E. Rothberg, Ordering sparse matrices using approximate minimum local fill, Silicon Graphics manuscript (Apr. 1996).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....A to construct an initial clique cover of A T A directly. Essentially, the nonzero structure of each row of A is interpreted as a bitmap that specifies membership of columns in one clique. Minimum degree algorithms are not the only heuristics for ordering matrices. Others include minimum fill [14, 15], nested dissection, and hybrid methods combining nested dissection and minimum degree [3, 9] Although other algorithms are easier to parallelize, A Column Minimum Degree Code for Multiprocessors 3 none is currently as widely used for column ordering nonsymmetric matrices as variants on minimum ....

E. Rothberg, Ordering sparse matrices using approximate minimum local fill, Silicon Graphics manuscript (Apr. 1996).

....an ordering, which minimizes the amount of fill in is an NP hard problem. Therefore, the problem can only be solved approximately in a reasonable amount of time. The approximate minimum degree heuristic (AMD) proposed by Davis, Amestoy, and Duff [1] including a modification suggested by Rothberg [27], is the default ordering method in the XPRESS interior point optimizer. Second, when an ordering has been determined the data structure for the Cholesky decomposition is initialized. This is referred to as the symbolic phase, because no numerical computations are involved. Finally, the actual ....

E. Rothberg. Ordering sparse matrices using approximate minimum local fill. Technical report, Silicon Graphics, Inc. Mountain View, CA 94043, April 1996.

....values in A are algebraic indeterminates. 2 Table 2 1 Algorithms that fit into the Minimum Priority family. Abbreviation Algorithm Name Primary Reference MMD Multiple Minimum Degree Liu [5] AMD Approximate Minimum Degree Amestoy, Davis and Du# [1] AMF Approximate Minimum Fill Rothberg [8] AMMF Approximate Minimum Mean Local Fill Rothberg and Eisenstat [9] AMIND Approximate Minimum Increase in Rothberg and Eisenstat [9] Neighbor Degree MMDF Modified Minimum Deficiency Ng and Raghavan [6] MMMD Modified Multiple Minimum Degree Ng and Raghavan [6] connecting vertices i and j in ....

Ed Rothberg, Ordering sparse matrices using approximate minimum local fill, Preprint, April 1996.

....independent set elimination) 1 The ETree object has the Tree object that defines the connectivity of the fronts, knows the internal and external size of each front, and has a map from the vertices to the fronts. 188 We intend to add more priorities, e.g. approximate deficiency from [18] [19] and [20] Choose a priority, then specify the definition of a step, how to choose an independent set of vertices to eliminate at a time. Then provide a map from each vertex to the stage at which it will be eliminated. Presently there is one ordering method, MSMD order( It orders the vertices by ....

E. Rothberg. Ordering sparse matrices using approximate minimum local fill. In Second SIAM Conference on Sparse Matrices, 1996. Conference presentation.

....two permutations, P 1 and P 2 . The first ordering, P 1 , reduces fill and the second ordering, P 2 , is a 1. Designing Object Oriented Sparse Direct Solvers 9 Abbreviation. Algorithm Name MMD Multiple Minimum Degree [LIU85] AMD Approximate Minimum Degree [ADD96] AMF Approximate Minimum Fill [ROT96] MMDF Modified Minimum Deficiency [NR97] MMMD Modified Multiple Minimum Degree [NR97] AMMF Approximate Minimum Mean Local Fill [RE98] AMIND Approximate Minimum Increase in Neighbor Degree [RE98] TABLE 1.1. Some of the algorithms in the Minimum Priority family. modification of P 1 to preserve ....

E. Rothberg. Ordering sparse matrices using approximate minimum local fill. Preprint, April 1996.

....of A to construct an initial clique cover of A T A directly. Essentially, the nonzero structure of each row of A is interpreted as a bitmap that specifies membership of columns in one clique. Minimum degree algorithms are not the only heuristics for ordering matrices. Others include minimum fill [14, 15], nested dissection, and hybrid methods combining nested dissection and minimum degree [3, 9] Although other algorithms are easier to parallelize, A Column Minimum Degree Code for Multiprocessors 3 none is currently as widely used for column ordering nonsymmetric matrices as variants on minimum ....

E. Rothberg, Ordering sparse matrices using approximate minimum local fill, Silicon Graphics manuscript (Apr. 1996).

.... accidental cancellations will occur during factorization if the numerical values in A are algebraic indeterminates. Abbreviation Algorithm Name Primary Reference MMD Multiple Minimum Degree Liu [5] AMD Approximate Minimum Degree Amestoy, Davis and Duff [1] AMF Approximate Minimum Fill Rothberg [8] AMMF Approximate Minimum Mean Local Fill Rothberg and Eisenstat [9] AMIND Approximate Minimum Increase in Rothberg and Eisenstat [9] Neighbor Degree MMDF Modified Minimum Deficiency Ng and Raghavan [6] MMMD Modified Multiple Minimum Degree Ng and Raghavan [6] Table 1. Algorithms that fit into ....

Ed Rothberg. Ordering sparse matrices using approximate minimum local fill. Preprint, April 1996.

....minimum degree algorithm [7] First, the minimum deficiency algorithm is typically much more expensive than the minimum degree algorithm. Second, it has been believed that the quality of minimum deficiency orderings is not much better than that of minimum degree orderings [7] Results by Rothberg [19] (and also by us [17] demonstrate that minimum deficiency leads to significantly better orderings than minimum degree. However, current implementations of the minimum deficiency algorithm require substantially more time than MMD. In this paper, we develop two greedy heuristics that are less ....

....4 we provide empirical results on the performance of MMDF and MMMD. Section 5 contains some concluding remarks. The remaining part of this section describes recent related work. Related work. Rothberg has investigated metrics for greedy ordering schemes based on approximations to the deficiency [19]. His work and our work [17] were done independently of each other. 1 Rothberg [19] ffl shows that the minimum deficiency algorithm is significantly superior to MMD in terms of the number of operations required to compute the Cholesky factor, ffl develops three approximate minimum fill ....

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E. Rothberg, Ordering sparse matrices using approximate minimum local fill. April 1996.

....independent set elimination) 1 The ETree object has the Tree object that defines the connectivity of the fronts, knows the internal and external size of each front, and has a map from the vertices to the fronts. 188 We intend to add more priorities, e.g. approximate deficiency from [18] [19] and [20] Choose a priority, then specify the definition of a step, how to choose an independent set of vertices to eliminate at a time. Then provide a map from each vertex to the stage at which it will be eliminated. Presently there is one ordering method, MSMD order( It orders the vertices by ....

E. Rothberg. Ordering sparse matrices using approximate minimum local fill. In Second SIAM Conference on Sparse Matrices, 1996. Conference presentation.

....specifically the multiple external minimum degree algorithm [21] was the preferred algorithm of choice for the better part of a decade. Alternative minimum priority codes have recently pushed multiple minimum degree aside, including approximate minimum degree [1] and approximate deficiency [23] [28]. They offer improved quality or improved run time, and on occasion, both. Nested dissection for regular grids [10] is within a factor of optimal with respect to factor entries and operation counts. One of the earliest attempts, automatic nested dissection [11] used a simple profile algorithm to ....

E. Rothberg. Ordering sparse matrices using approximate minimum local fill. In Second SIAM Conference on Sparse Matrices, 1996. Conference presentation.

....is a recent variant of minimum degree, called Approximate Minimum Degree (AMD) 1] AMD further reduces runtime by computing an inexpensive upper bound on a vertex s degree rather than the true degree. Another recently proposed variant of minimum degree, Approximate Minimum local Fill (AMF) [29], improves on minimum degree by modifying the strategy used to select vertices for elimination. The method uses a rough approximation of the fill that would be generated by eliminating a vertex rather than using the vertex degree. The runtime of AMF ordering is only slightly higher than that of ....

Rothberg, E., "Ordering sparse matrices using approximate minimum local fill", Silicon Graphics manuscript, submitted for publication, April, 1996.

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Rothberg, E. (1996b), Ordering sparse matrices using approximate minimum local fill, Technical Report Unnumbered, Silicon Graphics Inc.

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