An introduction to chordal graphs and clique trees.- Cutting down on fill using nested dissection: Provably good elimination orderings.- Automatic Mesh Partitioning.- Structural representations of Schur complements in sparse matrices.- Irreducibility and primitivity of Perron complements: Application of the compressed directed graph.- Predicting structure in nonsymmetric sparse matrix factorizations.- Highly parallel sparse triangular solution.- The fan-both family of column-based distributed Cholesky factorization algorithms.- Scalability of sparse direct solvers.- Sparse matrix factorization on SIMD parallel computers.- The efficient parallel iterative solution of large sparse linear systems.
This IMA Volume in Mathematics and its Appllcations GRAPH THEORY AND SPARSE MATRIX COMPUTATION is based on the proceedings of a workshop that was an integraI part of the 1991- 92 IMA program on "Applied Linear AIgebra." The purpose of the workshop was to bring together people who work in sparse matrix computation with those who conduct research in applied graph theory and grl:l,ph algorithms, in order to foster active cross-fertilization. We are grateful to Richard Brualdi, George Cybenko, Alan Geo~ge, Gene Golub, Mitchell Luskin, and Paul Van Dooren for planning and implementing the year-Iong program. We espeeially thank Alan George, John R. Gilbert, and Joseph W.H. Liu for organizing this workshop and editing the proceedings. The finaneial support of the National Science Foundation made the workshop possible. A vner Friedman Willard Miller. Jr. PREFACE When reality is modeled by computation, linear algebra is often the con nec tiori between the continuous physical world and the finite algorithmic one. Usually, the more detailed the model, the bigger the matrix, the better the answer. Efficiency demands that every possible advantage be exploited: sparse structure, advanced com puter architectures, efficient algorithms. Therefore sparse matrix computation knits together threads from linear algebra, parallei computing, data struetures, geometry, and both numerieal and discrete algorithms.
This book looks at graph theory as it connects to linear algebra, parallel computing, data structures, geometry, and both numerical and discrete algorithms. This book will be a resource for the researcher or advanced student of either graphs or sparse matrices; it will be useful to mathematicians, numerical analysis and theoretical computer scientists alike.