## Multifrontal Computation with the Orthogonal Factors of Sparse Matrices (1994)

Venue: | SIAM Journal on Matrix Analysis and Applications |

Citations: | 10 - 0 self |

### BibTeX

@ARTICLE{Lu94multifrontalcomputation,

author = {Szu-Min Lu and Jesse and Jesse L. Barlow},

title = {Multifrontal Computation with the Orthogonal Factors of Sparse Matrices},

journal = {SIAM Journal on Matrix Analysis and Applications},

year = {1994},

volume = {17}

}

### Years of Citing Articles

### OpenURL

### Abstract

. This paper studies the solution of the linear least squares problem for a large and sparse m by n matrix A with m n by QR factorization of A and transformation of the righthand side vector b to Q T b. A multifrontal-based method for computing Q T b using Householder factorization is presented. A theoretical operation count for the K by K unbordered grid model problem and problems defined on graphs with p n-separators shows that the proposed method requires O(NR ) storage and multiplications to compute Q T b, where NR = O(n log n) is the number of nonzeros of the upper triangular factor R of A. In order to introduce BLAS-2 operations, Schreiber and Van Loan's Storage-Efficient-WY Representation [SIAM J. Sci. Stat. Computing, 10(1989),pp. 55-57] is applied for the orthogonal factor Q i of each frontal matrix F i . If this technique is used, the bound on storage increases to O(n(logn) 2 ). Some numerical results for the grid model problems as well as Harwell-Boeing problems...

### Citations

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Citation Context ...-t Fig. 9. Unreduced frontal matrix We then have : C(u; v; t) = t X i=1 (i + 1) + u+v\Gammat X i=t+1 (t + 1) = t(t + 3) 2 + (t + 1)(u + v \Gamma 2t): We can use the concept of "bordered K by K gr=-=ids" [12]-=- to perform the merge operation. Let \Theta(k; i) be the number of nonzeros in the matrix Q, which is used to factor a K by K grid which is bordered on i sides. According to [20, 12], the following re... |

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Citation Context ...1 h 2 \Delta \Delta \Delta hn ) ; which is referred to as the Householder matrix. A matrix-vector product Q T b can be computed efficiently from H and b. The LINPACK routines SQRDC and SQRSL employ H =-=[7]. In [17],-=- Gilbert, Ng, and Peyton analyzed the nonzero counts of the factors Q, R, and H in terms of the sizes of separators in the column intersection graph G " (A) of A, where G " (A) is an undirec... |

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Citation Context ...n be stored and retrieved in a last-in/first-out (i.e. stack) manner if the nodes of the elimination tree are ordered by a postordering. The use of a stack for update matrices is due to Duff and Reid =-=[9]-=-. Below we outline the multifrontal QR factorization in an algorithm. Algorithm 2.3. (Multifrontal QR Factorization): For j = 1 To number of tree nodes Do 1. Assemble the frontal matrix for vertex j, ... |

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Citation Context ... The Extended Model Problem. We prove the bound of O(n log n) on jY j for p n-separator matrices. Let A be a p n-separator matrix whose columns are ordered by "generalized nested dissection"=-= ordering [21]-=-. If A has no more than n 0 = (fi=(1 \Gamma ff)) 2 columns, this recursive numbering algorithm numbers the unnumbered columns arbitrarily. In order to limit the initial jY i j, we assume that each s i... |

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Citation Context ...m, we assume all the matrices are ordered by that column ordering in our analysis for the p n-separator problem in section 4.2. On the other hand, George's original, simpler form of nested dissection =-=[10]-=-, which does not include the separators in the recursive call, is actually sufficient for some special classes of the p n-separable graphs: planar graphs, graphs of bounded genus or bounded excluded m... |

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Citation Context ...tion 2. An O(n log n) bound is proven on the number of nonzeros of all frontal Householder matrices Y i 's. We also count the number of nonzeros used in the WY representations of Bischof and Van Loan =-=[5]-=- and Schreiber and Van Loan [29]. We prove the bounds for the K by K grid model problem and problems which are defined on p n-separable graphs are O(n log n) and O(n(log n) 2 ), respectively. Note tha... |

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Citation Context ...oximation of the average value of s in section 4.2, since P n i s i = m, and s = max i s i and is presumed to be constant. We also solve the linear least squares problems given in (1) by QR method by =-=[18]-=- using our method for computing Q T b. We compare the QR method with the method of correct semi-normal equations method (CSNE) by [6]. The CSNE method 20 S.-M. LU AND J. L. BARLOW Table 3 Residuals fo... |

67 |
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Citation Context ...proven on the number of nonzeros of all frontal Householder matrices Y i 's. We also count the number of nonzeros used in the WY representations of Bischof and Van Loan [5] and Schreiber and Van Loan =-=[29]-=-. We prove the bounds for the K by K grid model problem and problems which are defined on p n-separable graphs are O(n log n) and O(n(log n) 2 ), respectively. Note that these bounds are valid even if... |

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Citation Context ...ly sufficient for some special classes of the p n-separable graphs: planar graphs, graphs of bounded genus or bounded excluded minor, and two-dimensional finite element meshes of bounded aspect ratio =-=[16]-=-. As a result, our analysis in section 4.1 for the model problem uses George's nested dissection ordering. 4.1. The Model Problem. If George's nested dissection ordering [10] is applied to the model p... |

28 | Stable numerical algorithms for equilibrium systems
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Citation Context ...report, we will test different representations of the orthogonal factors on p n-separator matrices and various practical problems such as the geodesy problems [19] and the equilibrium systems problem =-=[30] on -=-advanced architectures which support BLAS-2 and BLAS-3 operations. Acknowledgements. The authors would like to thanksAke Bj��orck and Pontus Matstoms for making the QR27 software available to us a... |

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Citation Context ...ed parameter NEMIN. Details of the implementation and complete study on the efficiency of the value of NEMIN on the performance of multifrontal QR factorization are given by Matstoms [24] and Puglisi =-=[27]-=-. We build a supernodal elimination tree by substituting a single node for all the nodes belonging to the same supernode in the original elimination tree. We formally define this elimination tree belo... |

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Citation Context ...ms: ` !A 1 A 2 ' x = ` !b 1 b 2 ' ; where the rows are of widely differing norms. It has been shown that the QR method performs well for "stiff" problems [1]. There is some comment on this p=-=roblem in [2, 3]. In order-=- to confirm that the QR method using the proposed method maintains this property, we apply the QR and CSNE methods on a sample "stiff" problem given in Figure 11. We take the exact solution ... |

13 |
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Citation Context ...and the orthogonal factor Q for a sparse matrix using G " (A) are, for example, given in [14, 26]. In this paper, we study the computation of orthogonal factor using the multifrontalsQR factoriza=-=tion [20, 24]-=-. Associated with each row of the upper triangular factor R , is a frontal matrix F i . Likewise for each F i , there is a frontal Householder matrix Y i . Note that Y i is the H matrix for F i . Figu... |

10 |
The direct solution of weighted and equality constrained least squares problems
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(Show Context)
Citation Context ...WITH THE ORTHOGONAL FACTORS 21 "stiff " problems: ` !A 1 A 2 ' x = ` !b 1 b 2 ' ; where the rows are of widely differing norms. It has been shown that the QR method performs well for "s=-=tiff" problems [1]. There is-=- some comment on this problem in [2, 3]. In order to confirm that the QR method using the proposed method maintains this property, we apply the QR and CSNE methods on a sample "stiff" proble... |

7 |
orck, Stability analysis of the method of seminormal equations for linear least squares problems
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Citation Context ...ve the linear least squares problems given in (1) by QR method by [18] using our method for computing Q T b. We compare the QR method with the method of correct semi-normal equations method (CSNE) by =-=[6]-=-. The CSNE method 20 S.-M. LU AND J. L. BARLOW Table 3 Residuals for the model problem. K QR CSNE 10 4.08 4.08 20 9.21 9.21 30 13.99 13.99 40 19.54 19.19 50 24.27 24.27 60 24.98 28.98 70 31.32 33.94 T... |

5 |
A Multifrontal Householder QR Factorization
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Citation Context ...and the orthogonal factor Q for a sparse matrix using G " (A) are, for example, given in [14, 26]. In this paper, we study the computation of orthogonal factor using the multifrontalsQR factoriza=-=tion [20, 24]-=-. Associated with each row of the upper triangular factor R , is a frontal matrix F i . Likewise for each F i , there is a frontal Householder matrix Y i . Note that Y i is the H matrix for F i . Figu... |

4 |
A note on deferred correction for equality constrained least squares problems
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(Show Context)
Citation Context ...ms: ` !A 1 A 2 ' x = ` !b 1 b 2 ' ; where the rows are of widely differing norms. It has been shown that the QR method performs well for "stiff" problems [1]. There is some comment on this p=-=roblem in [2, 3]. In order-=- to confirm that the QR method using the proposed method maintains this property, we apply the QR and CSNE methods on a sample "stiff" problem given in Figure 11. We take the exact solution ... |

4 | reflections versus Givens rotations in sparse orthogonal decomposition, Linear Algebra and its - Householder - 1987 |

4 |
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(Show Context)
Citation Context ...latively small if m is much larger than n. Other results on the nonzero structures of the Householder matrix H and the orthogonal factor Q for a sparse matrix using G " (A) are, for example, give=-=n in [14, 26]-=-. In this paper, we study the computation of orthogonal factor using the multifrontalsQR factorization [20, 24]. Associated with each row of the upper triangular factor R , is a frontal matrix F i . L... |

4 | Separators and structure prediction in sparse orthogonal factorization
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(Show Context)
Citation Context ...elta \Delta \Delta hn ) ; which is referred to as the Householder matrix. A matrix-vector product Q T b can be computed efficiently from H and b. The LINPACK routines SQRDC and SQRSL employ H [7]. In =-=[17], Gilbert,-=- Ng, and Peyton analyzed the nonzero counts of the factors Q, R, and H in terms of the sizes of separators in the column intersection graph G " (A) of A, where G " (A) is an undirected graph... |

3 |
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Citation Context ...der transformations. A theoretical operation count for the model problem is given by Lewis et.al. [20] which indicates the factorization algorithm requires half the multiplications as Liu's algorithm =-=[8, 22]-=-. We begin this subsection with the definition of the elimination tree. Definition 2.2 (Elimination Tree). Given an m by n matrix A, such that A T A is irreducible, the elimination tree of A T A is a ... |

3 | A tight and explicit representation of Q in sparse QR factorization
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(Show Context)
Citation Context ...latively small if m is much larger than n. Other results on the nonzero structures of the Householder matrix H and the orthogonal factor Q for a sparse matrix using G " (A) are, for example, give=-=n in [14, 26]-=-. In this paper, we study the computation of orthogonal factor using the multifrontalsQR factorization [20, 24]. Associated with each row of the upper triangular factor R , is a frontal matrix F i . L... |

1 |
A comparison between direct and iterative methods for certain large scale geodetic least squares problems
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(Show Context)
Citation Context ... as seen on the iPSC/2 [23]. In a future report, we will test different representations of the orthogonal factors on p n-separator matrices and various practical problems such as the geodesy problems =-=[19] and-=- the equilibrium systems problem [30] on advanced architectures which support BLAS-2 and BLAS-3 operations. Acknowledgements. The authors would like to thanksAke Bj��orck and Pontus Matstoms for m... |

1 |
Parallel computation of orthogonal factors of sparse matrices
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- 1993
(Show Context)
Citation Context ...graph; under which Q and its " StorageEfficient -WY Representation" have O(n p n) and O(n 2 ) nonzeros respectively. The proposed method has possibilities for parallel computing as seen on t=-=he iPSC/2 [23]-=-. In a future report, we will test different representations of the orthogonal factors on p n-separator matrices and various practical problems such as the geodesy problems [19] and the equilibrium sy... |