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**Solving nonlinear equations with Newton’s method.**
*(English)*
Zbl 1031.65069

Fundamentals of Algorithms 1. Philadelphia, PA: SIAM Society for Industrial and Applied Mathematics. xiii, 104 p. (2003).

This book – the first in a new series – is intended as a user-oriented guide to the selection of an appropriate Newton-type method for a given problem and of a corresponding solver. As such it complements the author’s earlier book “Iterative methods for solving linear and nonlinear equations” (1995; Zbl 0832.65046)].

The presentation is based on the use of MATLAB v6.5 and centers on three main solvers, namely (1) nsold.m a Newton-Armijo code using Gaussian elimination, (2) nsoli.m a Newton-Krylov code using several Krylov methods to satisfy the inexact Newton condition, and (3) brsola.m an implementation of Broyden’s method.

The introductory chapter discusses some of the relevant general theory, various things to consider when solving any problem, and certain difficulties that may arise. Three codes for scalar equations are also included here. The remaining three chapters cover the basic theory and implementation details of the three mentioned solvers and exhibit their use and effectiveness by means of several examples. Besides small simple problems there are examples concerning the Chandrasecar H-equation, convection-diffusion equation, Ornstein-Zernike equation, and some two-point boundary value and stiff initial value problems.

The book certainly provides a nice overview of some of the practical aspects of solving particular problems and the advantages and limitations of different methods and implementations.

The presentation is based on the use of MATLAB v6.5 and centers on three main solvers, namely (1) nsold.m a Newton-Armijo code using Gaussian elimination, (2) nsoli.m a Newton-Krylov code using several Krylov methods to satisfy the inexact Newton condition, and (3) brsola.m an implementation of Broyden’s method.

The introductory chapter discusses some of the relevant general theory, various things to consider when solving any problem, and certain difficulties that may arise. Three codes for scalar equations are also included here. The remaining three chapters cover the basic theory and implementation details of the three mentioned solvers and exhibit their use and effectiveness by means of several examples. Besides small simple problems there are examples concerning the Chandrasecar H-equation, convection-diffusion equation, Ornstein-Zernike equation, and some two-point boundary value and stiff initial value problems.

The book certainly provides a nice overview of some of the practical aspects of solving particular problems and the advantages and limitations of different methods and implementations.

Reviewer: W.C.Rheinboldt (Pittsburgh)

### MSC:

65H10 | Numerical computation of solutions to systems of equations |

65-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis |