Book summary
Local convergence results on Newton-type methods for nonlinear systems of equations are studied.
Solving of large linear systems by Krylov methods (GMRES, GMBACK, MINPERT) are also dealt with, as well as their behavior when used in Newton methods.
Book cover
Introduction
Notations
Ch. 1 Krylov methods for large linear systems
1.1 Motivation
1.2 Backward error minimization Krylov solvers
1.2.1 The GMRES method
1.2.2 The GMBACK method
1.2.3 The MINPERT method
Ch. 2 Local convergence of the Newton methods
2.1 Historical notes
2.2 The local convergence of the Newton method
2.3 Local convergence of the quasi-Newton methods
2.4 Inexact Newton methods
2.4.1 The local convergence of the IN methods
2.4.2 Choosing the forcing terms in the IN methods
2.4.3 Global convergence of the IN methods
2.5 Inexact(ly solved) Perturbed Newton methods
2.6 Newton-Krylov methods
2.6.1 The Newton-GMRES method
2.6.2 The Newton-GMBACK method
2.6.3 The Newton-MINPERT method
2.6.4 Numerical examples
2.7 Finite difference Newton-Krylov methods
2.8 The high convergence orders of the successive approximations
Ch. 3 Semilocal convergence results
3.1 The eigenpair problem
3.2 The Chebyshev method
3.3 A Chebyshev-type method
3.4 A Newton-type method
3.5 Numerical examples
Ch. 4 The chord method
4.1 Definitions and examples
4.2 The local convergence of the chord method
4.3 The local convergence of the inexact chord method
4.4 The chord method for second degree polynomial operators
4.5 Hybrid methods
Appendix. Rates of convergence
A.1 The quotient convergence order
A.2 The root convergence order
A.3 Relations between Q and R-factors
Notes
- The Appendix contains a synthesis of the results from the book of Ortega and Rheinboldt. New directions have been obtained in the recent paper
E. Cătinaş, A survey on the high convergence orders and computational convergence orders of sequences, Appl. Math. Comput., 343 (2019) 1-20.
and in other papers submitted for publication; - The historical notes have been updated and extended in a submitted paper (to be posted soon);
- Ch. 4 was entitled as such for keeping notations from previous works. Perhaps more appropriate is to be entitled Secant methods (as we considered, e.g., in [51]).
Keywords
Newton method; nonlinear systems in Rn; local convergence; Newton-Krylov methods; convergence order; linear systems in Rn; Krylov methods; backward errors.
pdf file (soon)
References
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Cite this book as
E. Catinas, Methods of Newton and Newton-Krylov type, Risoprint, Cluj-Napoca, Romania, 2007.
Book Title
Methods of Newton and Newton-Krylov type
Publisher
Risoprint
Print ISBN
978-973-751-533-9
Google scholar
The book on google scholar.
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