## Diagonal markowitz scheme with local symmetrization (2003)

Venue: | SIAM J. Matrix Anal. Appl |

Citations: | 8 - 2 self |

### BibTeX

@TECHREPORT{Amestoy03diagonalmarkowitz,

author = {Patrick R. Amestoy and Xiaoye S. Li and Esmond G. Ng},

title = {Diagonal markowitz scheme with local symmetrization},

institution = {SIAM J. Matrix Anal. Appl},

year = {2003}

}

### OpenURL

### Abstract

y work of this author was performed while he was on a sabbatical visit to NERSC.

### Citations

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Citation Context ...merical values into consideration. The most common approach is to pick a nonzero entry as the next pivot as long as its Markowitz weight is reasonably small and its magnitude is reasonably large. See =-=[15]-=- for further details. Alternatively, one can use the approach proposed by George and Ng [21]. In this case, a permutation P is chosen to reduce fill in the Cholesky factorization of A T A; some exampl... |

502 |
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Citation Context ...rage required to implement the elimination model. It is possible, however, to characterize the elimination graphs in terms of the original graph G 0 using the notion of reachable sets. George and Liu =-=[20]-=- gave such a characterization for the symmetric case with undirected graphs. Pagallo and Maulino [31] extended it to the nonsymmetric case with bipartite graphs. The "interior" vertices on a reachable... |

282 |
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Citation Context ...ults from numerical simulations to show that indeed the approach is reasonable. In our simulations, we considered all the square nonsymmetric matrices from the Harwell-Boeing Sparse Matrix Collection =-=[11]-=-. Four matrices (MBEACXC, MBEAFLW, MBEAUSE, and SAYLR3) are excluded in the following discussion since they are rank-deficient. For each matrix A in the collection, we performed the following experime... |

255 | An approximate minimum degree ordering algorithm
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Citation Context ...ph model is bounded above by the size of the original symmetric matrix [20]. The quotient graph model has been used extensively in many minimum-degree type of ordering codes, such as MMD [28] and AMD =-=[1]-=-. The key idea is a compact representation of the subgraph (of the original graph) induced by the vertices that have been eliminated. Suppose G is a (undirected) graph corresponding to a sparse symmet... |

216 |
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Citation Context ...ike the two-phase approach, iterative refinements may be included in the triangular solution. In this class of methods we will consider the multifrontal and the supernodal algorithms (see for example =-=[2, 6, 9, 17]-=-). The analysis phase involves preprocessing of the matrix that may consider numerical values and a symbolic phase that builds the computational graph for the numerical factorization phase. Both algor... |

192 | A supernodal approach to sparse partial pivoting
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Citation Context ...ike the two-phase approach, iterative refinements may be included in the triangular solution. In this class of methods we will consider the multifrontal and the supernodal algorithms (see for example =-=[2, 6, 9, 17]-=-). The analysis phase involves preprocessing of the matrix that may consider numerical values and a symbolic phase that builds the computational graph for the numerical factorization phase. Both algor... |

177 |
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Citation Context ...the approach proposed by George and Ng [21]. In this case, a permutation P is chosen to reduce fill in the Cholesky factorization of A T A; some examples are minimum degree [18] and nested dissection =-=[22]-=-. Then P is applied to the columns of A before Gaussian elimination with partial pivoting is performed. Following is the theoretical basis. Assume that the diagonal entries of A are nonzero 1 . Denote... |

164 |
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Citation Context ...properties with an example. In Figure 3, we apply the DMLS algorithm assuming that pivots are in the natural ordering. The matrix on the right is the structure of the LU factors. The elimination tree =-=[18]-=- built by the DMLS algorithm is shown in Figure 4. Each node of the tree corresponds to the elimination of a pivot. The nonsymmetric frontal matrix of each node corresponds to the structure of Up and ... |

139 |
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Citation Context ... quotient graph model is bounded above by the size of the original symmetric matrix [20]. The quotient graph model has been used extensively in many minimum-degree type of ordering codes, such as MMD =-=[28]-=- and AMD [1]. The key idea is a compact representation of the subgraph (of the original graph) induced by the vertices that have been eliminated. Suppose G is a (undirected) graph corresponding to a s... |

128 |
The evolution of the minimum degree ordering algorithm
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Citation Context ...Alternatively, one can use the approach proposed by George and Ng [21]. In this case, a permutation P is chosen to reduce fill in the Cholesky factorization of A T A; some examples are minimum degree =-=[18]-=- and nested dissection [22]. Then P is applied to the columns of A before Gaussian elimination with partial pivoting is performed. Following is the theoretical basis. Assume that the diagonal entries ... |

117 | An unsymmetric-pattern multifrontal method for sparse LU factorization
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Citation Context ...mise fill-in reduction with numerical stability. In this context, there are generally two approaches for solving nonsymmetric sparse linear systems. For the first approach, such as those described in =-=[7, 8, 16]-=-), pivots are chosen based on criteria that take both numerical stability and fill-in reduction into consideration. This may take place prior to or during numerical factorization. Then the LU factors ... |

89 | J.W.: SuperLU DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems
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- 2003
(Show Context)
Citation Context ... less than 0.5. We consider two state-of-the-art sparse direct solvers. One is a multifrontal solver with nonsymmetric fronts, called MA41 UNS [4]. Another is a supernodal solver, called SuperLU DIST =-=[26, 27]-=-, which first permutes A so that the diagonal entries of the permuted matrix have large magnitude, and then computes a triangular factorization without pivoting for numerical stability. 5 EU ESPRIT IV... |

87 |
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Citation Context ...to account the nonsymmetric structure of the matrix. In Section 2, we first review existing approaches for permuting a nonsymmetric matrix in the context of Gaussian elimination. The Markowitz scheme =-=[29]-=- is probably the most well-known approach for reordering the rows and columns of a sparse matrix. We then analyze the potential of a so called so called diagonal Markowitz ordering that limits its sea... |

73 | On algorithms for permuting large entries to the diagonal of a sparse matrix
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Citation Context ...this numerical preprocessing is more crucial since it will reduce the number of small pivots that are modified and set to " 1 2 jjAjj 1 . In practice, it has been observed in [3] that using MA64 code =-=[13, 14]-=- from [25] one can very significantly reduce the amount of numerical problems during factorization. In general, the two-phase approach has been shown to be best on very nonsymmetric (and often very sp... |

72 |
LAPACK User’s Guide, Third Edition
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Citation Context ...symmetric permutation P so that PCP T has a sparse LU factorization when Gaussian elimination without pivoting is applied. 2 The algorithm we used in the simulations is the same as that in the LAPACK =-=[5]-=- routine DGEEQU. 6s4. Determine the number of nonzero entries in the LU factorization and the number of operations required to compute the factorization. Steps 1 and 2 require the numerical values of ... |

72 | A combined unifrontal/multifrontal method for unsymmetric sparse matrices
- Davis, Duff
- 1999
(Show Context)
Citation Context ...mise fill-in reduction with numerical stability. In this context, there are generally two approaches for solving nonsymmetric sparse linear systems. For the first approach, such as those described in =-=[7, 8, 16]-=-), pivots are chosen based on criteria that take both numerical stability and fill-in reduction into consideration. This may take place prior to or during numerical factorization. Then the LU factors ... |

71 | The design and use of algorithms for permuting large entries to the diagonal of sparse matrices
- Duff, Koster
- 1999
(Show Context)
Citation Context ...this numerical preprocessing is more crucial since it will reduce the number of small pivots that are modified and set to " 1 2 jjAjj 1 . In practice, it has been observed in [3] that using MA64 code =-=[13, 14]-=- from [25] one can very significantly reduce the amount of numerical problems during factorization. In general, the two-phase approach has been shown to be best on very nonsymmetric (and often very sp... |

59 | A column pre-ordering strategy for the unsymmetricpattern multifrontal method
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(Show Context)
Citation Context ...7.1×10 8 and 20.7×10 8 using AMD. Moreover, for this class of matrices, MA41 UNS combined with the AMD ordering applied to A + A T significantly outperforms all the nonsymmetric solvers considered in =-=[7]-=-. Using AMDF thus further reduced the number of operations, and the attempt to exploit the asymmetry of the original matrix did not improve the ordering quality (as shown by the UMPFPACK code which at... |

52 |
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Citation Context ...he choice of the pivotal sequence in 1 It is well known that the rows (or the columns) of a nonsingular matrix can be permuted so that the diagonal entries of the permuted matrix are all nonzero. See =-=[10]-=- for details. 5sGaussian elimination. Thus, the approach due to George and Ng [19] reduces fill in L and U by reducing fill in the Cholesky factor of A T A. However, the strategy is often too generous... |

49 |
Vectorization of a multiprocessor multifrontal code
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Citation Context ...ximate minimum degree and minimum deficiency algorithms on A + A T . 3.1. Testing environment. To experiment with our ordering algorithm, we will consider the unsymmetrized multifrontal code MA41 UNS =-=[2, 6]-=-, which automatically detects and exploits the structural asymmetry of the submatrices involved when processing the elimination tree associated with the pattern of the symmetric matrix A + A T . In [7... |

43 |
A collection of Fortran codes for large scale scientific computation
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Citation Context ... preprocessing is more crucial since it will reduce the number of small pivots that are modified and set to " 1 2 jjAjj 1 . In practice, it has been observed in [3] that using MA64 code [13, 14] from =-=[25]-=- one can very significantly reduce the amount of numerical problems during factorization. In general, the two-phase approach has been shown to be best on very nonsymmetric (and often very sparse) matr... |

39 | Symbolic factorization for sparse Gaussian elimination with partial pivoting - George, Ng - 1987 |

36 | Elimination structures for unsymmetric sparse LU factors
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- 1993
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Citation Context ...rical values and a symbolic phase that builds the computational graph for the numerical factorization phase. Both algorithms (supernodal and multifrontal) can be described by a directed acyclic graph =-=[23]-=- whose nodes represent computations and whose edges represent transfer of data. This graph reduces to a tree in the case of the multifrontal method. In a multifrontal algorithm, some steps of Gaussian... |

32 |
Gaussian Elimination with Partial Pivoting and Load Balancing on a Multiprocessor
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- 1987
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Citation Context ...ssume that the diagonal entries of A are nonzero 1 . Denote by L and U the lower and upper triangular factors obtained via Gaussian elimination with partial pivoting, respectively. Then George and Ng =-=[19]-=- have showed that the sparsity structure of L + U is always contained in the sparsity structure of L C + L T C , where L C denotes the Cholesky factor of A T A, irrespective of the choice of the pivot... |

32 |
Node selection strategies for bottom-up sparse matrix ordering
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Citation Context ...based approximate degrees _ d r i and _ d c i . We have experimented different approximations of the deficiency (i.e., the number of nonzero entries introduced). Most basic heuristics experimented in =-=[33, 30]-=- to approximate the deficiency can be adapted in a simple way to our nonsymmetric case. Furthermore, we have also considered a new deficiency heuristic that results from discussion with T. Davis and I... |

27 |
The Rutherford-Boeing Sparse Matrix Collection
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Citation Context ...this section, we compare the ordering quality of our DMLS algorithm with that obtained by applying AMD on A+A T . The test matrices are from the forthcoming Rutherford-Boeing Sparse Matrix Collection =-=[12]-=-, the industrial partners of the PARASOL Project 5 , Tim Davis' collection 6 , and SPARSEKIT2 7 . There are altogether 98 structurally nonsymmetric matrices in our testbed. Among them, 67 are very non... |

25 | The design of MA48: A code for the direct solution of sparse unsymmetric linear systems of equations
- Duff, Reid
- 1996
(Show Context)
Citation Context ...mise fill-in reduction with numerical stability. In this context, there are generally two approaches for solving nonsymmetric sparse linear systems. For the first approach, such as those described in =-=[7, 8, 16]-=-), pivots are chosen based on criteria that take both numerical stability and fill-in reduction into consideration. This may take place prior to or during numerical factorization. Then the LU factors ... |

20 | Analysis and comparison of two general sparse solvers for distributed memory comoputers
- Amestoy, Duff, et al.
- 2001
(Show Context)
Citation Context ...phase. For SuperLU DIST, this numerical preprocessing is more crucial since it will reduce the number of small pivots that are modified and set to " 1 2 jjAjj 1 . In practice, it has been observed in =-=[3]-=- that using MA64 code [13, 14] from [25] one can very significantly reduce the amount of numerical problems during factorization. In general, the two-phase approach has been shown to be best on very n... |

19 | An unsymmetrized multifrontal LU factorization
- Amestoy, Puglisi
(Show Context)
Citation Context ... assembly at the parent node. To continue our description of the three-phase approach we will focus on two state-of-the-art direct solvers representative of this class: the multifrontal code MA41 UNS =-=[2, 4]-=- and the supernodal code SuperLU DIST [26, 26]. Both solvers can benefit from a numerical preordering (row or column permutations) to maximize entries on the diagonal and from a numerical scaling of t... |

17 |
The use of linear graphs in Gaussian elimination
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Citation Context ...graph model for symmetric Gaussian elimination. The bi-cliques in the bipartite graph model would correspond to the cliques in the symmetric case. Lemma 3.1 is in fact analogous to a result by Parter =-=[32]-=-. As in the symmetric case, the number of newly added edges in the bipartite graph model may be larger than the number of edges removed. Thus, in general we cannot predict a priori the amount of stora... |

16 | Tarjan, Algorithmic aspects of vertex elimination on graphs - Rose, Lueker, et al. - 1976 |

13 | Performance of greedy ordering heuristics for sparse Cholesky factorization
- Ng, Raghavan
- 1999
(Show Context)
Citation Context ...he asymmetry in the matrix. A secondary contribution is to adapt and extend the metrics to select pivots based on approximate degree [1] to metrics based on approximate Markowitz count and deficiency =-=[24, 20]-=-. Indeed, in our context all metrics have to anticipate the effect that local symmetrization would have on the pivot to be selected. Our algorithm has the same asymptotic complexity as the minimumdegr... |

12 | A.: The Computational Complexity of the Minimum Degree Algorithm
- Heggernes, Eisenstat, et al.
- 2001
(Show Context)
Citation Context ...than jEj. Overall, there are at most n elimination steps. The total time complexity is given by O(n jEj). This complexity is the same as that of the AMD code [1], see the analysis by Heggernes et al. =-=[24]-=-. We remark that, as a first order approximation, we essentially double the work of AMD if DMLS is applied to a symmetric matrix. 4.3 Numerical Results In this section, we compare the ordering quality... |

11 |
Performance of greedy heuristics for sparse Cholesky factorization
- Ng, Raghavan
- 1999
(Show Context)
Citation Context ...based approximate degrees _ d r i and _ d c i . We have experimented different approximations of the deficiency (i.e., the number of nonzero entries introduced). Most basic heuristics experimented in =-=[33, 30]-=- to approximate the deficiency can be adapted in a simple way to our nonsymmetric case. Furthermore, we have also considered a new deficiency heuristic that results from discussion with T. Davis and I... |

6 |
SPOOLES: An object oriented sparse matrix library
- Ashcraft, Grimes
- 1999
(Show Context)
Citation Context ...ike the two-phase approach, iterative refinements may be included in the triangular solution. In this class of methods we will consider the multifrontal and the supernodal algorithms (see for example =-=[2, 6, 9, 17]-=-). The analysis phase involves preprocessing of the matrix that may consider numerical values and a symbolic phase that builds the computational graph for the numerical factorization phase. Both algor... |

4 |
A bipartite quotient graph model for unsymmetric matrices
- Pagallo, Maulino
- 1983
(Show Context)
Citation Context ...ation graphs in terms of the original graph G 0 using the notion of reachable sets. George and Liu [20] gave such a characterization for the symmetric case with undirected graphs. Pagallo and Maulino =-=[31]-=- extended it to the nonsymmetric case with bipartite graphs. The "interior" vertices on a reachable path are the vertices that have been eliminated. The following result, which is taken from [31], is ... |

4 |
Unsymmetric ordering using a constrained markowitz scheme
- Amestoy, Li, et al.
(Show Context)
Citation Context ...tuation is different for at least two reasons. First, taking into account the asymmmetry of the matrix significantly improves the quality of the ordering. Second, it has been shown in our recent work =-=[5]-=- (generalization of the DMLS approach to allow off-diagonal and numerical-based pivot selection) that using separate row and column supervariables, one can significantly decrease the ordering time on ... |

2 |
A new pivoting strategy for Gaussian elimination
- OLSHOWKA, A
- 1996
(Show Context)
Citation Context ... delayed eliminations will result in an increase in the size of the LU factors estimated in the analysis and an increase in the number of operations. In practice, it has been observed that using MC64 =-=[21, 9, 10]-=- from HSL [15] as preordering can significantly reduce the number of delayed pivots during factorization [3]. This preordering will thus be applied on all our test matrices. Our test matrices are from... |