author.
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| Other Form: |
Print version: Petkovi, Miodrag. Multipoint methods for solving nonlinear equations. Amsterdam : Elsevier/Academic Press, [2013] 9780123970138 (DLC) 2012022098 (OCoLC)803465369 |
| ISBN |
9780123970138 |
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012397013X |
| Standard No. |
AU@ 000050608168 |
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CHBIS 009942994 |
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CHVBK 303569980 |
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DEBBG BV042310467 |
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DEBSZ 405346212 |
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NZ1 15192385 |
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CHNEW 001010844 |
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AU@ 000067106983 |
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