This book aims to give an encyclopedic overview of the state-of-the-art
of Krylov subspace iterative methods for solving nonsymmetric systems of
algebraic linear equations and to study their mathematical properties.
Solving systems of algebraic linear equations is among the most frequent
problems in scientific computing; it is used in many disciplines such as
physics, engineering, chemistry, biology, and several others. Krylov
methods have progressively emerged as the iterative methods with the
highest efficiency while being very robust for solving large linear
systems; they may be expected to remain so, independent of progress in
modern computer-related fields such as parallel and high performance
computing. The mathematical properties of the methods are described and
analyzed along with their behavior in finite precision arithmetic. A
number of numerical examples demonstrate the properties and the behavior
of the described methods. Also considered are the methods'
implementations and coding as Matlab(R)-like functions. Methods which
became popular recently are considered in the general framework of Q-OR
(quasi-orthogonal )/Q-MR (quasi-minimum) residual methods.
This book can be useful for both practitioners and for readers who are
more interested in theory. Together with a review of the
state-of-the-art, it presents a number of recent theoretical results of
the authors, some of them unpublished, as well as a few original
algorithms. Some of the derived formulas might be useful for the design
of possible new methods or for future analysis. For the more applied
user, the book gives an up-to-date overview of the majority of the
available Krylov methods for nonsymmetric linear systems, including
well-known convergence properties and, as we said above, template codes
that can serve as the base for more individualized and elaborate
implementations.
