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Reviewer: Ralph Walter Wilkerson

This monograph presents basic theory, algorithms, and data structures for sparse matrix computations. The book begins with the fundamentals, and builds on the C programming language in order to construct a downloadable software library, CSparse, which contains the basic features of a sparse matrix package. The book covers basic decompositions, such as Cholesky factorization, orthogonal-triangular factorization, and lower-upper factorization, and uses them to solve sparse linear systems. One of the book’s stated goals is to give the reader a better understanding of what takes place in Matlab’s sparse matrix library of functions. The book is divided into 10 chapters, and the first two are devoted to fundamental matrix data structures and matrix operations in the C programming language. Chapter 3 is a short chapter devoted to solving sparse triangular linear systems (skills that are used in the next three chapters). Chapters 5, 6, and 7 concentrate on the three major decomposition methods: Cholesky, QR, and LU. Of course, the goal is to develop methods for sparse linear systems in C. Chapter 7 is devoted to fill-reducing orderings, which are used to minimize the work done by the previous algorithms. Chapters 8, 9, and 10 bring all these methods together to solve general sparse linear systems (with chapters 9 and 10 being specific to CSparse and Matlab implementations). The book is based on the author’s coursework, and each chapter ends with a number of exercises designed to reinforce the methods under consideration. To best appreciate this book, readers should complete the programming exercises and study the (included) code for CSparse. Online Computing Reviews Service

Reviewer: Mike Minkoff

The solution of sparse linear systems is at the core of much of computational science, since such systems arise via approximations of operator equations (for example, fluid-flow and computational-mechanics models). This book specifically addresses the direct solution of sparse linear systems by placing itself in the interface of numerical theory, algorithmic practice, programming software (via C) and commonly used turnkey packages (specifically, MATLAB). Additionally, it provides a thorough collection of software (CSparse) to illustrate the features presented in the book. This software is completely available, and is maintained on a Web site. CSparse is written in C and is about 2200 lines in length. It presents many of the elements of faster production and commercial codes, such as the over 100,000 lines in MATLAB for solving such systems. The author clearly states his objectives in the preface. He limits himself to direct methods, and does not address iterative techniques. He addresses the solution of linear systems by using sparse LU, Cholesky, and QR factorization methods; he does not address eigenvalue problems. The book proceeds in an orderly manner of presentation, carefully discussing C, MATLAB, and theoretical results in a combined and clear manner. Each chapter includes a thorough set of exercises, references, and suggestions for further reading. After a brief introductory chapter that addresses the fundamentals of linear algebra, algorithms, graph theory, and data structures, Davis addresses pseudocode and equivalent C code for basic algorithms (for example, matrix-vector multiplication and addition, and reading and printing a matrix). He also discusses data representation (triplet form and compressed-column form). Using these foundations, he continues on to the solution of triangular systems, Cholesky factorization, orthogonalization methods, LU factorization, and fill-reducing orderings. This material is thoroughly presented, and comprises more than two-thirds of the book. Using this foundation, the author presents the solution of sparse linear systems for all three factorizations and the Dulmage-Mendelsohn decomposition. Additionally, he relates the presentation to the "standard" MATLAB command "x=A\b," and provides a thorough survey of available software for solving sparse systems, as of April 2006. These results are, however, updated on the Web site, where the CSparse software is also available (www.siam.org/books/fa02). The remaining two chapters are software oriented, examining CSparse in detail (all of the code appears in the book as well as on the Web site), and ending with a chapter on handling sparse matrices via MATLAB as well as an interface between CSparse and MATLAB. The book concludes with an appendix on C programming, for use as a reference to the language, and a bibliography of 200 references. This is an excellent book, addressing the theory and practice of solving sparse linear systems. While one can use MATLAB and be confident that one is accessing top-quality linear algebra software "under the hood," it is important for the serious student of numerical linear algebra and computational software to know what makes the racecar work; this book does that excellently. Online Computing Reviews Service