## Finding Good Column Orderings for Sparse QR Factorization (1996)

Venue: | In Second SIAM Conference on Sparse Matrices |

Citations: | 17 - 0 self |

### BibTeX

@TECHREPORT{Heggernes96findinggood,

author = {Pinar Heggernes and Pontus Matstoms},

title = {Finding Good Column Orderings for Sparse QR Factorization},

institution = {In Second SIAM Conference on Sparse Matrices},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

For sparse QR factorization, finding a good column ordering of the matrix to be factorized, is essential. Both the amount of fill in the resulting factors, and the number of floating-point operations required by the factorization, are highly dependent on this ordering. A suitable column ordering of the matrix A is usually obtained by minimum degree analysis on A T A. The objective of this analysis is to produce low fill in the resulting triangular factor R. We observe that the efficiency of sparse QR factorization is also dependent on other criteria, like the size and the structure of intermediate fill, and the size and the structure of the frontal matrices for the multifrontal method, in addition to the amount of fill in R. An important part of this information is lost when A T A is formed. However, the structural information from A is important to consider in order to find good column orderings. We show how a suitable equivalent reordering of an initial fill-reducing ordering can...

### Citations

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Citation Context ...same. The filled graph is reordered, and hence the sparsity pattern of the new resulting Cholesky factor is changed, although its graph might be the same as before. Following the definition by Tarjan =-=[33]-=-, a topological ordering of the elimination tree is an ordering that numbers the children nodes before their parent node. Therefore, topological orderings preserve the elimination tree. Postorderings,... |

537 |
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Citation Context ...f 2, the remaining graph in the elimination process is complete, and all three of the vertices 3, 4 and 5 are simplicial. Thus no further fill is introduced. The filled graph G + is chordal, and ff = =-=[1; 5; 4; 2; 3]-=- and fi = [1; 2; 3; 4; 5] are two of its perfect elimination orderings. As defined by Rose, Tarjan and Lueker [31], a minimal fill ordering on a graph is an ordering such that no proper subgraph of th... |

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Citation Context ...s no further fill is introduced. The filled graph G + is chordal, and ff = [1; 5; 4; 2; 3] and fi = [1; 2; 3; 4; 5] are two of its perfect elimination orderings. As defined by Rose, Tarjan and Lueker =-=[31]-=-, a minimal fill ordering on a graph is an ordering such that no proper subgraph of the resulting filled graph is chordal. Note that the initial ordering of M in the example of Figure 2 is a minimal f... |

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Citation Context ...ch the reader is assumed to be familiar with. Table 1 shows the test matrices we use for our numerical experiments. Some of the matrices are from the Harwell-Boeing collection (Duff, Grimes and Lewis =-=[5]-=-). The others are artificially made by concatenating two Harwell-Boeing matrices. The matrices that are marked with an asterisk do not have the assumed Strong Hall Property, explained in Section 2. Ma... |

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Citation Context ...ination ordering of a chordal graph can be found by eliminating a simplicial vertex at each step of the elimination game. These results on chordal graphs are due to Dirac [2], and Fulkerson and Gross =-=[11]-=-. Let us show these graph issues on an example. Figure 2 shows a symmetric matrix M and its Cholesky factor L T . The graph G = G(M) and the filled graph G + = G + (M) are also shown. In G, vertex 1 i... |

129 | Sparse matrices in MATLAB: design and implementation
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Citation Context ...is, however, not true in implementations where computed rows of R overwrite eliminated elements of A. A better alternative, proposed by George and Liu [14] and implemented in matlab by Gilbert et al. =-=[18]-=-, is to operate directly on the matrix A. Then the nonzero structure of A T A need not be explicitly formed, and savings in efficiency can be expected. An even more important argument, mentioned in Se... |

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Citation Context ...rtices of G. We use G ff to denote the renumbered graph, and G + ff to denote the resulting filled graph. A perfect elimination ordering of G is an elimination ordering which results in no fill. Rose =-=[30]-=- has shown that all filled graphs are chordal. Chordal graphs are exactly the class of graphs which have perfect elimination orderings. A simplicial vertex in a chordal graph is a vertex whose neighbo... |

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Citation Context ...m (Tinney scheme 2) by Tinney and Walker [34] is a well-known and often used symmetric fill-reducing method. It is a greedy heuristic, and is a symmetric version of the Markowitz algorithm (Markowitz =-=[25]-=-) for unsymmetric matrices. Suppose that the first k columns of the symmetric matrix M 2 R n\Thetan have been eliminated. In the partially eliminated matrix, M k = ` R (k) 1 R (k) 2 0 M (k) ' ; the fi... |

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Citation Context ...o the above criteria, the resulting triangular matrix R should be sparse, while at the same time the factorization cost is reduced. The minimum degree algorithm (Tinney scheme 2) by Tinney and Walker =-=[34]-=- is a well-known and often used symmetric fill-reducing method. It is a greedy heuristic, and is a symmetric version of the Markowitz algorithm (Markowitz [25]) for unsymmetric matrices. Suppose that ... |

60 |
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Citation Context .... Accordingly, a perfect elimination ordering of a chordal graph can be found by eliminating a simplicial vertex at each step of the elimination game. These results on chordal graphs are due to Dirac =-=[2]-=-, and Fulkerson and Gross [11]. Let us show these graph issues on an example. Figure 2 shows a symmetric matrix M and its Cholesky factor L T . The graph G = G(M) and the filled graph G + = G + (M) ar... |

59 |
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Citation Context ...P, then symbolic Cholesky factorization, and also symbolic Givens and Householder factorization, correctly predict the nonzero structure of R. By a reordering scheme studied by Dulmage and Mendelsohn =-=[8]-=-, [9], [10], a general matrix can be permuted into block triangular form with diagonal blocks having the SHP. In the solution of a least squares problem where the coefficient matrix has this block tri... |

51 |
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Citation Context ...f 2, the remaining graph in the elimination process is complete, and all three of the vertices 3, 4 and 5 are simplicial. Thus no further fill is introduced. The filled graph G + is chordal, and ff = =-=[1; 5; 4; 2; 3]-=- and fi = [1; 2; 3; 4; 5] are two of its perfect elimination orderings. As defined by Rose, Tarjan and Lueker [31], a minimal fill ordering on a graph is an ordering such that no proper subgraph of th... |

50 |
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Citation Context ...term G(L + L T ). When M and L are clear from the context, we will only use G and G + . The filled graph G + can be obtained from G without computing L, with help of the following algorithm by Parter =-=[28]-=-, called the elimination game: for i = 1 : n, add edges to make all neighbors of vertex i pairwise adjacent; delete vertex i from the graph; end The graph G + is obtained by adding to G all the new ed... |

49 |
A new implementation of sparse Gaussian elimination
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Citation Context ... i j r ij 6= 0g: In the elimination tree, node i corresponds to the elimination of column i in A. Let i and j ? i be two nodes in the tree and let T [j] denote the subtree rooted at node j. Schreiber =-=[32]-=- shows that r ij = 0 if i = 2 T [j]. It follows that the columns corresponding to nodes in disjoint subtrees are independent, and can be eliminated in any order. Note, however, that r ij might be zero... |

48 |
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(Show Context)
Citation Context ...actorization. Researchers This research is partly supported by The Research Council of Norway. 1 have concentrated on finding column orderings that result in as sparse R as possible. George and Heath =-=[12]-=- have observed that a suitable column ordering of A can be found by symmetric fill-reducing analysis of A T A. Efficient methods, like minimum degree and nested dissection, exist for fill-reduction in... |

39 |
Symbolic factorization for sparse Gaussian elimination with partial pivoting
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(Show Context)
Citation Context ...ing strategies. This paper proposes methods for tie-breaking in minimum degree, which study the structure of A to produce column orderings that are suitable for sparse QR factorization. George and Ng =-=[16]-=- show a connection between structural Householder QR factorization and structural LU factorization. For this purpose, it is interesting to find good orderings not only for the multifrontal algorithm, ... |

35 |
Sparse QR Factorization in MATLAB
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(Show Context)
Citation Context ...entries. The introduction of multifrontal methods have made direct methods based on sparse QR factorization attractive and competitive to previously recommended alternatives, as explained by Matstoms =-=[26]-=-. In this paper, we consider the QR factorization, A = Q ` R 0 ' ; of a large and sparse Strong Hall matrix A 2 R m\Thetan of full column rank, where msn. The column ordering on A is decisive for the ... |

30 | Predicting structure in nonsymmetric sparse matrix factorizations
- Gilbert, Ng
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(Show Context)
Citation Context ...nal Householder algorithm introduces too much intermediate fill, and can in practice therefore not be used. However, an interesting connection to sparse LU factorization (Gilbert [17], Gilbert and Ng =-=[19]-=-) makes the traditional Householder algorithm still interesting in real applications. The main result is presented in Theorem 5.1. Theorem 5.1 (George and Ng [16]) Let M 2 R n\Thetan be a nonsingular ... |

25 |
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Citation Context ...: 4 The matrix A 2 R m\Thetan and its bipartite graph have the Strong Hall Property (SHP) if every m \Theta k submatrix, for 1sk ! n, has at least k + 1 nonzero rows. Coleman, Edenbrandt, and Gilbert =-=[1]-=- show that, if A has the SHP, then symbolic Cholesky factorization, and also symbolic Givens and Householder factorization, correctly predict the nonzero structure of R. By a reordering scheme studied... |

20 |
A compact row storage scheme for Cholesky factors using elimination trees
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(Show Context)
Citation Context ... numbers the children nodes before their parent node. Therefore, topological orderings preserve the elimination tree. Postorderings, mentioned in Section 2, are a subset of topological orderings. Liu =-=[21]-=- observes that any topological ordering of the elimination tree corresponds to an equivalent reordering of the matrix M . However, as we have mentioned above, the opposite is not true. All the leaves ... |

16 |
QR Factorization of large sparse overdetermined and square matrices using a multifrontal method in a multiprocessor environment
- Puglisi
- 1993
(Show Context)
Citation Context ...n important starting point for today's algorithms. References on the way from George and Heath's paper to modern multifrontal methods include Liu [22], George and Liu [13], Lewis et al. [20], Puglisi =-=[29]-=- and Matstoms [26]. There are essentially three important features characterizing the multifrontal method for sparse QR factorization. First, flexibility in the elimination order of the columns is pro... |

12 |
private communication
- Gilbert
- 1998
(Show Context)
Citation Context ...atrices. The traditional Householder algorithm introduces too much intermediate fill, and can in practice therefore not be used. However, an interesting connection to sparse LU factorization (Gilbert =-=[17]-=-, Gilbert and Ng [19]) makes the traditional Householder algorithm still interesting in real applications. The main result is presented in Theorem 5.1. Theorem 5.1 (George and Ng [16]) Let M 2 R n\The... |

7 |
A comparison of some methods for the solution of sparse overdetermined systems of linear equations
- Duff, Reid
- 1976
(Show Context)
Citation Context ...teresting and recommended for sparse least squares problems (Matstoms [26]). We should keep in mind that, until the early 80s, QR factorization was rejected for general sparse problems (Duff and Reid =-=[6]-=-). The main explanation for why QR factorization has become useful, is the multifrontal technique. This technique makes it possible to use Householder transformations efficiently also for sparse matri... |

6 | Householder reflections versus Givens rotations in sparse orthogonal decomposition - George, Liu - 1987 |

5 |
sparse matrix reordering by elimination tree rotations
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(Show Context)
Citation Context ...or each ordering ff, there can be found a permutation matrix P such that G ff = G(P T MP ). In matlab notation, P can be expressed as P = I n (:; ff), where I n is the n \Theta n identity matrix. Liu =-=[23]-=- gives the following definition for equivalent reorderings. Definition 3.1 An elimination ordering ff on G is an equivalent reordering of G if G + ff = G + , that is, the new ordering ff produces a fi... |

3 |
general row merging schemes for sparse Givens transformations
- On
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(Show Context)
Citation Context ...ivens rotations, makes the paper by George and Heath an important starting point for today's algorithms. References on the way from George and Heath's paper to modern multifrontal methods include Liu =-=[22]-=-, George and Liu [13], Lewis et al. [20], Puglisi [29] and Matstoms [26]. There are essentially three important features characterizing the multifrontal method for sparse QR factorization. First, flex... |

2 |
Row ordering schemes for sparse Givens transformations
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(Show Context)
Citation Context ...s to make all vertices in S adjacent to all vertices in T ; delete row-vertex i and column-vertex i from the graph; end A similar process is described for Givens transformations by George, Liu and Ng =-=[15]-=-. The filled graph B + is obtained by adding to B all the edges suggested by this algorithm. The bipartite elimination graph B i is the resulting graph after the elimination of column and row i. Thus,... |

2 | QR Factorization with Applications to Linear Least Squares Problems - Sparse - 1994 |

1 |
On some numerical methods for solving large sparse linear least squares problems
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(Show Context)
Citation Context ... patterns. George and Heath [12] show that Givens prediction is always at least as good as symbolic Cholesky factorization. The same is true for symbolic Householder prediction, as shown by Manneback =-=[24]-=-. Altogether this can be summarized as follows: struct(R) ` ae pred(Householder on A) pred(Givens on A) oe ` pred(Cholesky on A T A): 4 The matrix A 2 R m\Thetan and its bipartite graph have the Stron... |