© 2011

Newton Methods for Nonlinear Problems

Affine Invariance and Adaptive Algorithms


Part of the Springer Series in Computational Mathematics book series (SSCM, volume 35)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Peter Deuflhard
    Pages 7-41
  3. Algebraic Equations

    1. Front Matter
      Pages 43-43
    2. Peter Deuflhard
      Pages 45-107
    3. Peter Deuflhard
      Pages 109-172
    4. Peter Deuflhard
      Pages 173-231
  4. Differential Equations

    1. Front Matter
      Pages 283-283
    2. Peter Deuflhard
      Pages 285-314
    3. Peter Deuflhard
      Pages 315-368
    4. Peter Deuflhard
      Pages 369-404
  5. Back Matter
    Pages 405-424

About this book


This book deals with the efficient numerical solution of challenging nonlinear problems in science and engineering, both in finite dimension (algebraic systems) and in infinite dimension (ordinary and partial differential equations). Its focus is on local and global Newton methods for direct problems or Gauss-Newton methods for inverse problems. The term 'affine invariance' means that the presented algorithms and their convergence analysis are invariant under one out of four subclasses of affine transformations of the problem to be solved. Compared to traditional textbooks, the distinguishing affine invariance approach leads to shorter theorems and proofs and permits the construction of fully adaptive algorithms. Lots of numerical illustrations, comparison tables, and exercises make the text useful in computational mathematics classes. At the same time, the book opens many directions for possible future research.


Gauss-newton methods Newton methods affine invariance continuation methods differential equations

Authors and affiliations

  1. 1.Zuse-Institut Berlin (ZIB)BerlinGermany

About the authors

Peter Deuflhard is founder and head of the internationally renowned Zuse Institute Berlin (ZIB) and full professor of Numerical Analysis and Scientific Computing at the Free University of Berlin. He is a regular invited speaker at international conferences and universities as well as industry places all over the world.

Bibliographic information

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From the reviews:

“This monograph covers a multitude of Newton methods and presents the algorithms and their convergence analysis from the perspective of affine invariance, which has been the subject of research by the author since 1970. … The book is intended for graduate students of mathematics and computational science and also for researchers in the area of numerical analysis and scientific computing. … As a research monograph, the book not only assembles the current state of the art, but also points to future research prospects.” (Gudula Runger, ACM Computing Reviews, June, 2012)