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This book offers the first comprehensive account of the new method of density matrix renormalization. Recent years have seen enormous progress in the numerical treatment of low-dimensional quantum sytems. With this new technique, which selects a reduced set of basis states via density matrices, it has become possible to treat large systems with amazing accuracy. The method has been applied successfully to a variety of important one-dimensional problems such as spin chains, Kondo models, and correlated electron systems. Extensions to other systems and higher dimensions are currently being developed. The contributions to this book are written by leading experts in the field. The two parts contain an introduction to the subject and a review of physical applications. As a combination of advanced textbook and guide to current research the book should become a standard source for everyone interested in the topic. TOC:Introductory Lectures: Wilson's Numerical Renormalization Group; The Density Matrix Renormalization Group; Thermodynamic Limit and Matrix-Product States; A Recurrent Variational Approach; Transfer-Matrix Approach to Classical Systems; Transfer-Matrix and Momentum-Space DMRG; Calculation of Dynamical Properties.- Physical Applications: Chains and Ladders; Molecules and Polymers; Classical Statistical Physics; Thermodynamic Properties; Phonons and Disorder.