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35
A column approximate minimum degree ordering algorithm
, 2000
"... Sparse Gaussian elimination with partial pivoting computes the factorization PAQ = LU of a sparse matrix A, where the row ordering P is selected during factorization using standard partial pivoting with row interchanges. The goal is to select a column preordering, Q, based solely on the nonzero patt ..."
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Cited by 274 (52 self)
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Sparse Gaussian elimination with partial pivoting computes the factorization PAQ = LU of a sparse matrix A, where the row ordering P is selected during factorization using standard partial pivoting with row interchanges. The goal is to select a column preordering, Q, based solely on the nonzero pattern of A such that the factorization remains as sparse as possible, regardless of the subsequent choice of P. The choice of Q can have a dramatic impact on the number of nonzeros in L and U. One scheme for determining a good column ordering for A is to compute a symmetric ordering that reduces fillin in the Cholesky factorization of ATA. This approach, which requires the sparsity structure of ATA to be computed, can be expensive both in
A supernodal approach to sparse partial pivoting
 SIAM Journal on Matrix Analysis and Applications
, 1999
"... We investigate several ways to improve the performance of sparse LU factorization with partial pivoting, as used to solve unsymmetric linear systems. To perform most of the numerical computation in dense matrix kernels, we introduce the notion of unsymmetric supernodes. To better exploit the memory ..."
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Cited by 209 (24 self)
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We investigate several ways to improve the performance of sparse LU factorization with partial pivoting, as used to solve unsymmetric linear systems. To perform most of the numerical computation in dense matrix kernels, we introduce the notion of unsymmetric supernodes. To better exploit the memory hierarchy, weintroduce unsymmetric supernodepanel updates and twodimensional data partitioning. To speed up symbolic factorization, we use Gilbert and Peierls's depth rst search with Eisenstat and Liu's symmetric structural reductions. We have implemented a sparse LU code using all these ideas. We present experiments demonstrating that it is signi cantly faster than earlier partial pivoting codes. We also compare performance with Umfpack, which uses a multifrontal approach; our code is usually faster.
Communicationoptimal parallel and sequential QR and LU factorizations
, 2008
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Sparse Gaussian Elimination on High Performance Computers
, 1996
"... This dissertation presents new techniques for solving large sparse unsymmetric linear systems on high performance computers, using Gaussian elimination with partial pivoting. The efficiencies of the new algorithms are demonstrated for matrices from various fields and for a variety of high performan ..."
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Cited by 36 (6 self)
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This dissertation presents new techniques for solving large sparse unsymmetric linear systems on high performance computers, using Gaussian elimination with partial pivoting. The efficiencies of the new algorithms are demonstrated for matrices from various fields and for a variety of high performance machines. In the first part we discuss optimizations of a sequential algorithm to exploit the memory hierarchies that exist in most RISCbased superscalar computers. We begin with the leftlooking supernodecolumn algorithm by Eisenstat, Gilbert and Liu, which includes Eisenstat and Liu's symmetric structural reduction for fast symbolic factorization. Our key contribution is to develop both numeric and symbolic schemes to perform supernodepanel updates to achieve better data reuse in cache and floatingpoint register...
Recent Advances in Direct Methods for Solving Unsymmetric Sparse Systems of Linear Equations
, 2001
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Partitioning mathematical programs for parallel solution
, 1994
"... This paper describes heuristics for partitioning a general M x N matrix into arrowhead form. Such heuristics are useful for decomposing large, constrained, optimization problems into forms that are amenable to parallel processing. The heuristics presented can be easily implemented using publicly ava ..."
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Cited by 28 (0 self)
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This paper describes heuristics for partitioning a general M x N matrix into arrowhead form. Such heuristics are useful for decomposing large, constrained, optimization problems into forms that are amenable to parallel processing. The heuristics presented can be easily implemented using publicly available graph partitioning algorithms. The application of such techniques for solving large linear programs is described. Extensive computational results on the effectiveness of our partitioning procedures and their usefulness for parallel optimization are presented. @ 1998 The
Finding Good Column Orderings for Sparse QR Factorization
 In Second SIAM Conference on Sparse Matrices
, 1996
"... 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 floatingpoint operations required by the factorization, are highly dependent on this ordering. A suitable column ordering of ..."
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Cited by 18 (0 self)
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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 floatingpoint 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 fillreducing ordering can...
NestedDissection Orderings For Sparse Lu With Partial Pivoting
 SIAM J. Matrix Anal. Appl
, 2000
"... . We describe the implementation and performance of a novel fillminimization ordering technique for sparse LU factorization with partial pivoting. The technique was proposed by Gilbert and Schreiber in 1980 but never implemented and tested. Like other techniques for ordering sparse matrices for ..."
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Cited by 18 (5 self)
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. We describe the implementation and performance of a novel fillminimization ordering technique for sparse LU factorization with partial pivoting. The technique was proposed by Gilbert and Schreiber in 1980 but never implemented and tested. Like other techniques for ordering sparse matrices for LU with partial pivoting, our new method preorders the columns of the matrix (the row permutation is chosen by the pivoting sequence during the numerical factorization). Also like other methods, the column permutation Q that we select is a permutation that minimizes the fill in the Cholesky factor of Q T A T AQ. Unlike existing columnordering techniques, which all rely on minimumdegree heuristics, our new method is based on a nesteddissection ordering of A T A. Our algorithm, however, never computes a representation of A T A, which can be expensive. We only work with a representation of A itself. Our experiments demonstrate that the method is e#cient and that it can reduce fill significantly relative to the best existing methods. The method reduces the LU running time on some very large matrices (tens of millions of nonzeros in the factors) by more than a factor of 2. 1.
Computing Row and Column Counts for Sparse QR and LU Factorization
 BIT
, 2001
"... We present algorithms to determine the number of nonzeros in each row and column of the factors of a sparse matrix, for both the QR factorization and the LU factorization with partial pivoting. The algorithms use only the nonzero structure of the input matrix, and run in time nearly linear in th ..."
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Cited by 14 (2 self)
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We present algorithms to determine the number of nonzeros in each row and column of the factors of a sparse matrix, for both the QR factorization and the LU factorization with partial pivoting. The algorithms use only the nonzero structure of the input matrix, and run in time nearly linear in the number of nonzeros in that matrix. They may be used to set up data structures or schedule parallel operations in advance of the numerical factorization. The row and column counts we compute are upper bounds on the actual counts. If the input matrix is strong Hall and there is no coincidental numerical cancellation, the counts are exact for QR factorization and are the tightest bounds possible for LU factorization. These algorithms are based on our earlier work on computing row and column counts for sparse Cholesky factorization, plus an ecient method to compute the column elimination tree of a sparse matrix without explicitly forming the product of the matrix and its transpose. ...