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39
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 319 (54 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
Algorithm 887: Cholmod, supernodal sparse cholesky factorization and update/downdate
 ACM Transactions on Mathematical Software
, 2008
"... CHOLMOD is a set of routines for factorizing sparse symmetric positive definite matrices of the form A or A A T, updating/downdating a sparse Cholesky factorization, solving linear systems, updating/downdating the solution to the triangular system Lx = b, and many other sparse matrix functions for b ..."
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Cited by 109 (8 self)
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CHOLMOD is a set of routines for factorizing sparse symmetric positive definite matrices of the form A or A A T, updating/downdating a sparse Cholesky factorization, solving linear systems, updating/downdating the solution to the triangular system Lx = b, and many other sparse matrix functions for both symmetric and unsymmetric matrices. Its supernodal Cholesky factorization relies on LAPACK and the Level3 BLAS, and obtains a substantial fraction of the peak performance of the BLAS. Both real and complex matrices are supported. CHOLMOD is written in ANSI/ISO C, with both C and MATLAB TM interfaces. It appears in MATLAB 7.2 as x=A\b when A is sparse symmetric positive definite, as well as in several other sparse matrix functions.
A column preordering strategy for the unsymmetricpattern multifrontal method
 ACM Transactions on Mathematical Software
, 2004
"... A new method for sparse LU factorization is presented that combines a column preordering strategy with a rightlooking unsymmetricpattern multifrontal numerical factorization. The column ordering is selected to give a good a priori upper bound on fillin and then refined during numerical factoriza ..."
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Cited by 94 (5 self)
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A new method for sparse LU factorization is presented that combines a column preordering strategy with a rightlooking unsymmetricpattern multifrontal numerical factorization. The column ordering is selected to give a good a priori upper bound on fillin and then refined during numerical factorization (while preserving the bound). Pivot rows are selected to maintain numerical stability and to preserve sparsity. The method analyzes the matrix and automatically selects one of three preordering and pivoting strategies. The number of nonzeros in the LU factors computed by the method is typically less than or equal to those found by a wide range of unsymmetric sparse LU factorization methods, including leftlooking methods and prior multifrontal methods.
Strategies for scaling and pivoting for sparse symmetric indefinite problems
, 2004
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Towards a tighter coupling of bottomup and topdown sparse matrix ordering methods
 BIT
, 2001
"... Most stateoftheart ordering schemes for sparse matrices are a hybrid of a bottomup method such as minimum degree and a top down scheme such as George's nested dissection. In this paper we present an ordering algorithm that achieves a tighter coupling of bottomup and topdown methods. In our ..."
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Cited by 35 (0 self)
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Most stateoftheart ordering schemes for sparse matrices are a hybrid of a bottomup method such as minimum degree and a top down scheme such as George's nested dissection. In this paper we present an ordering algorithm that achieves a tighter coupling of bottomup and topdown methods. In our methodology vertex separators are interpreted as the boundaries of the remaining elements in an unfinished bottomup ordering. As a consequence, we are using bottomup techniques such as quotient graphs and special node selection strategies for the construction of vertex separators. Once all separators have been found, we are using them as a skeleton for the computation of several bottomup orderings. Experimental results show that the orderings obtained by our scheme are in general better than those obtained by other popular ordering codes.
Multilevel preconditioners constructed from inversebased ILUs
, 2004
"... This paper analyzes dropping strategies in a multilevel incomplete LU decomposition context and presents a few of strategies for obtaining related ILUs with enhanced robustness. The analysis shows that the Incomplete LU factorization resulting from dropping small entries in Gaussian elimination prod ..."
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Cited by 32 (9 self)
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This paper analyzes dropping strategies in a multilevel incomplete LU decomposition context and presents a few of strategies for obtaining related ILUs with enhanced robustness. The analysis shows that the Incomplete LU factorization resulting from dropping small entries in Gaussian elimination produces a good preconditioner when the inverses of these factors have norms that are not too large. As a consequence a few strategies are developed whose goal is to achieve this feature. A number of “templates” for enabling implementations of these factorizations are presented. Numerical experiments show that the resulting ILUs offer a good compromise between robustness and efficiency.
The Design of Sparse Direct Solvers using ObjectOriented Techniques
, 1999
"... We describe our experience in designing objectoriented software for sparse direct solvers. We discuss, a library of sparse matrix ordering codes, and, a package that implements the factorization and triangular solution steps of a direct solver. We discuss the goals of our design: managing complex ..."
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Cited by 18 (4 self)
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We describe our experience in designing objectoriented software for sparse direct solvers. We discuss, a library of sparse matrix ordering codes, and, a package that implements the factorization and triangular solution steps of a direct solver. We discuss the goals of our design: managing complexity, simplicity of interface, exibility, extensibility, safety, and efficiency. High performance is obtained by carefully implementing the computationally intensive kernels and by making several tradeo s to balance the conflicting demands of efficiency and good software design. Some of the missteps that we made in the course of this work are also described.
Performance Of Greedy Ordering Heuristics For Sparse Cholesky Factorization
, 1997
"... . Greedy algorithms for ordering sparse matrices for Cholesky factorization can be based on different metrics. Minimum degree, a popular and effective greedy ordering scheme, minimizes the number of nonzero entries in the rank1 update (degree) at each step of the factorization. Alternatively, minim ..."
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Cited by 17 (3 self)
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. Greedy algorithms for ordering sparse matrices for Cholesky factorization can be based on different metrics. Minimum degree, a popular and effective greedy ordering scheme, minimizes the number of nonzero entries in the rank1 update (degree) at each step of the factorization. Alternatively, minimum deficiency minimizes the number of nonzero entries introduced (deficiency) at each step of the factorization. In this paper we develop two new heuristics: "modified minimum deficiency" (MMDF) and "modified multiple minimum degree" (MMMD). The former uses a metric similar to deficiency while the latter uses a degreelike metric. Our experiments reveal that on the average MMDF has 21% fewer operations to factor than minimum degree; MMMD has 15% fewer operations to factor than minimum degree. MMMD is no more expensive to compute than minimum degree while MMDF requires on the average 30% more time than minimum degree. Key words. sparse matrix ordering, minimum degree, minimum deficiency, gre...
Algorithm 8xx: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate
, 2006
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Task scheduling in an asynchronous distributed memory multifrontal solver
 SIAM Journal on Matrix Analysis and Applications
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