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The Resource Advances in Iterative Methods for Nonlinear Equations

Advances in Iterative Methods for Nonlinear Equations

Label
Advances in Iterative Methods for Nonlinear Equations
Title
Advances in Iterative Methods for Nonlinear Equations
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Subject
Language
eng
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Cataloging source
MiAaPQ
Literary form
non fiction
Nature of contents
dictionaries
Series statement
SEMA SIMAI Springer Ser.
Series volume
v.10
Advances in Iterative Methods for Nonlinear Equations
Label
Advances in Iterative Methods for Nonlinear Equations
Link
http://libproxy.rpi.edu/login?url=https://ebookcentral.proquest.com/lib/rpi/detail.action?docID=4699882
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Copyright
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Carrier category
online resource
Carrier category code
cr
Carrier MARC source
rdacarrier
Color
multicolored
Content category
text
Content type code
txt
Content type MARC source
rdacontent
Contents
  • Contents -- 1 Introduction -- 2 An Overview on Steffensen-Type Methods -- 2.1 Introduction -- 2.2 The Real Case -- 2.2.1 Semismooth Equations -- 2.2.1.1 A Modification of Steffensen's Method and Convergence Analysis -- 2.2.1.2 Numerical Experiments and Conclusions -- 2.3 Dynamics -- 2.4 Extension to Banach Space Setting -- 2.4.1 Convergence Analysis -- 2.5 Application to Boundary Value Problems -- 2.6 Other Contributions -- References -- 3 Newton's Method for Convex Optimization -- 3.1 Newton's Method -- 3.2 Introduction -- 3.3 Generalized Lipschitz Conditions and Majorizing Sequences -- 3.4 Background on Regularities -- 3.5 Semilocal Convergence Analysis for (GNA) -- 3.6 General Majorant Conditions -- 3.7 Applications -- 3.8 Conclusion -- References -- 4 Inexact Newton Methods on Riemannian Manifolds -- 4.1 Introduction -- 4.2 Background -- 4.3 Local Convergence Analysis -- 4.4 Special Cases -- References -- 5 On the Design of Optimal Iterative Methods for Solving Nonlinear Equations -- 5.1 Introduction -- 5.2 Optimal Fourth-Order Methods -- 5.3 High Order Optimal Multipoint Methods -- 5.4 General Classes of Optimal Multipoint Methods -- 5.5 Dynamical Behavior of Optimal Methods -- 5.5.1 Conjugacy Classes -- 5.5.2 Optimal Methods on Quadratic Polynomials -- 5.5.3 Optimal Methods on Cubic Polynomials -- 5.6 Numerical Performances -- 5.7 Conclusion -- References -- 6 The Theory of Kantorovich for Newton's Method: Conditions on the Second Derivative -- 6.1 Introduction -- 6.2 The Theory of Kantorovich -- 6.3 F'' is a Lipschitz-Type Operator -- 6.3.1 F'' is a Center Lipschitz Operator -- 6.3.2 F'' is a Center Hölder Operator -- 6.3.3 F'' is a Center }-Lipschitz Operator -- 6.3.3.1 Semilocal Convergence Result -- 6.3.3.2 Uniqueness of Solution -- 6.3.3.3 A Priori Error Estimates -- 6.3.3.4 Particular Cases -- 6.4 F'' is an }-Bounded Operator
  • 6.4.1 Semilocal Convergence Result -- 6.4.2 Uniqueness of Solution -- 6.4.3 Improvement of the Domain of Starting Points, the Domains of Existence and Uniqueness of Solution and the Error Bounds -- 6.4.4 A Priori Error Estimates -- References -- 7 Complexity of an Homotopy Method at the Neighbourhood of a Zero -- 7.1 Introduction -- 7.2 Survey and Remarks -- 7.3 Ü and Þ Theorems -- 7.4 Homotopy and Existence of a Zero -- 7.5 Proof of Theorem 2 -- 7.6 Study of Special Cases -- Appendix 1: Proof of Ü-Theorem -- Appendix 2: Proof of Þ-Theorem -- Appendix 3: Separation Theorem -- References -- 8 A Qualitative Analysis of a Family of Newton-Like Iterative Process with R-Order of Convergence At Least Three -- 8.1 Introduction -- 8.2 A Family of Iterative Processes with R-Order of Convergence At Least Three -- 8.3 Semilocal Convergence -- 8.3.1 Semilocal Convergence Under Kantorovich's Conditions -- 8.3.2 Semilocal Convergence for Operators with Second Derivative }-Conditioned -- 8.3.3 Semilocal Convergence Under Weak Conditions -- 8.4 A Study of the Accessibility Domain for the Iterative Processes of the Family -- 8.4.1 Attraction Basins -- 8.4.2 Regions of Accessibility -- 8.4.3 Domains of Parameters -- 8.5 Improvement of the Accessibility: Hybrid Method -- 8.5.1 Application -- 8.6 An Improvement of the Efficiency -- 8.6.1 Application -- References -- 9 Measures of the Basins of Attracting n-Cycles for the Relaxed Newton's Method -- 9.1 Introduction -- 9.2 Mathematical Framework -- 9.2.1 Discrete Semi-flows on Metric Spaces and Basins -- 9.2.2 Basins of Rational Functions on the Riemann Sphere -- 9.2.3 Spherical Multipliers -- 9.2.4 Lebesgue Measures on the 2-Sphere -- 9.3 Algorithms for Computing Basins of Non-repelling Cyclic Points, Measures and Initial Efficiency, Julia Sets and Attracting Cyclic Points
  • 9.3.1 Algorithm 1: Spherical Plots of Basins of Fixed and Non-repelling Cyclic Points -- 9.3.2 Algorithm 2: Areas of the Basins of End Points Associated with Non-repelling Cyclic Points -- 9.3.3 Algorithm 3: The Julia Set of a Rational Function in the Riemann Sphere -- 9.3.4 Algorithm 4: The Parameter Plane -- 9.4 Applications of the Algorithms to the Relaxed Newton's Method -- 9.4.1 Attracting p-Cyclic Points for the Relaxed Newton's Method -- 9.4.1.1 Dynamic of the Relaxed Newton's Method Applied to q1(z)=z3+z -- 9.4.1.2 Dynamic of the Relaxed Newton's Method Applied to q2(z)=(z-1)2(z+1) -- 9.4.2 Initial Efficiency and Areas of Basins of the Relaxed Newton's Method -- 9.4.2.1 A Study of Initial Efficiency and Areas Associated to z3+z -- 9.4.2.2 A Study of Initial Efficiency and Areas Associated to (z-1)2(z+1) -- 9.5 Conclusion -- References -- 10 On Convergence and Efficiency in the Resolution of Systems of Nonlinear Equations from a Local Analysis -- 10.1 Iteration Functions -- 10.1.1 One-Dimensional Case -- 10.1.2 Multidimensional Case -- 10.2 Computational Estimations of the Order -- 10.2.1 Computational Order of Convergence and Its Variants -- 10.2.2 New Parameters to Compute the Local Order of Convergence -- 10.2.3 Multidimensional Case -- 10.3 The Vectorial Error Difference Equation -- 10.3.1 Notation -- 10.3.2 Symbolic Computation of the Inverse of a Function of Several Variables -- 10.3.3 A Development of the Inverse of the First Order Divided Differences of a Function of Several Variables -- 10.4 Efficiency Indices -- 10.4.1 Efficiency Index and Computational Efficiency -- 10.4.2 Computational Efficiency Index -- 10.4.3 Examples of Iterative Methods -- 10.4.4 Comparisons Between These Methods -- 10.4.5 Numerical Results -- 10.5 Theoretical Numerical Considerations -- 10.5.1 Theoretical Estimations -- 10.6 Ball of Local Convergence
  • 10.7 Adaptive Arithmetic -- 10.7.1 Iterative Method -- 10.7.2 Numerical Example -- 10.7.3 Practical Result -- References
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Extent
1 online resource (286 pages)
Form of item
online
Isbn
9783319392288
Media category
computer
Media MARC source
rdamedia
Media type code
c
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Specific material designation
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