Predicting fill for sparse orthogonal factorization

In solving large sparse linear least squares problems <italic>A</italic> x ≃ b, several different numeric methods involve computing the same upper triangular factor <italic>R</italic> of <italic>A</italic>. It is of interest to be able to compute the nonzero structure of <italic>R</italic>, given only the structure of <italic>A</italic>. The solution to this problem comes from the theory of matchings in bipartite graphs. The structure of <italic>A</italic> is modeled with a bipartite graph, and it is shown how the rows and columns of <italic>A</italic> can be rearranged into a structure from which the structure of its upper triangular factor can be correctly computed. Also, a new method for solving sparse least squares problems, called block back-substitution, is presented. This method assures that no unnecessary space is allocated for fill, and that no unnecessary space is needed for intermediate fill.

[1]  P. Hall.  On Representatives of Subsets , 1935 .

[2]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[3]  N. S. Mendelsohn,et al.  Two Algorithms for Bipartite Graphs , 1963 .

[4]  H. Fédérer.  Geometric Measure Theory , 1969 .

[5]  Alan George,et al.  Computer Solution of Large Sparse Positive Definite , 1981 .

[6]  Iain S. Duff,et al.  On Algorithms for Obtaining a Maximum Transversal , 1981, TOMS.

[7]  M. Heath.  Some Extensions of an Algorithm for Sparse Linear Least Squares Problems , 1982 .

[8]  M. Heath.  Numerical Methods for Large Sparse Linear Least Squares Problems , 1984 .

[9]  M. Heath.  Sparse matrix computations , 1984, The 23rd IEEE Conference on Decision and Control.

[10]  R. Willoughby,et al.  Some results on sparse matrices , 1970 .

[11]  A. George,et al.  Solution of sparse linear least squares problems using givens rotations , 1980 .

[12]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[13]  N. S. Mendelsohn,et al.  Coverings of Bipartite Graphs , 1958, Canadian Journal of Mathematics.

[14]  A. George,et al.  An Implementation of Gaussian Elimination with Partial Pivoting for Sparse Systems , 1985 .

[15]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[16]  Journal of the Association for Computing Machinery , 1961, Nature.

[17]  W. Morven Gentleman,et al.  Row elimination for solving sparse linear systems and least squares problems , 1976 .

[18]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[19]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[20]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[21]  F. Gustavson.  FINDING THE BLOCK LOWER TRIANGULAR FORM OF A SPARSE MATRIX , 1976 .

[22]  D. R. Fulkerson,et al.  Flows in Networks. , 1964 .

[23]  A. George,et al.  Row-ordering schemes for sparse givens transformations. II. implicit graph model , 1984 .

[24]  T. D. Howell.  Partitioning Using PAQ , 1975 .

[25]  G. Golub,et al.  Linear least squares solutions by householder transformations , 1965 .

[26]  N. S. Mendelsohn,et al.  Connectivity and Reducibility of Graphs , 1962, Canadian Journal of Mathematics.