#### Document Type

Article

#### Publication Date

2005

#### Abstract

In the context of solving nonlinear equations, the term "affine invariance" was introduced to describe the fact that when a function *F: R ^{n} → R^{n}* is transformed to

*G*=

*AF*,where

*A*is an invertible matrix, then the equation

*F*(

*x*) = 0 has the same solutions as

*G*(

*x*) = 0, and the Newton iterates

*X*

_{k+1}=

*X*-

_{k}*F'*(

*X*)

_{k}^{-1}

*F*(

*X*) remain unchanged when

_{k}*F*is replaced by

*G*. The idea was that this property of Newton's method should be reflected in its convergence analysis and practical implementation, not only on aesthetic grounds but also because the resulting algorithms would likely be less sensitive to scaling, conditioning, and other numerical issues.

#### Publication Title

SIAM Review

#### Volume

47

#### Issue

2

#### First Page

401

#### Last Page

403

#### Required Publisher's Statement

Published by the Society for Industrial and Applied Mathematics

Courtesy of JSTOR

Stable URL: http://www.jstor.org/stable/20453655

#### Recommended Citation

Ypma, Tjalling J., "Review of: Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, by P. Deuflhard" (2005). *Mathematics.* Paper 94.

http://cedar.wwu.edu/math_facpubs/94