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Solving Linear Systems

An Analysis of Matrix Prefactorization Iterative Methods

by Zbigniew Ignacy Woźnicki


Iterative methods for solving systems of linear equation  form a  beautiful, living, and useful field  of numerical linear algebra. Beautiful, because it is full of powerful ideas and theoretical results, and living, because it is a rich source of well-established algorithms for accurate solutions of many large and sparse linear systems.  Useful,  because solutions of large systems of linear equations are essential in many fields.

 The recent literature on iterative methods has been dominated by Krylov subspace methods, currently the most common group of techniques used in applications. The combination of preconditioning and Krylov subspace iterations for solving nonsymmetric linear systems has become a central area of research and new techniques are still emerging. In the vast literature there are a thousand different algorithms, and many derivations and  estimates  of error bounds;  it is difficult for a typical reader or user (and sometimes even the specialist) to identify the basic principles involved and estimate the performance of particular algorithms.

These new techniques are called parameter-free methods, because they can be applied without   
knowing the inner properties of the matrices that one must know in the case of traditional methods.

There is a general feeling that traditional iterative methods, based on matrix splittings, are usually less efficient than the Krylov methods, and that to use these methods effectively one must resort to rather complicated procedures   for determining optimal acceleration parameters.  Therefore, in current applications, the traditional methods have generally been relegated to  the role of preconditioners. However, as can be concluded from numerical experiments described in this book, such an opinion is unjustified in many cases.

 This book is devoted to the description and convergence analysis of iterative methods based on matrix splittings and their implementation in mesh structures. It is  in some sense a summary of the author's results and experience gained in this important field of numerical linear algebra.

Matrix splitting iterative methods are especially attractive for
solving linear equations with nonsymmetric matrices because these methods  yield more
accurate solutions, with  significantly less computational work, than   Krylov subspace methods. Special attention is paid to
developing efficient techniques for {\itshape a priori} estimations of optimal
acceleration parameters, as useful tools when solving many problems of
practical interest.

The   book is organized as follows. Chapter 1   summarizes fundamentals of numerical linear algebra and matrix computations. Chapter 2 discusses matrix splitting theory, providing comparison theorems proved for different types of splittings, as useful tools in convergence analysis of iterative methods.  Chapter 3 deals with discretization aspects of elliptic partial differential equations. Standard iterative methods and useful procedures for determining optimal acceleration parameters are presented in chapter 4. Chapters 5--7 are devoted to the study of a large family of prefactorization iterative methods, introduced by the author, and divided into
three groups: explicit  prefactorization  algorithms, semiexplicit algorithms  with implicit backward sweep, and  semiexplicit algorithms  with implicit forward sweep. Recent advances obtained by the author for solving large Sylvester and algebraic Riccati equations are discussed in chapter 8.

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