Double Ordering and Fill-In for the <i>LU</i> Factorization

Baumann, Markus; Fleischmann, Peter; Mutzbauer, Otto

doi:10.1137/s0895479802392989

Cited by 4 publications

(7 citation statements)

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“…For the node-centroid algorithm we have experimented with using values of λ in the range [0. 8,1] and values of α in the range [1.5, 2.5]. Our experience was that the bandwidths were not very sensitive to the precise choice of λ, and for most examples 0.85 gave results that were within three percent of the best.…”

Section: Test Resultsmentioning

confidence: 88%

“…The algorithm proceeds by alternating between ordering the rows in decreasing order and ordering the columns in decreasing order. The authors [1] show that this converges to a limit and call it a "double ordering." Following reverse CuthillMcKee, they reverse the converged ordering.…”

Section: Relaxed Double Orderingmentioning

confidence: 99%

“…Before presenting numerical results for our proposed algorithm, in this section we briefly discuss the recently published algorithm of Baumann, Fleischmann, and Mutzbauer [1] for reducing the bandwidth of an unsymmetric matrix. The sparsity pattern of the matrix is represented by a (0,1)-matrix, that is, a matrix which is the same as the original matrix except that each nonzero entry is replaced by 1.…”

Section: Relaxed Double Orderingmentioning

confidence: 99%

“…Unfortunately, there are huge numbers of double orderings, and Baumann, Fleischmann, and Mutzbauer [1] have no strategy for choosing a good one. For example, a Cuthill-McKee ordering produces a relaxed double ordering regardless of the starting node, since the leading entries of the rows (or columns) form a monotonic sequence.…”

Section: Relaxed Double Orderingmentioning

confidence: 99%

“…In section 6, we look at modifying the hill-climbing algorithm of Lim, Rodrigues, and Xiao [10] to improve a given ordering, and in section 7, we propose a variant of the node-centroid algorithm of [10] for the unsymmetric case. In section 8, we discuss the recently published algorithm of Baumann, Fleischmann, and Mutzbauer [1] for reducing the bandwidth of an unsymmetric matrix. In section 9, we use our proposed algorithms to reorder a set of matrices that arise from a range of practical problems; we report the total bandwidths before and after reordering, and we summarize our findings in section 10.…”

Section: Introduction Ifmentioning

confidence: 99%

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Reducing the Total Bandwidth of a Sparse Unsymmetric Matrix

Reid¹,

Scott²

2006

SIAM J. Matrix Anal. & Appl.

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Abstract. For a sparse symmetric matrix, there has been much attention given to algorithms for reducing the bandwidth. As far as we can see, little has been done for the unsymmetric matrix A, which has distinct lower and upper bandwidths l and u. When Gaussian elimination with row interchanges is applied, the lower bandwidth is unaltered, while the upper bandwidth becomes l + u. With column interchanges, the upper bandwidth is unaltered, while the lower bandwidth becomes l + u. We therefore seek to reduce min(l, u) + l + u, which we call the total bandwidth. We compare applying the reverse Cuthill-McKee algorithm to A + A T , to the row graph of A, and to the bipartite graph of A. We also propose an unsymmetric variant of the reverse Cuthill-McKee algorithm. In addition, we have adapted the node-centroid and hill-climbing ideas of Lim, Rodrigues, and Xiao to the unsymmetric case. We have found that using these to refine a Cuthill-McKee-based ordering can give significant further bandwidth reductions. Numerical results for a range of practical problems are presented and comparisons made with the recent lexicographical method of Baumann, Fleischmann, and Mutzbauer. 1. Introduction. If Gaussian elimination is applied without interchanges to an unsymmetric matrix A = {a ij } of order n, each fill-in takes place between the first entry of a row and the diagonal or between the first entry of a column and the diagonal. It is therefore sufficient to store all the entries in the lower triangle from the first entry in each row to the diagonal and all the entries in the upper triangle from the first entry in each column to the diagonal. This simple structure allows straightforward code using static data structures to be written. We will call the sum of the lengths of the rows the lower profile and the sum of the lengths of the columns the upper profile.We will also use the term lower bandwidth for l = max aij =0 (i − j) and the term upper bandwidth for u = max aij =0 (j − i). For a symmetric matrix, these are the same and are called the semibandwidth. A particularly simple data structure is available by taking account of only the bandwidths l and u. If row interchanges are used for stability reasons during the factorization, it may be readily verified that the lower bandwidth remains l but the upper bandwidth may increase to l + u. With column interchanges (or row interchanges applied while factorizing A T ), the upper bandwidth is unaltered, while the lower bandwidth becomes l + u. We may therefore always have one triangular factor of bandwidth min(l, u) and the other of bandwidth l + u. Thus we seek to reduce min(l, u) + l + u, which we call the total bandwidth.Many algorithms for reducing the bandwidth of a sparse symmetric matrix A have been proposed in the literature, most of which make extensive use of the adjacency graph G of the matrix. This is an undirected graph that has a node for each row (or

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Section: Test Resultsmentioning

confidence: 88%

Section: Relaxed Double Orderingmentioning

confidence: 99%

Section: Relaxed Double Orderingmentioning

confidence: 99%

Section: Relaxed Double Orderingmentioning

confidence: 99%

Section: Introduction Ifmentioning

confidence: 99%

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Reducing the Total Bandwidth of a Sparse Unsymmetric Matrix

Reid¹,

Scott²

2006

SIAM J. Matrix Anal. & Appl.

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Software for Numerical Linear Algebra

2007

Springer Texts in Statistics

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On the design of interfaces to sparse direct solvers

Sala

Stanley

Heroux

2008

ACM Trans. Math. Softw.

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We discuss the design of general, flexible, consistent, reusable, and efficient interfaces to software libraries for the direct solution of systems of linear equations on both serial and distributed memory architectures. We introduce a set of abstract classes to access the linear system matrix elements and their distribution, access vector elements, and control the solution of the linear system. We describe a concrete implementation of the proposed interfaces, and report examples of applications and numerical results showing that the overhead induced by the object-oriented design is negligible under typical conditions of usage. We include examples of applications, and we comment on the advantages and limitations of the design.

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Double Ordering and Fill-In for the LU Factorization

Cited by 4 publications

References 1 publication

Reducing the Total Bandwidth of a Sparse Unsymmetric Matrix

Reducing the Total Bandwidth of a Sparse Unsymmetric Matrix

Software for Numerical Linear Algebra

On the design of interfaces to sparse direct solvers

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