Abstract

. Many sparse matrix algorithms---for example, solving a sparse system of linear equations---begin by predicting the nonzero structure of the output of a matrix computation from the nonzero structure of its input. This paper is a catalog of ways to predict nonzero structure. It contains known results for problems including various matrix factorizations, and new results for problems including some eigenvector computations. Key words. sparse matrix algorithms, graph theory, matrix factorization, systems of linear equations, eigenvectors AMS(MOS) subject classifications. 15A18, 15A23, 65F50, 68R10 1. Introduction. A sparse matrix algorithm is an algorithm that performs a matrix computation in such a way as to take advantage of the zero/nonzero structure of the matrices involved. Usually this means not explicitly storing or manipulating some or all of the zero elements; sometimes sparsity can also be exploited to work on different parts of a matrix problem in parallel. Large sparse matr...

Citations

883	Graph theory - Harary - 1972
428	Computer Solution of Large Sparse Positive Definite Systems - George, Liu - 1981
415	Matching Theory - Lovász, Plummer - 1986
261	Algorithmic aspects of vertex elimination on graphs - Rose, Tarjan, et al. - 1976
88	A Graph-Theoretic Study of the Numerical Solution of Sparse Positive Definite Systems of Linear Equations - Rose - 1972
49	Sparse partial pivoting in time proportional to artithmetic operations - Gilbert, Peierls - 1988
44	Solution of Sparse Linear Least Squares Problems Using Givens Rotations - George, Heath - 1980
43	On algorithms for obtaining a maximum transversal - Duff - 1981
38	The use of linear graphs in Gauss elimination - Parter - 1961
35	Symbolic factorization for sparse Gaussian elimination with partial pivoting - George, Ng - 1987
33	An implementation of Gaussian elimination with partial pivoting for sparse systems - George, Ng - 1985
31	Elimination structures for unsymmetric sparse LU factors - R, Liu - 1993
25	Some design features of a sparse matrix code - DUFF, REID - 1979
25	Computational models and task scheduling for parallel sparse Cholesky factorization - Liu - 1986
23	Algorithms and data structures for sparse symmetric Gaussian elimination - Eisenstat, Schultz, et al. - 1981
21	Predicting fill for sparse orthogonal factorization - COLEMAN, EDENBRANDT, et al. - 1986
13	Exploiting structural symmetry in unsymmetric sparse symbolic factorization - Eisenstat, Liu - 1992
10	E.: On the complexity of sparse QR and LU factorization of finite-element matrices - George, Ng - 1988
10	Computing a sparse basis for the null space - Gilbert, Heath - 1987
10	Tarjan and Mihalis Yannakakis. Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs - Robert - 1984
9	The null space problem II: Algorithms - Coleman, Pothen - 1987
9	An efficient parallel sparse partial pivoting algorithm - Gilbert - 1988
9	den Driessche, Sparsity analysis of the QR factorization - Hare, Johnson, et al. - 1993
9	Endre Tarjan. A unified approach to path problems - Robert - 1981
8	The null space problem I: Complexity - Coleman, Pothen - 1986
8	A combinatorial approach to some sparse matrix problems - McCormick - 1983
5	Predicting the structure of sparse orthogonal factors - Pothen - 1991
4	Some results on sparse matrices - BRAYTON, GUSTAVSON, et al. - 1970
3	and Robert Endre Tarjan. Algorithmic aspects of vertex elimination on directed graphs - Rose - 1978
2	Some remarks on inverses of sparse matrices - Duff, Erisman, et al. - 1985
2	Inversion of bigraphs and connection with the Gauss elimination - FIEDLER - 1976
1	Optimal parallel solution of triangular sparse systems - Alvarado, Schreiber - 1992
1	den Driessche. Inherited matrix entries: Principal submatrices of the inverse - Barrett, Johnson, et al. - 1987
1	Gunnar Edenbrandt. Combinatorial Problems in Matrix Computation - Anders - 1985
1	Notes on the prediction of fill in some sparse matrix factorizations - Gilbert, Ng - 1992
1	den Driessche. Inherited matrix entries: LU factorization - Johnson, Olesky, et al.
1	Predicting structure in eigenvector computation - Mascarenhas - 1992
1	Sparse matrix techniques in geothermal reservoir modelling - Sigursson - 1991