Sign in
Help
View Cart
Manage this Book
Recommend & Share
Notify Me!
Title Information
Since the first edition of this book was published in 1996, tremendous progress has been made in the scientific and engineering disciplines regarding the use of iterative methods for linear systems. The size and complexity of the new generation of linear and nonlinear systems arising in typical applications has grown. Solving the three-dimensional models of these problems using direct solvers is no longer effective. At the same time, parallel computing has penetrated these application areas as it became less expensive and standardized. Iterative methods are easier than direct solvers to implement on parallel computers but require approaches and solution algorithms that are different from classical methods.
Iterative Methods for Sparse Linear Systems, Second Edition gives an in-depth, up-to-date view of practical algorithms for solving large-scale linear systems of equations. These equations can number in the millions and are sparse in the sense that each involves only a small number of unknowns. The methods described are iterative, i.e., they provide sequences of approximations that will converge to the solution.
This new edition includes a wide range of the best methods available today. The author has added a new chapter on multigrid techniques and has updated material throughout the text, particularly the chapters on sparse matrices, Krylov subspace methods, preconditioning techniques, and parallel preconditioners. Material on older topics has been removed or shortened, numerous exercises have been added, and many typographical errors have been corrected. The updated and expanded bibliography now includes more recent works emphasizing new and important research topics in this field.
In the six years that have passed since the publication of the first edition of this book, iterative methods for linear systems have made good progress in scientific and engineering disciplines. This is due in great part to the increased complexity and size of the new generation of linear and nonlinear systems that arise from typical applications. At the same time, parallel computing has penetrated the same application areas, as inexpensive computer power has become broadly available and standard communication languages such as MPI have provided a much needed standardization. This has created an incentive to utilize iterative rather than direct solvers, because the problems solved are typically from threedimensional models for which direct solvers often become ineffective. Another incentive is that iterative methods are far easier to implement on parallel computers.
Although iterative methods for linear systems have seen a significant maturation, there are still many open problems. In particular, it still cannot be stated that an arbitrary sparse linear system can be solved iteratively in an efficient way. If physical information about the problem can be exploited, more effective and robust methods can be tailored to the solutions. This strategy is exploited by multigrid methods. In addition, parallel computers necessitate different ways of approaching the problem and solution algorithms that are radically different from classical ones.
Several new texts on the subject of this book have appeared since the first edition. Among these are the books by Greenbaum [154] and Meurant [208]. The exhaustive five-volume treatise by G. W. Stewart [273] is likely to become the de facto reference in numerical linear algebra in years to come. The related multigrid literature has also benefited from a few notable additions, including a new edition of the excellent Multigrid Tutorial [65] and a new title by Trottenberg et al. [285].
Most notable among the changes from the first edition is the addition of a sorely needed chapter on multigrid techniques. The chapters that have seen the biggest changes are Chapters 3, 6, 10, and 12. In most cases, the modifications were made to update the material by adding topics that have been developed recently or gained importance in the last few years. In some instances some of the older topics were removed or shortened. For example, the discussion on parallel architecture has been shortened. In the mid-1990s hypercubes and “fat-trees” were important topics to teach. This is no longer the case, since manufacturers have taken steps to hide the topology from the user, in the sense that communication has become much less sensitive to the underlying architecture.
