J. R. Gilbert and T. Peierls, Sparse partial pivoting in time proportional to arithmetic operations, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 862--874.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the sparse LU solver provided in MATLAB. For general nonsymmetric matrices, a strategy that this method implements is to reorder the columns according to their minimum degree in order to minimize the fill in. On the other hand, a sparse partial pivoting technique proposed by Gilbert and Peierls [45] is used to prevent losses in the stability of the method. In Table 5.7, we present some results obtained with the method we have just described. m = 10 m = 20 m = 30 tolerance tolerance tolerance T 10 7 76 130 170 74 107 135 76 109 145 8 117 170 220 103 125 190 110 150 200 10 ....

J. R. Gilbert and T. Peierls. Sparse Partial Pivoting in Time Proportional to Arithmetic Operations. SIAM J. Sci. Statist. Comput., 9:862--874, 1988.

....method to nonsymmetric matrices and explores e#ective dropping strategies that are enabled by the Crout variant. 2 The Crout LU and ILU The main disadvantage of the standard delayed update IKJ factorization is that it requires access to the entries in the current row of L by topological order [8]. One topological order, which is perhaps most appropriate for threshold based incomplete factorizations, is increasing order by column number (since the nonzero pattern of the factorization is not known beforehand) The entries in this order must be found via searches, which are further ....

J. R. Gilbert and T. Peierls. Sparse partial pivoting in time proportional to artithmetic operations. SIAM Journal on Scientific Computing, 9:862--874, 1988.

....the sparse LU solver provided in MATLAB. For general nonsymmetric matrices, a strategy that this method implements is to reorder the columns according to their minimum degree in order to minimize the fill in. On the other hand, a sparse partial pivoting technique proposed by Gilbert and Peierls [14] is used to prevent losses in the stability of the method. We present some results obtained with the method just described in Table 4. T Mflops 7 91 8 160 10 2950 15 7300 20 30 50 The symbol indicates that the memory capacity (40 Mbytes) has been exceeded. Out of memory in the ....

J. R. Gilbert and Peierls. Sparse Partial Pivoting in Time Proportional to Arithmetic Operations. SIAM J. Sci. Statist. Comput., 9:862--874, 1988.

....W. HAGER Theorem 5.2. During symbolic downdate AA T = AA T ww T (where w is a column of A) the nonzero pattern of v = L 1 w is equal to the path P(k) in the (old) elimination tree of L where k = min i : w i #= 0 . 5.1) Proof. Let W = i : w i #= 0 . Theorem 5. 1 of Gilbert [15, 16, 19] states that the nonzero pattern of v is the set of nodes reachable from the nodes in W by paths in the directed graph G(L T ) By Algorithm 1, W # L k . Hence, each element of W is reachable from k by a path of length one, and the nodes reachable from W are a subset of the nodes ....

J. R. Gilbert and T. Peierls, Sparse partial pivoting in time proportional to arithmetic operations, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 862--874. MODIFYING A SPARSE CHOLESKY FACTORIZATION 627

....sections. In Section 6, we compare the performance of our algorithm with two algorithms based on the classical multifrontal method: MUPS [1, 2] and SSGETRF [3] and two algorithms based on conventional (compressed sparse vector) data structures: Gilbert and Peierls partial pivoting code (GPLU [23]) and MA48 [16] a successor to MA28 [13] GPLU does not use dense matrix kernels. MA48 uses dense matrix kernels only after switching to a dense factorization code towards the end of factorization when the active submatrix is fairly dense. 2. The basic approach. Our goal with the UMFPACK ....

....matrix. i R i j C j 4 (1,3) 4 (1,1) 2,1) 5 (2,1) 5 (1,2) 2,2) 6 6 7 (2,2) 7 (2,3) 5. Algorithm. Algorithm 3 is a full outline of the UMFPACK (version 1.0) algorithm. 6. Performance results. In this section, we compare the performance of UMFPACK version 1. 0 with MUPS [1] MA48 [16] GPLU [23], and SSGETRF [3] on a single processor of a CRAY YMP (although MUPS and SSGETRF are parallel codes) Each method has a set of input parameters that control its behavior. We used the recommended defaults for most of these, with a few exceptions that we indicate below. All methods can factorize ....

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J. R. Gilbert and T. Peierls, Sparse partial pivoting in time proportional to arithmetic operations, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 862--874.

....has been made to the leading columns of PAQ, or both. Solve. MA50C uses the factorization produced by MA50B to solve the equation Ax = b or the T equation A x = b. A significant change from MA30 is the inclusion of row interchanges for stability in the factorize subroutine, based on the work of Gilbert and Peierls (1988), which allows these to be included without an increase in overall complexity. In turn, this has allowed us to simplify the analyse subroutine so that it provides the permutations without the actual factors. This saves storage during analyse since only the active submatrix need be stored and will ....

....may be factorized more economically if only the numerical values have changed or if the changes are confined to late columns. 2.2.1 First factorization We begin by considering a first factorization when the rank is n. The operations are performed column by column because the technique of Gilbert and Peierls (1988) then allows row interchanges to be introduced while ensuring that the organizational overheads are proportional to the number of floating point operations. It also means that the factorization, including fill ins, can be built progressively by columns with very simple data management. A packed ....

Gilbert, J. R. and Peierls, T. (1988). Sparse partial pivoting in time proportional to arithmetic operations. SIAM J. Sci. Stat. Comput. 9, 862-874.

....a distributed memory implementation is possible as well because we have a coarse grained algorithm that does not need complicated communication. The direct solver we used (for the direct part of our method) was our own implementation of GP Mod. This is the sparse LU method of Gilbert and Peierls [9] extended with the symmetric reductions of Eisenstat and Liu [8] 7] see also [4] Any sparse LU method for the Schur complement is allowed as a direct solver, including, for example, multifrontal methods. We choosed GP Mod because it is relatively easy to implement. Moreover, in [4] it is ....

J.R. Gilbert, T. Peierls, Sparse partial pivoting in time proportional to arithmetic operations. SIAM J. Sci. Comput., 9(5) (1988), pp. 862-874.

....and many solves for each factorize. Since it is normal for factorize to be substantially faster than analyse factorize and solve to be substantially faster than solve, this subdivision fits well in such an environment. The most exciting algorithmic development in recent years is the work of Gilbert and Peierls (1988) for economically generating the patterns of the columns of the factors when factorizing with a given column sequence but allowing for row interchanges. They use a depth first search of the directed graph of the previous row operations to generate the pattern of the current column. The overall ....

....performing the stability test (3.1.2) by columns, and we in fact store the reciprocal of the pivot to avoid excessive divisions. 3.2.1 First factorization We begin by considering a first factorization when the rank is n. The operations are performed column by column because the technique of Gilbert and Peierls (1988) then allows row interchanges to be introduced while ensuring that the organizational overheads are proportional to the number of floating point operations. It also means that the factorization, including fill ins, can be built progressively by columns with very simple data management. If, ....

Gilbert, J. R. and Peierls, T. (1988). Sparse partial pivoting in time proportional to arithmetic operations. SIAM J. Sci. Stat. Comput. 9, 862-874.

....for each iteration: ffl Symbolic factorization (line 3) Initial stage, where the nonzero structure for the column is predicted. This prediction is not more timeconsuming than the numerical factorization when we use depth first search and topological ordering, such as proven by Gilbert and Peierls [10]. This simbolic factorization was speeded up by Eisenstat and Liu [8] who designed a pruning technique (line 10) to reduce the amount of structural information required for the symbolic factorization. ffl Numerical factorization (lines 5, 8 and 9) They perform the arithmetic operations ....

J.R. Gilbert and T. Peierls, Sparse Partial Pivoting in Time Proportional to Arithmetic Operations. SIAM J. Sientific and Estatistical Computing, 9(5):862-874, Sep. 1988.

....C , low level compiling, and linking. Results The goal of AML is to allow sparse matrix code that is both easy to understand and modify, and efficient. This section measures the expression of LU in AML against those goals. We compare sparse LU written in AML with two other implementations: GPLU [10], a well known Fortran version of the algorithm, also implemented (in C) in the Matlab version 4 internal library [5] and SuperLU [11] the best performing research code, which uses much more Created 08 20 97 4:02 PM John Lamping Last saved 09 05 97 10:14 AM This copy printed 09 05 97 10:25 AM ....

Gilbert, J.R. and T. Peierls, Sparse Partial Pivoting in Time Proportional to Arithmetic Operations. SIAM J. Sci. Statist. Comput., 1988. 9: p. 862-874.

....ordering methods 2 1 Introduction Sparse Gaussian elimination with partial pivoting computes the factorization PAQ = LU for the sparse nonsymmetric matrix A, where P and Q are permutation matrices, L is a lower triangular matrix, and U is an upper triangular matrix. Gilbert and Peierls [30] have shown that sparse partial pivoting can be implemented in time proportional to the number of oatingpoint operations required. The method is used by Matlab when solving a system of linear equations Ax = b when A is sparse [27] An improved implementation is in a sparse matrix package, SuperLU ....

J. R. Gilbert and T. Peierls. Sparse partial pivoting in time proportional to arithmetic operations. SIAM J. Sci. Statist. Comput., 9:862-874, 1988.

.... Mathematical Software algorithm analysis, eciency General terms: Algorithms, Experimentation, Performance Keywords: sparse unsymmetric matrices, linear equations, ordering methods 1 Overview Sparse partial pivoting methods compute the factorization PAQ = LU by rst nding a column ordering Q [3, 4, 5]. The column ordering Q is selected without regard to the numerical values. The row permutation P is found via standard partial pivoting, without regard to sparsity. The goal is to nd Q to limit the worst case ll in, regardless of how P is subsequently chosen. Such an ordering Q also limits the ....

J. R. Gilbert and T. Peierls. Sparse partial pivoting in time proportional to arithmetic operations. SIAM J. Sci. Statist. Comput., 9:862-874, 1988. 5

....multiplication that computes the jth column of L. The algorithm determines which columns of L and U need to update v, as well as an admissible order for the updates, using a depth first search (DFS) on the directed graph that underlies L. This technique was developed by Gilbert and Peierls [7]. Once all the updates have been applied to v, the algorithm factors v, using threshold partial pivoting. Specifically, the algorithm searches for the largest entry v m in v L , the lower part of v (we use v U to denote the upper part of v) If v d # v m , where 0 # # # 1 is ....

....have increased the density of the factors, and we estimated that the increased density would o#set the savings in running time gained from supernodes. Still, this technique could perhaps enhance performance on some matrices. We have implemented this algorithm as a modification to the GP code [7]. The modifications are mostly restricted to the driver subroutine and to the column factorization subroutine. The subroutines that perform the DFS and update the current column are essentially unmodified. We had to slightly modify all the routines in order to implement a column ordering ....

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J. R. Gilbert and T. Peierls. Sparse partial pivoting in time proportional to arithmetic operations. SIAM Journal on Scientific and Statistical Computing, 9:862--874, 1988.

....: ng; there is an edge from i to j (for i 6= j) if and only if the entry a ij is nonzero. We shall write the directed edge from i to j as hi; ji. Notice that the edges hi; ji and hj; ii are different; the former means that a ij 6= 0 and the latter means that a ji 6= 0. Many sparse matrix codes [6, 9] use a data structure that lists the nonzeros of a matrix A in column major order. In graph terms, this is the adjacency list structure for G(A T ) Representing A by G(A T ) instead of by G(A) seems to be necessary for some efficient algorithms [9] This is the reason that this paper ....

....that a ji 6= 0. Many sparse matrix codes [6, 9] use a data structure that lists the nonzeros of a matrix A in column major order. In graph terms, this is the adjacency list structure for G(A T ) Representing A by G(A T ) instead of by G(A) seems to be necessary for some efficient algorithms [9]. This is the reason that this paper sometimes states results and algorithms in terms of the graph of the transpose of the matrix in question. 2 L = 0 B B B B B B 1 2 ffl 3 ffl ffl 4 ffl ffl 5 ffl ffl ffl 6 1 C C C C C C A j i j j i j i j j i 3 4 5 1 6 2 oe 6 ....

[Article contains additional citation context not shown here]

John R. Gilbert and Tim Peierls. Sparse partial pivoting in time proportional to arithmetic operations. SIAM Journal on Scientific and Statistical Computing, 9:862--874, 1988.

....The asymptotically fastest algorithms to compute the Cholesky factorization of a symmetric positive definite matrix are those of the Yale Sparse Matrix Package [12] and Sparspak [15] which predict the structure of the triangular factor by a version of Theorem 4.3 below. Gilbert and Peierls [24] have used prediction of the structure of the solution of a triangular system of equations, a special case of Theorem 5.1 below, to develop the first algorithm that performs sparse LU factorization with partial pivoting in worst case time proportional to the number of real arithmetic operations. ....

....in Duff and Reid s MA28 code [9] Curiously enough, the most important applications of the results in this section are at the opposite extreme, for triangular systems. Structure prediction for sparse triangular systems is used in efficient algorithms for LU factorization with partial pivoting [24] and in parallel triangular solution [1] Throughout this section A is an n by n matrix with nonzero diagonal, and the graph in question is the directed graph G(A) Theorem 5.1. Let the structures of A and b be given. i) Whatever the values of the nonzeros in A and b, if A is nonsingular then ....

[Article contains additional citation context not shown here]

John R. Gilbert and Tim Peierls. Sparse partial pivoting in time proportional to arithmetic operations. SIAM Journal on Scientific and Statistical Computing, 9:862--874, 1988.

....Q for sparsity can, of course, ask for lu(A Q ) or lu(A( q) where q is an integer permutation vector. Section 3.3 describes a few ways to find such a permutation. The matrix division operators and do pivot for sparsity by default; see Section 3.4. We use a version of the GPLU algorithm [18] to compute the LU factorization. This computes one column of L and U at a time by solving a sparse triangular system with the already finished columns of L. Section 3.4.2 describes the sparse triangular solver that does most of the work. The total time for the factorization is proportional to the ....

....division is nearly singular. Sparse MATLAB should do the same, but the current version does not yet implement it. 3.4. 2 Sparse triangular solution The sparse triangular linear equation solver, which is also the main step of LU factorization, is based on an algorithm of Gilbert and Peierls [18]. When A is triangular and b is a sparse vector, x = Anb is computed in two steps. First, the nonzero structures of A and b are used (as described below) to make a list of the nonzero indices of x. This list is also the list of columns of A that participate nontrivially in the triangular ....

John R. Gilbert and Tim Peierls. Sparse partial pivoting in time proportional to arithmetic operations. SIAM Journal on Scientific and Statistical Computing, 9:862--874, 1988.

.... of complete pivoting choose 19 a pivot at each step to minimize operation count from among candidates that are not too far from maximum magnitude [6] Another approach is to preorder the matrix columns purely to preserve sparsity, and then use partial pivoting to reorder the rows for stability [13, 16]. This section parallels Section 3 in outline. In Section 4.1, we review a graph model of Gaussian elimination with row and column interchanges, and we prove some results on the structure of the matrix during elimination. These results are symbolic; that is, they assume that zeros are introduced ....

John R. Gilbert and Tim Peierls. Sparse partial pivoting in time proportional to arithmetic operations. SIAM Journal on Scientific and Statistical Computing, 9:862--874, 1988.

....multiplication that computes the jth column of L. The algorithm determines which columns of L and U need to update v, as well as an admissible order for the updates, using a depth first search (DFS) on the directed graph that underlies L. This technique was developed by Gilbert and Peierls [7]. Once all the updates have been applied to v, the algorithm factors v, using threshold partial pivoting. Specifically, the algorithm searches for the largest entry v m in vL , the lower part of v (we use vU to denote the upper part of v) If jv d j jv m j, where 0 1 is the pivoting ....

....have increased the density of the factors, and we estimated that the increased density would offset the savings in running time gained from supernodes. Still, this technique could perhaps enhance performance on some matrices. We have implemented this algorithm as a modification to the GP code [7]. The 6 J. R. GILBERT AND S. TOLEDO modifications are mostly restricted to the driver subroutine and to the column factorization subroutine. The subroutines that perform the DFS and update the current column are essentially unmodified. We had to slightly modify all the routines in order to ....

[Article contains additional citation context not shown here]

J. R. Gilbert and T. Peierls. Sparse partial pivoting in time proportional to arithmetic operations. SIAM Journal on Scientific and Statistical Computing, 9:862-- 874, 1988.

....di#erences, however, are in the numerical factorization phase. Perhaps the most significant di#erence is the use of a priority queue in our algorithm versus the use of depth first search (dfs) in SuperLU. To determine which columns of L must update the active panel, SuperLU uses a dfs technique [6]. The graph being searched represents a subset of the nonzeros of L [3] Earlier algorithms searched in a graph that represents the entire nonzero structure of L, in which case the cost of the searches is proportional to the number of floating point operations in the algorithm. Searching in the ....

J. R. Gilbert and T. Peierls, Sparse partial pivoting in time proportional to arithmetic operations, SIAM J. Scientific and Statistical Computing, 9 (1988), pp. 862--874.

....Q for sparsity can, of course, ask for lu(A Q ) or lu(A( q) where q is an integer permutation vector. Section 3.3 describes a few ways to find such a permutation. The matrix division operators and do pivot for sparsity by default; see Section 3.4. We use a version of the GPLU algorithm [15] to compute the LU factorization. This computes one column of L and U at a time by solving a sparse triangular system with the already finished columns of L. Section 3.4.2 describes the sparse triangular solver that does most of the work. The total time for the factorization is proportional to the ....

....in matrix division is nearly singular. Sparse Matlab should do the same, but the current version does not yet implement it. 3.4.2. Sparse triangular systems. The triangular linear system solver, which is also the main step of LU factorization, is based on an algorithm of Gilbert and Peierls [15]. When A is triangular and b is a sparse vector, x = Anb is computed in two steps. First, the nonzero structures of A and b are used (as described below) to make a list of the nonzero indices of x. This list is also the list of columns of A that participate nontrivially in the triangular solution. ....

J. R. Gilbert and T. Peierls, Sparse partial pivoting in time proportional to arithmetic operations, SIAM Journal on Scientific and Statistical Computing, 9 (1988), pp. 862--874.

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J. R. Gilbert and T. Peierls, Sparse partial pivoting in time proportional to arithmetic operations, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 862--874.

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J. R. Gilbert and T. Peierls. Sparse partial pivoting in time proportional to artithmetic operations. SIAM Journal on Scientific Computing, 9:862--874, 1988.

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J. R. Gilbert and T. Peierls, Sparse partial pivoting in time proportional to arithmetic operations, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 862-874.

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J. R. Gilbert and T. Peierls (1988), Sparse Partial Pivoting in Time Proportional to Arithmetic Operations, SJSSC, 9, pp. 862--874.

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J. R. Gilbert and T. Peierls. Sparse partial pivoting in time proportional to arithmetic operations. SIAM Journal on Scienti#c and Statistical Computing, 9:862#874, 1988.

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