# Methods of Newton and Newton-Krylov type

## 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

## Keywords

Newton method; nonlinear systems in Rn; local convergence; Newton-Krylov methods; convergence order; linear systems in Rn; Krylov methods; backward errors.

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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

Risoprint

##### Print ISBN

978-973-751-533-9

##### Authors

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