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This book has two faces: the first one is that of a textbook addressing itself to graduate students of mathematics and computational sciences, the second one is that of a research monograph addressing itself to numerical analysts and computational scientists working on the subject.
As a textbook, selected chapters may be useful in classes on Numerical Analysis, Nonlinear Optimization, Numerical ODEs, or Numerical PDEs. The presentation is striving for structural simplicity, but not at the expense of precision. It contains a lot of theorems and proofs, from affine invariant versions of the classical Newton-Kantorovich and Newton-Mysovskikh theorem (with proofs simpler than the traditional ones) up to new convergence theorems that are the basis for advanced algorithms in large scale scientific computing. I confess that I did not work out all details of all proofs, if they were folklore or if their structure appeared repeatedly. More elaboration on this aspect would have unduly blown up the volume without adding enough value for the construction of algorithms. However, I definitely made sure that each section is self-contained to a reasonable extent. At the end of each chapter, exercises are included. Web addresses for related software are given.
As a research monograph, the presentation (a) quite often goes into the depth covering a large amount of otherwise unpublished material, (b) is open in many directions of possible future research, some of which are explicitly indicated in the text. Even though the experienced reader will have no difficulties in identifying further open topics, let me mention a few of them: There is no complete coverage of all possible combinations of local and global, exact and inexact Newton or Gauss-Newton methods in connection with continuation methods—let alone of all their affine invariant realizations in other words, the above structural matrix is far from being full. Moreover, apart from convex optimization and constrained nonlinear least squares problems, general
optimization and optimal control is left out. Also not included are recent results on interior point methods as well as inverse problems in L2, even though affine invariance has just started to play a role in these fields.
Systems of equations: local Newton methods
Systems of equations: global Newton methods
Least squares problems: Gauss-Newton methods
Parameter dependent systems: continuation methods
Stiff ODE initial value problems
ODE boundary value problems
PDE boundary value problems