Matrix
Editionsserious mathematics,
written with the reader in mind
Matrix
Editions
serious mathematics,
written with the reader in mind
Please consider recommending that your university or department librarian order this book. Thank you!
"Polish mathematician Woznicki (1937-2008) was one of the founders of incomplete factorization algorithms
and the associated iterative methods, and this account of his later work was in nearly finished form when
he died unexpectedly. It is devoted to the description and convergence analysis of iterative methods based on matrix
splittings and their implementation in mesh structures. Matrix splitting iterative methods are especially attractive for
solving linear equations with non-symmetric matrices, he says, because they yield more accurate solutions with significantly
less computations work than Krylov subspace methods. He pays special attention to developing efficient techniques
for a priori estimations of optimal acceleration parameters, which can be useful tools for solving many practical problems.
A new small publisher specializing in mathematics, Matrix Editions has scored a coup with this title." — © www.booknews.com
"This book contains a detailed treatment of linear algebra, and how it can be applied to the iterative solution of
elliptic boundary-value problems. In Chapter 5 the author attacks the interesting and difficult problem of comparing
the numerical work required to solve large elliptic problems by his SOR-like AGA method, with the numerical work required
using newer computational methods, such as Krylov subspace methods and GMRES.
His conclusions here are quite unexpected! Indeed, for some problems, he found that the SOR-like methods were superior to Krylov subspace methods by several orders of magnitude. This chapter and the corresponding numerical experiments detailed in the appendix form one of the most interesting contributions of the book.
He also gives a very complete study of the theory of matrix splittings
in Chapter 2, which includes his own research in this area. Finally,
he includes a new and interesting chapter on the use of iterative methods in solving linear control problems, such as the Sylvester equation and continuous-time algebraic Riccati equations.
This book is not a textbook with exercises for the reader; rather, it is a book that
focuses on the particular problem of solving elliptic boundary-value problems by iterative methods. It should interest many readers." — Richard Varga, Emeritus
University Professor of Mathematics, Kent State University
This book is devoted to the description and convergence analysis of iterative methods based on matrix splitting, and their implementation in mesh structures. It is, in some sense, a summary of the author’s results and experience gained in the field of numerical linear algebra.
Special attention is paid to developing efficient techniques for a priori
estimates of optimal acceleration parameters, as useful tools when solving many problems of practical interest.
First, the book summarizes fundamentals of numerical linear algebra and matrix computations.
Then, it discusses matrix splitting theory, providing comparison
theorems proved for different types of splitting, as useful tools in convergence analysis of iterative methods. Next,
the book deals with discretization aspects of elliptic partial differential equations. Standard
iterative methods and useful procedures for determining optimal acceleration parameters are then
presented. A significant portion of the text is devoted to the study of a large family of prefactorization iterative methods,
introduced by the author, and divided into three groups: explicit prefactorization algorithms, semi-explicit algorithms with implicit backward sweep, and semi-explicit algorithms with implicit forward sweep.
The book concludes with a discussion of recent results
obtained by the author for solving large Sylvester and algebraic Riccati equations. The numerical results produce the unexpected conclusion that for some problems, the author’s SOR-like AGA method is superior to Krylov subspace methods by several orders of magnitude.
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.
Zbigniew Woźnicki died in March, 2008, before editing was completed. Two of his friends and colleagues, Janusz Mika of the Institute of Nuclear Research in Otwock, Poland, and Robert Beauwens of the Université Libre de Bruxelles, wrote the following notice.
Zbigniew Woźnicki (Institute of Atomic Energy of Poland) passed away early morning on Saturday 1 March, 2008, as the result of a mesenteric embolism, leaving his wife Ewa and two daughters Agnieszka and Ola.
Together with Varga, van der Vorst and Jennings, he was one of the fathers of the incomplete factorization algorithms and of the associated iterative methods. His PhD thesis, published in 1973, although at the beginning not appropriately acknowledged, represented a true breakthrough in the development of these methods, as well as in the general theory of regular splitting, paving the way to countless extensions and leading to many present day applications of ILU factorization algorithms and of their numerous variants.
His recent works included the use of the iterative solution of large linear problems arising in the field of control system design and analysis. The community of matrix iterative analysts lost one of its most prominent members. We, his colleagues, lost a warm and devoted friend.