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Designed for use by first-year graduate students from a variety of engineering and scientific disciplines, this comprehensive textbook covers the solution of linear systems, least squares problems, eigenvalue problems, and the singular value decomposition. The author, who helped design the widely-used LAPACK and ScaLAPACK linear algebra libraries, draws on this experience to present state-of-the-art techniques for these problems, including recommendations of which algorithms to use in a variety of practical situations. This is the book for you if you are looking for a textbook that:* Teaches state-of-the-art techniques for solving linear algebra problems. * Covers the most important methods for dense and sparse problems.* Presents both the mathematical background and good software techniques. * Is self-contained, assuming only a good undergraduate background in linear algebra. Algorithms are derived in a mathematically illuminating way, including condition numbers and error bounds. Direct and iterative algorithms, suitable for dense and sparse matrices, are discussed. Algorithm design for modern computer architectures, where moving data is often more expensive than arithmetic operations, is discussed in detail, using LAPACK as an illustration. There are many numerical examples throughout the text and in the problems at the ends of chapters, most of which are written in Matlab and are freely available on the Web. Material either not available elsewhere, or presented quite differently in other textbooks, includes:* A discussion of the impact of modern cache-based computer memories on algorithm design.* Frequent recommendations and pointers in the text to the best software currently available, including a detailed performance comparison of state-of-the-art software for eigenvalue and least squares problems, and a description of sparse direct solvers for serial and parallel machines. * A discussion of iterative methods ranging from Jacobi's method to multigrid and domain decomposition, with performance comparisons on a model problem.* A great deal of Matlab-based software, available on the Web, which either implements algorithms presented in the book, produces the figures in the book, or is used in homework problems.* Numerical examples drawn from fields ranging from mechanical vibrations to computational geometry.* High-accuracy algorithms for solving linear systems and eigenvalue problems, along with tighter "relative" error bounds. * Dynamical systems interpretations of some eigenvalue algorithms. Demmel discusses several current research topics, making students aware of both the lively research taking place and connections to other parts of numerical analysis, mathematics, and computer science. Some of this material is developed in questions at the end of each chapter, which are marked Easy, Medium, or Hard according to their difficulty. Some questions are straightforward, supplying proofs of lemmas used in the text. Others are more difficult theoretical or computing problems. Questions involving significant amounts of programming are marked Programming. The computing questions mainly involve Matlab programming, and others involve retrieving, using, and perhaps modifying LAPACK code from NETLIB.

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 have 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 the 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.