Mathematics of Computing -- General.

Applied Iterative Methods

In the past decade, primal-dual algorithms have emerged as the most important and useful algorithms from the interior-point class. This book presents the major primal-dual algorithms for linear programming in straightforward terms. A thorough description of the theoretical properties of these methods is given, as are a discussion of practical and computational aspects and a summary of current software. This is an excellent, timely, and well-written work. The major primal-dual algorithms covered in this book are path-following algorithms (short- and long-step, predictor-corrector), potential-reduction algorithms, and infeasible-interior-point algorithms. A unified treatment of superlinear convergence, finite termination, and detection of infeasible problems is presented. Issues relevant to practical implementation are also discussed, including sparse linear algebra and a complete specification of Mehrotra's predictor-corrector algorithm. Also treated are extensions of primal-dual algorithms to more general problems such as monotone complementarity, semidefinite programming, and general convex programming problems.

Linear and nonlinear systems of equations are the basis for many, if not most, of the models of phenomena in science and engineering, and their efficient numerical solution is critical to progress in these areas. This is the first book to be published on nonlinear equations since the mid-1980s. Although it stresses recent developments in this area, such as Newton-Krylov methods, considerable material on linear equations has been incorporated. This book focuses on a small number of methods and treats them in depth. The author provides a complete analysis of the conjugate gradient and generalized minimum residual iterations as well as recent advances including Newton-Krylov methods, incorporation of inexactness and noise into the analysis, new proofs and implementations of Broyden's method, and globalization of inexact Newton methods. Examples, methods, and algorithmic choices are based on applications to infinite dimensional problems such as partial differential equations and integral equations. The analysis and proof techniques are constructed with the infinite dimensional setting in mind and the computational examples and exercises are based on the MATLAB environment.

Iterative methods use successive approximations to obtain more accurate solutions. This book gives an introduction to iterative methods and preconditioning for solving discretized elliptic partial differential equations and optimal control problems governed by the Laplace equation, for which the use of matrix-free procedures is crucial. All methods are explained and analyzed starting from the historical ideas of the inventors, which are often quoted from their seminal works. Iterative Methods and Preconditioners for Systems of Linear Equations grew out of a set of lecture notes that were improved and enriched over time, resulting in a clear focus for the teaching methodology, which derives complete convergence estimates for all methods, illustrates and provides MATLAB codes for all methods, and studies and tests all preconditioners first as stationary iterative solvers. This textbook is appropriate for undergraduate and graduate students who want an overview or deeper understanding of iterative methods. Its focus on both analysis and numerical experiments allows the material to be taught with very little preparation, since all the arguments are self-contained, and makes it appropriate for self-study as well. It can be used in courses on iterative methods, Krylov methods and preconditioners, and numerical optimal control. Scientists and engineers interested in new topics and applications will also find the text useful.

Solving nonlinear equations in Banach spaces (real or complex nonlinear equations, nonlinear systems, and nonlinear matrix equations, among others), is a non-trivial task that involves many areas of science and technology. Usually the solution is not directly affordable and require an approach using iterative algorithms. This Special Issue focuses mainly on the design, analysis of convergence, and stability of new schemes for solving nonlinear problems and their application to practical problems. Included papers study the following topics: Methods for finding simple or multiple roots either with or without derivatives, iterative methods for approximating different generalized inverses, real or complex dynamics associated to the rational functions resulting from the application of an iterative method on a polynomial. Additionally, the analysis of the convergence has been carried out by means of different sufficient conditions assuring the local, semilocal, or global convergence. This Special issue has allowed us to present the latest research results in the area of iterative processes for solving nonlinear equations as well as systems and matrix equations. In addition to the theoretical papers, several manuscripts on signal processing, nonlinear integral equations, or partial differential equations, reveal the connection between iterative methods and other branches of science and engineering.

Iterative Methods for Linear Systems?offers a mathematically rigorous introduction to fundamental iterative methods for systems of linear algebraic equations. The book distinguishes itself from other texts on the topic by providing a straightforward yet comprehensive analysis of the Krylov subspace methods, approaching the development and analysis of algorithms from various algorithmic and mathematical perspectives, and going beyond the standard description of iterative methods by connecting them in a natural way to the idea of preconditioning.??

Mathematics of Computing -- Numerical Analysis.

In this book, which focuses on the use of iterative methods for solving large sparse systems of linear equations, templates are introduced to meet the needs of both the traditional user and the high-performance specialist. Templates, a description of a general algorithm rather than the executable object or source code more commonly found in a conventional software library, offer whatever degree of customization the user may desire. Templates offer three distinct advantages: they are general and reusable; they are not language specific; and they exploit the expertise of both the numerical analyst, who creates a template reflecting in-depth knowledge of a specific numerical technique, and the computational scientist, who then provides "value-added" capability to the general template description, customizing it for specific needs. For each template that is presented, the authors provide: a mathematical description of the flow of algorithm; discussion of convergence and stopping criteria to use in the iteration; suggestions for applying a method to special matrix types; advice for tuning the template; tips on parallel implementations; and hints as to when and why a method is useful.

Iterative Solution of Large Linear Systems describes the systematic development of a substantial portion of the theory of iterative methods for solving large linear systems, with emphasis on practical techniques. The focal point of the book is an analysis of the convergence properties of the successive overrelaxation (SOR) method as applied to a linear system where the matrix is "consistently ordered". Comprised of 18 chapters, this volume begins by showing how the solution of a certain partial differential equation by finite difference methods leads to a large linear system with a sparse matrix. The next chapter reviews matrix theory and the properties of matrices, as well as several theorems of matrix theory without proof. A number of iterative methods, including the SOR method, are then considered. Convergence theorems are also given for various iterative methods under certain assumptions on the matrix A of the system. Subsequent chapters deal with the eigenvalues of the SOR method for consistently ordered matrices; the optimum relaxation factor; nonstationary linear iterative methods; and semi-iterative methods. This book will be of interest to students and practitioners in the fields of computer science and applied mathematics.