J. R. Gilbert and E. G. Ng. Predicting structure in nonsymmetric sparse matrix factorizations. In Alan George, John R. Gilbert, and Joseph W. H. Liu, editors, Graph Theory and Sparse Matrix Computation, pages 107-139. Springer-Verlag, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....to balance considerations of stability and sparsity by using approximate Markowitz counts with a pivot threshold. A directed acyclic graph is implicitly used to drive the numerical factorization. In both SuperLU and UMFPACK3, a preordering of the columns is computed. The column elimination tree [19] is used during the analysis phase to get memory estimates and during the factorization phase to drive the computations. All solvers factorize general unsymmetric matrices and use dense matrix kernels. In each code, ll in reduction has been set to use some variant of the minimum degree ordering ....

J. R. Gilbert and E. G. Ng. Predicting structure in nonsymmetric sparse matrix factorizations. In J.R. Gilbert A. George and J.W.H Liu, editors, Graph Theory and Sparse Matrix Computations, pages 107-140. SpringerVerlag NY, 1993.

....like the multifrontal scheme and complete factorization of the frontal matrices. The traditional Householder algorithm introduces too much intermediate fill, and can in practice therefore not be used. However, an interesting connection to sparse LU factorization (Gilbert [17] Gilbert and Ng [19]) makes the traditional Householder algorithm still interesting in real applications. The main result is presented in Theorem 5.1. Theorem 5.1 (George and Ng [16] Let M 2 R be a nonsingular square matrix with nonzero diagonal. Suppose M is factorized by Gaussian elimination with row ....

J. R. Gilbert and E. G. Ng, Predicting structure in nonsymmetric sparse matrix factorization, tech. rep., Xerox Palo Alto Research Center, Palo Alto, California 94304-1314, 1992.

...., and updating that uses submatrices L i;k and U k;j to modify A i;j for k 1 i N . Figure 2 outlines the partitioned LU factorization algorithm with partial pivoting. 3. Elimination forests and non symmetric supernode partitioning. In this section, we study properties of elimination forests [1, 15, 16, 25] 2 and use them to design more robust strategies for supernode partitioning and parallelism detection. As a result, both sequential and parallel versions of our code can be improved. We will use the following notations in our discussion. Let A be the given n Theta n sparse matrix. Notice that ....

....The definition of elimination forests. We study the elimination forest of a matrix which may or may not be reducible. Previous research on elimination forests has shown that an elimination forest contains information about all potential dependency if the corresponding sparse matrix is irreducible [1, 15, 16, 25]. Although it is always possible to decompose a reducible matrix into several smaller irreducible matrices, the decomposition introduces extra burden on software design and implementation. Instead, we generalize the original definition of elimination tree to reducible matrices. Our definition, ....

J. R. Gilbert and E. Ng, Predicting structure in nonsymmetric sparse matrix factorizations, Graph Theory and Sparse Matrix Computation (Edited by Alan George and John R. Gilbert and Joseph W. H. Liu), Springer-Verlag, 1993.

....schemes. The graph theoretic model of nonsymmetric sparse matrix factorization evolved as a tool for understanding the exact nonzero structure of the factor and several variants are possible. We refer the reader our earlier report [38] where we use the characterization developed by Gilbert and Ng [20] to reveal the similarities for either numeric scheme and to motivate the development of our algorithms in Section 2. We formulate algorithms for distributed memory machines in Section 2. Section 3 contains key empirical results. We first show that our formulation of CND for nonsymmetric matrices ....

J. R. Gilbert and E. Ng, Predicting structure in nonsymmetric sparse matrix factorizations, Tech. Rep. ORNL/TM-12204, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8083, 1992.

....like the multifrontal scheme and complete factorization of the frontal matrices. The traditional Householder algorithm introduces too much intermediate fill, and can in practice therefore not be used. However, an interesting connection to sparse LU factorization (Gilbert [17] Gilbert and Ng [19]) makes the traditional Householder algorithm still interesting in real applications. The main result is presented in Theorem 5.1. Theorem 5.1 (George and Ng [16] Let M 2 R n Thetan be a nonsingular square matrix with nonzero diagonal. Suppose M is factorized by Gaussian elimination with row ....

J. R. Gilbert and E. G. Ng, Predicting structure in nonsymmetric sparse matrix factorization, tech. rep., Xerox Palo Alto Research Center, Palo Alto, California 94304-1314, 1992.

....and that for strong Hall matrices (that is, they cannot be permuted to nontrivial block triangular forms) no stronger information is obtainable from the nonzero structure of A. Note that column i updates column j in LU factorization if and only if u ij 6= 0. Theorem 1 (Column Elimination Tree) [63] Let A be a square, nonsingular, possibly unsymmetric matrix, and let PA = LU be any factorization of A with pivoting by row interchanges. Let T be the column elimination tree of A. 1. If vertex i is an ancestor of vertex j in T , then i j. 2. If l ij 6= 0, then vertex i is an ancestor of ....

....sparse Cholesky factorization algorithms take advantage of this type of parallelism, referred to as tree or task parallelism. In unsymmetric LU factorization with partial pivoting, we also wish to determine column dependencies prior to the factorization. It has been shown in a series of studies [50, 54, 63, 65] that the column elimination tree gives the information about all potential dependencies. We herein simply state the most relevant results. The interested reader can consult Gilbert and Ng [63] for a complete and rigorous treatment of this topic. Recall that column i of L and or U modifies column ....

[Article contains additional citation context not shown here]

J. R. Gilbert and E. Ng. Predicting structure in nonsymmetric sparse matrix factorizations. In Alan George, John R. Gilbert, and Joseph W. H. Liu, editors, Graph Theory and Sparse Matrix Computation. Springer-Verlag, 1993.

....a procedure whereby all of the linking columns can be removed by adding some columns to various blocks and extra coupling constraints, thus transforming an arrowhead form into a singly bordered blockdiagonal form. An important concept in what follows is that of the associated graph of a matrix [10]. Definition 4 Given a matrix AM ThetaN , the associated graph of A, denoted by G(A) is the pair (V; E) satisfying: i) V = R [ C, R = fr 1 ; r 2 ; r M g, C = fc 1 ; c 2 ; c N g. ii) r i ; c j ) 2 E if r i 2 R, c j 2 C, and a i;j 6= 0. Note that the G(A) is a bipartite graph, ....

J. R. Gilbert and E. Ng, "Predicting structure in nonsymmetric sparse matrix factorizations", in: A. George, J. Gilbert and J. Liu eds., Graph Theory and Sparse Matrix Computation (Springer-Verlag, 1993) .

....theoretic concepts; a good reference is [15] The graph theoretic model of nonsymmetric sparse matrix factorization evolved in the process of determining the exact nonzero structure of the factor. Although several variants are possible, our presentation is based on a characterization developed in [10] which is best suited for the development of our algorithms in Section 3. The bipartite graph of A is denoted by H(A) and has m row vertices and n column vertices. The row vertices are labeled 1; 2; m and the column vertices are labeled 1; 2; n. The graph has an edge (r; c) from ....

....factor R. Let f i denote the structure of the i th row of either factor. Observe that f i CH i where CH i is as defined earlier with respect to H i . An interesting fact is that the structure estimated by this process is actually quite tight for matrices with the combinatorial Hall property [10]. 3. Algorithms. We now present our distributed algorithms to compute the structure of the factor and to perform numeric computations for either scheme, i.e, Gaussian elimination or orthogonal factorization. A column order of A based on nested dissection of the graph of A T A is central to the ....

J. R. Gilbert and E. Ng, Predicting structure in nonsymmetric sparse matrix factorizations, Tech. Rep. ORNL/TM-12204, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8083, 1992.

....similar) numerical entries. Reusing P c saves the sometimes quite expensive operation of computing it. 3. Reuse P c , P r and data structures allocated for L and U . If P r and P c do not change, then the work of building the data structures associated with L and U (including the elimination tree [14]) can be avoided. This is most useful when A has the same sparsity structure and similar numerical entries as A . When the numerical entries are not similar, one can still use this option, but at a higher risk of numerical instability (BERR will always report whether or not the solution was ....

J. R. Gilbert and E. G. Ng. Predicting structure in nonsymmetric sparse matrix factorizations. In Alan George, John R. Gilbert, and Joseph W. H. Liu, editors, Graph Theory and Sparse Matrix Computation, pages 107-139. Springer-Verlag, 1993.

....rows and columns cannot be permuted to yield a nontrivial block upper triangular form [2, 24] 3. No coincidental numerical cancellation takes place during any arithmetic on A. Formally, this is guaranteed in exact arithmetic if the nonzeros of A are algebraically independent. The references [2, 12, 14, 16, 23] contain more complete discussions of these assumptions and their consequences. These three assumptions imply that (1) the model we use in this paper for the nonzero structure of H and R is exact, not merely an upper bound, and (2) the nonzero structures of H and R are the tightest upper bounds on ....

J. R. Gilbert and E. G. Ng. Predicting structure in nonsymmetric sparse matrix factorizations. In George et al. [8].

....Let L C denote the Cholesky factor of A T A. Let L and U be the LU factors of A obtained by partial pivoting. Then Struct(U) Struct(L T C ) and the entries within each column of L can be rearranged to yield a triangular matrix L with Struct( L) Struct(L C ) Gilbert and Ng [28] also showed that the bound on U is tight when A is a strong Hall matrix. 1 Theorem 2.2 (Gilbert and Ng [28] Let A be a nonsingular and nonsymmetric matrix that has a zero free diagonal. Assume that A is strong Hall. Let L C denote the Cholesky factor of A T A. Let L and U be the LU factors ....

....pivoting. Then Struct(U) Struct(L T C ) and the entries within each column of L can be rearranged to yield a triangular matrix L with Struct( L) Struct(L C ) Gilbert and Ng [28] also showed that the bound on U is tight when A is a strong Hall matrix. 1 Theorem 2. 2 (Gilbert and Ng [28]) Let A be a nonsingular and nonsymmetric matrix that has a zero free diagonal. Assume that A is strong Hall. Let L C denote the Cholesky factor of A T A. Let L and U be the LU factors of A obtained by partial pivoting. For any choice of (i; j) 2 Struct(L T C ) there exists an assignment of ....

J. R. Gilbert and E. G. Ng. Predicting structure in nonsymmetric sparse matrix factorizations. In A. George, J. R. Gilbert, and J. W.H. Liu, editors, Graph Theory and Sparse Matrix Computation, Volume 56 of the IMA Volumes in Mathematics and its Applications, pages 107-139. Springer-Verlag, 1993.

....hence predicting and minimizing fill is more di#cult. Both colamd and colmmd compute a symmetric ordering of A T A; the permutation P is then applied only to the columns of A. The fill in an LL T factorization of PA T AP T is an upper bound on the fill in an LU factorization of PA [8], and reducing fill in PA T AP T tends to reduce fill in the LU factors of PA. We refer to the permutation P as a column ordering. State of the art column minimum degree algorithms do not compute A T A. Instead, they use the nonzero structure of A to construct an initial clique cover of A ....

J. R. Gilbert and E. G. Ng, Predicting structure in nonsymmetric sparse matrix factorizations, in Graph Theory and Sparse Matrix Computation, Springer-Verlag, 1993.

....numerical entries. Reusing P c saves the sometimes quite expensive operation of computing it. 9 3. Reuse P c , P r and data structures allocated for L and U . If P r and P c do not change, then the work of building the data structures associated with L and U (including the elimination tree [13]) can be avoided. This is most useful when A (2) has the same sparsity structure and similar numerical entries as A (1) When the numerical entries are not similar, one can still use this option, but at a higher risk of numerical instability (BERR will always report whether or not the ....

.... 13.26, 7.58, 21.00 ] rowind = 0, 1, 2, 0 ] colptr = 0, 0, 0, 1, 4, 4 ] ffl L = Stype = SC; Dtype = D; Mtype = TRLU; nrow = 5; ncol = 5; Store = nnz = 11; nsuper = 2; nzval = 19.00, 0.63, 0.63, 21.00, 0.57, 0.57, 13.26, 23.58, 0.24, 5.00, 0. 77, 21.00, 34.20 ] nzvalcolptr = [ 0 3, 6, 9, 11, 13 ]; rowind = 0, 1, 4, 1, 2, 4, 3, 4 ] rowindcolptr = 0, 3, 6, 6, 8, 8 ] coltosup = 0, 1, 1, 2, 2 ] suptocol = 0, 1, 3, 5 ] Figure 2.3: The data structures for a 5 Theta 5 matrix and its LU factors, as represented in the SuperMatrix data structure. Zero based indexing is used. 20 ....

J. R. Gilbert and E. G. Ng. Predicting structure in nonsymmetric sparse matrix factorizations. In Alan George, John R. Gilbert, and Joseph W. H. Liu, editors, Graph Theory and Sparse Matrix Computation, pages 107--139. Springer-Verlag, 1993.

....as H and as the narrow part of Q. Finally, Section 6 concludes with a few remarks. 2 Graph models of QR factorization Three graphs model sparsity structure in QR factorization: the column intersection graph, the filled column intersection graph, and the column elimination tree. Gilbert and Ng [16] discuss these graphs in detail; here we review them briefly. All three graphs have the same set of vertices, namely the integers 1, n, representing the columns of A. Matrix A = a ij ) is m by n, with m n and rank n. For a matrix B and a graph G, we write B G if for each entry b ij 6= ....

....diagonal. George and Ng [11] prove that the structure of these rows of H 7 is a subset of the structure of the symbolic Cholesky factor of P T A T AP , which is G (AP ) George and Ng actually consider only the case of square A, but their result extends easily to the non square case [16]. Therefore these n rows of H have O(n log n) nonzeros in all. Finally, consider the last m Gamma n rows of H . The structure of each such row H i is a subset of the path in T (AP ) from the first nonzero in row A i to a root. Therefore jH i j = O( p n ) for each row i, n 1 i m, and ....

[Article contains additional citation context not shown here]

John R. Gilbert and Esmond Ng. Predicting structure in nonsymmetric sparse matrix factorizations. In Alan George, John R. Gilbert, and Joseph W. H. Liu, editors, Graph Theory and Sparse Matrix Computation. Springer-Verlag, 1993. 15

....hence predicting and minimizing fill is more difficult. Both colamd and colmmd compute a symmetric ordering of A T A; the permutation P is then applied only to the columns of A. The fill in an LL T factorization of PA T AP T is an upper bound on the fill in an LU factorization of PA [8], and reducing fill in PA T AP T tends to reduce fill in the LU factors of PA. We refer to the permutation P as a column ordering. State of the art column minimum degree algorithms do not compute A T A. Instead, they use the nonzero structure of A to construct an initial clique cover of A ....

J. R. Gilbert and E. G. Ng, Predicting structure in nonsymmetric sparse matrix factorizations, in Graph Theory and Sparse Matrix Computation, Springer-Verlag, 1993.

....of this tree are the integers 1 through n, representing the columns of A. The column etree of A is the (symmetric) elimination tree of the column intersection graph of A, or equivalently the elimination tree of A T A provided there is no cancellation in computing A T A. See Gilbert and Ng [27] for complete definitions. The column etree can be computed from A in time almost linear in the number of nonzeros of A by a variation of an algorithm of Liu [35] The following theorem says that the column etree represents potential dependencies among columns in LU factorization, and that (for ....

....represents potential dependencies among columns in LU factorization, and that (for strong Hall matrices) no stronger information is obtainable from the nonzero structure of A. Note that column i updates column j in LU factorization if and only if u ij 6= 0. Theorem 2. 1 (Column Elimination Tree) [27] Let A be a square, nonsingular, possibly unsymmetric matrix, and let PA = LU be any factorization of A with pivoting by row interchanges. Let T be the column elimination tree of A. 1. If vertex i is an ancestor of vertex j in T , then i j. 2. If l ij 6= 0, then vertex i is an ancestor of ....

J. R. Gilbert and E. Ng, Predicting structure in nonsymmetric sparse matrix factorizations, in Graph Theory and Sparse Matrix Computation, A. George, J. R. Gilbert, and J. W. H. Liu, eds., Springer-Verlag, 1993.

....of this tree are the integers 1 through n, representing the columns of A. The column etree of A is the (symmetric) elimination tree of the column intersection graph of A, or equivalently the elimination tree of A T A provided there is no cancellation in computing A T A. See Gilbert and Ng [19] for complete definitions. The column etree can be computed from A in time almost linear in the number of nonzeros of A by a variation of an algorithm of Liu [24] The following theorem says that the column etree represents potential dependencies among columns in LU factorization, and that (for ....

....represents potential dependencies among columns in LU factorization, and that (for strong Hall matrices) no stronger information is obtainable from the nonzero structure of A. Note that column i updates column j in LU factorization if and only if u ij 6= 0. Theorem 1 (Column Elimination Tree) [19] Let A be a square, nonsingular, possibly unsymmetric matrix, and let PA = LU be any factorization of A with pivoting by row interchanges. Let T be the column elimination tree of A. 1. If vertex i is an ancestor of vertex j in T , then i j. 2. If l ij 6= 0, then vertex i is an ancestor of ....

J. R. Gilbert and E. Ng. Predicting structure in nonsymmetric sparse matrix factorizations. In Alan George, John R. Gilbert, and Joseph W. H. Liu, editors, Graph Theory and Sparse Matrix Computation. Springer-Verlag, 1993.

No context found.

J. R. Gilbert and E. G. Ng, Predicting structure in nonsymmetric sparse matrix factorizations, in Graph theory and sparse matrix computation, Springer, New York, 1993, pp. 107-- 139.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC