Iterative methods for linear systems: theory and applications

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.

Audience: The book supplements standard texts on numerical mathematics for first-year graduate and advanced undergraduate courses and is suitable for advanced graduate classes covering numerical linear algebra and Krylov subspace and multigrid iterative methods. It will be useful to researchers interested in numerical linear algebra and engineers who use iterative methods for solving large algebraic systems.

Contents: Chapter 1: Krylov Subspace Methods; 1.1: Simple iterative methods; 1.2: Subspaces and iterative methods; 1.3: Analysis of the minimal residual method; 1.4: Analysis of the conjugate gradient method; Chapter 2: Toeplitz Matrices and Preconditioners; 2.1: Introduction to Toeplitz Matrices; 2.2: Preconditioners and applications; Chapter 3: Multigrid Preconditioners; 3.1: The introductory section; 3.2: Two-grid iteration; 3.3: Multigrid iteration; 3.4: Convergence analysis; Chapter 4: Preconditioners by Space Decomposition; 4.1: Space decomposition framework; 4.2: Grid decomposition methods; 4.3: Domain decomposition methods; 4.4: Convergence analysis for the Poisson problem; Chapter 5: Some Applications; 5.1: Multigrid preconditioners for singular-perturbed problems; 5.2: Preconditioners for certain problems of fluid mechanics; Bibliography; Index