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.
