I. Duff. On algorithms for obtaining a maximum transversal. ACM Trans. Math. Software, 7:315--330, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for finding set representation [MH56] or solutions to the assignment problem[Kuh55] could be used to find the transversal. An alternative algorithm involves finding maximal matchings in bipartite graphs[HK73] The algorithm chosen for the transversal is based on work of Duff and Gustavson [Duf81b, Duf81a, Gus76]. The algorithm uses a depth first search of the matrix to determine a series of column interchanges. The algorithm creates a transversal by assigning a unique diagonal position to each column of the matrix. These assignments determine a column permutation which places nonzero elements on the ....

I.S. Duff. On algorithms for obtaining a maximum transversal. ACM Trans. Math. Software, 7(3):315-330, September 1981. 63

....the set of all higher numbered neighbors of i in G A) G(R) For 1 i n, the set T is also the set of all neighbors of vertex i in the elimination graph G i (A A) For the bipartite elimination game algorithm to work, the matrix must have only nonzero entries in its main diagonal. Duff [3] shows that the rows of a Hall matrix can always be permuted to result in a zero free diagonal. The total fill involved in the elimination process is the total of all the edges added during this algorithm. The fill edges (i; j) where i j, correspond to fill appearing in A A and R. The number ....

I. S. Duff, On algorithms for obtaining a maximum transversal, ACM Trans. Math. Software, 7 (1981), pp. 315--330.

....system) and that each variable can be paired with an equation in which it appears. Otherwise, the system is said to be structurally singular and cannot be solved. This step consists of permuting the rows of the incidence matrix so that the incidence is obtained for all diagonal elements [4][5] If it is not possible to obtain a zero free diagonal, then the system is structurally singular. 5.3 Causal ordering Once the model has been checked, the variables and equations which describe the continuous dynamics of each local model are ordered in a computational order. This is an ....

I.S. Duff, On Algorithm for Obtaining a Maximum Transversal, ACM transactions on Mathematical Software, 7(3) (1981) 315-330.

....4 is not zero after symbolic factorization. We assume that every diagonal element in the original sparse matrix is nonzero. Notice that for any nonsingular matrix which does not have a zero free diagonal, it is always possible to permute the rows of A to obtain a matrix with zero free diagonal [8]. Let l k be the index set of nonzeros in l k , i.e. fi j a i;k 6=0 ikg. Similarly, let u k be the index set of nonzeros in u k , i.e. fj j a k;j 6=0 jkg. Symbol j l k j (or ju k j) denotes the cardinality of l k (or u k ) 3.1. The definition of elimination forests. We study the ....

....But the A T A approach overestimates substantially more nonzeros, which indicates that the elimination tree of A T A can introduce too many false dependency edges. All matrices are ordered using the minimum degree algorithm 3 on A T A and the permutation algorithm for zero free diagonal [8]. factor entries jAj Matrix Order jAj Dynamic Static A T A Application domain sherman5 3312 20793 12.03 15.70 20.42 Oil reservoir modeling lnsp3937 3937 25407 17.87 27.33 36.76 Fluid flow modeling lns3937 3937 25407 18.07 27.92 37.21 Fluid flow modeling sherman3 5005 20033 22.13 31.20 ....

I. S. Duff, On Algorithms for Obtaining a Maximum Transversal, ACM Transactions on Mathematical Software, 7 (1981), pp. 315--330.

....step k. In this way we can decrease the number of candidate rows at each step. Therefore a transversal algorithm to transverse the rows to produce a zero free diagonal is necessary for the matrix being ordered by minimum degree ordering. The most popular transversal algorithm is Duff s algorithm [9] which ensures the transverse matrix will have a zero free diagonal. First let s describe the basic techniques of the algorithm, making use of some terminology from graph theory. The transversal is constructed in n major steps, after the kth of which we have a transversal for a submatrix of ....

....forests. Considering an n Theta n sparse matrix A, we assume that every diagonal element of A is nonzero. Notice that for any nonsingular matrix which does not have a zero free diagonal, it is always possible to permute the rows of the matrix so that the permuted matrix has a zero free diagonal [9]. We will use the following notations in the rest of this section. We will still call the matrix after symbolic factorization as A since this paper assumes the symbolic factorization is conducted first. Let a i;j be the element of row i and column j in A and a i:j;s:t be the submatrix of A from ....

I. S. Duff. On Algorithms for Obtaining a Maximum Transversal. ACM Transactions on Mathematical Software, 7(3):315--330, September 1981.

....integers. Permuting the rows and columns of A only relabels the vertices of the bipartite graph: if P and Q are row and column permutation matrices, then H(PAQ T ) is isomorphic to H(A) Several structure prediction problems use matchings and alternating paths in the bipartite graph of a matrix [4, 6, 7, 20, 23, 21, 32]. This paper does not consider such problems in detail, but we include enough definitions here to state some of these results in later sections. Let A be an m by n matrix with m n. We say that A has the Hall property if, for every k with 0 k n, every set of k columns of A contains nonzeros in ....

I. S. Duff. On algorithms for obtaining a maximum transversal. ACM Transactions on Mathematical Software, 7:315--330, 1981.

....by a method analogous to block back substitution; see the references for more details. The decomposition is named for A. L. Dulmage and N. S. Mendelsohn, whose work on matchings and decompositions of bipartite graphs in the late 1950s has found application in several sparse matrix settings [5, 7, 16, 23]. Pothen and Fan [24] survey this theory in sparse matrix language. In graphtheoretic terms, dmperm computes a maximum matching on the bipartite graph of A, permutes the matched edges onto the diagonal of the matrix, and then finds the strongly connected components of the directed graph of the ....

....0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o Figure 3: The Cholesky factor of a matrix and its elimination tree. is based on Iain Duff s MC21A code from the Harwell Subroutine Library [7]. It uses depth first search through alternating paths in the graph with one step of breadth first lookahead. The theoretically fastest algorithm known is due to Hopcroft and Karp [19] but Duff s experiments indicate that MC21A is superior in practice. Duff and Wiberg [11] discuss more ....

I. S. Duff. On algorithms for obtaining a maximum transversal. ACM Transactions on Mathematical Software, 7:315--330, 1981.

....linear system solver. The key component in the solver is the reordering step, which transforms the matrix into a bordered block upper triangular form. Their reordering first uses an unsymmetric ordering to put relatively large entries on the diagonal. The algorithm is a modified version of Duff [11, 12]. After this unsymmetric ordering, they use several symmetric permutations, which preserve the diagonal, to order the matrix into the desired form. With large diagonal entries, there is a better chance of obtaining a stable factorization by pivoting only within the diagonal blocks. The number of ....

I. S. Duff. On algorithms for obtaining a maximum transversal. ACM Trans. Mathematical Software, 7:315--330, 1981.

....matching of a bipartite graph where the rows and columns form the set of vertices and the edges are represented by the nonzero entries. If there is not a perfect matching, where all diagonal positions have a nonzero entry, then the matrix is symbolically singular. This idea was first used by Duff [15] and it is applied as a first step for permuting a matrix to block triangular form. In [15] several matching algorithms are compared. A Fortran code for the best of them is given. The implementation used in the numerical experiments presented later is based on this code. 82 In [12] this idea is ....

....edges are represented by the nonzero entries. If there is not a perfect matching, where all diagonal positions have a nonzero entry, then the matrix is symbolically singular. This idea was first used by Duff [15] and it is applied as a first step for permuting a matrix to block triangular form. In [15] several matching algorithms are compared. A Fortran code for the best of them is given. The implementation used in the numerical experiments presented later is based on this code. 82 In [12] this idea is extended to rectangular matrices. They were concerned in finding a sparse basis for the ....

DUFF, I. S. On Algorithms for Obtaining a Maximum Transversal. ACM Transactions on Mathematical Software, Vol. 7, pp.315-330, 1981.

....Our experiments will indicate which, among the different algorithms implemented in MC64, are the most effective from the point of view of preconditioning. The use of one sided permutations to obtain a zero free diagonal for the purpose of preconditioning is not new. The Harwell Subroutine MC21 [23], 24] which permutes a matrix A to one with a zero free diagonal (or else concludes that the matrix is structurally singular) was used for example in [18] However, MC21 does not take into account the numerical values of the entries that are permuted to the diagonal. In many cases this results ....

....Q = q ij ) where ae q ji = 1; for (i; j) 2 M q ji = 0; otherwise, and thus AQ and QA are matrices with the transversal entries on the diagonal. In this paper, we limit our discussion to row permutations, i.e. permutations of the form QA. MC64 uses the algorithm MC21 implemented by Duff [23], 24] MC21 is a depthfirst search algorithm with look ahead; for a sparse matrix with nonzero entries, the algorithm has worst case complexity of O(n ) but in practice exhibits O(n ) behavior. The use of such a reordering strategy is fundamental as the first step of permuting sparse ....

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I. S. Duff, On algorithms for obtaining a maximum transversal, ACM Trans. Math. Software, 7 (1981), pp. 315--330.

....which a nonzero entry could not be assigned we have a symbolically dependent column. This problem is equivalent to find a matching of a bipartite graph where the rows and columns form the set of vertices and the edges are represented by the nonzero entries. This idea was first used by Duff [4] where several matching algorithms are compared. A Fortran code for the best of them is given. Our implementation is based on Duff s code. In [3] this idea is extended to rectangular matrices. They were concerned in finding a sparse basis for the null space of the matrix. In order to obtain a ....

I. S. Duff, On Algorithms for Obtaining a Maximum Transversal, ACM Trans. Mathe. Software, 7 (1981), pp. 315--330.

....forests Considering an n Thetan sparse matrix A, we assume that every diagonal element of A is nonzero. Notice that for any nonsingular matrix which does not have a zero free diagonal, it is always possible to permute the rows of the matrix so that the permuted matrix has a zero free diagonal [6]. We will use the following notations in the rest of this section. We will still call the matrix after symbolic factorization as A since this paper assumes the symbolic factorization is conducted first. Let a i;j be the element of row i and column j in A and a i:j;s:t be the submatrix of A from ....

....SuperLU does. But the A T A approach overestimates substantially more nonzeros, which also indicates that the elimination tree of A T A introduces too many false dependency edges. All matrices are ordered using the minimum degree algorithm and the permutation algorithm for zero free diagonal [6]. In subsection 6.3, we will also report performance of S for circuit simulation matrices. 6.1 Overall code performance Our previous study [8, 10] shows that even with the introduction of extra nonzero elements by static symbolic factorization, the performance of the S sequential code can ....

I. S. Duff. On Algorithms for Obtaining a Maximum Transversal. ACM Transactions on Mathematical Software, 7(3):315--330, September 1981.

....k as follows: P k = f i j i k; and a k Gamma1 ik is structurally nonzerog: We assume that a kk is always a nonzero. For any nonsingular matrix which does not have a zero free diagonal, it is always possible to permute the rows of the matrix so that the permuted matrix has a zero free diagonal [9]. Though the symbolic factorization does work on a matrix that contains zero entries in the diagonal, it is not preferable because it makes the overestimation too generous. The symbolic factorization process will iterate n steps and at step k, for each row i 2 P k , its structure will be updated ....

....degree ordering for A T A. We Nonzero Fill in A k=1 k=2 k=3 Figure 2: The first 3 steps of the symbolic factorization on a sample 5 Theta 5 sparse matrix. The structure remains unchanged at steps 4 and 5. also permute the rows of the matrix using a transversal obtained from Duff s algorithm [9] to make A have a zero free diagonal. The transversal can often help reduce fill ins [10] We have tested the storage impact of overestimation for a number of nonsymmetric testing matrices from various sources. The results are listed in Table 1. The fourth column in the table is original number ....

I. S. Duff. On Algorithms for Obtaining a Maximum Transversal. ACM Transactions on Mathematical Software, 7(3):315--330, September 1981.

....forests. Considering an n Theta n sparse matrix A, we assume that every diagonal element of A is nonzero. Notice that for any nonsingular matrix which does not have a zero free diagonal, it is always possible to permute the rows of the matrix so that the permuted matrix has a zero free diagonal [7]. In [16] we propose the following definitions which will also be used in the rest of this paper. We still call the matrix after symbolic factorization as A since this paper assumes the symbolic factorization is conducted first. Let a i;j be the element of row i and column j in A and a i:j;s:t be ....

I. S. Duff. On Algorithms for Obtaining a Maximum Transversal. ACM Transactions on Mathematical Software, 7(3):315--330, September 1981.

....k as follows: P k = fiji k; and a k Gamma1 ik is structurally nonzerog: We assume that a kk is always a nonzero. For any nonsingular matrix which does not have a zero free diagonal, it is always possible to permute the rows of the matrix so that the permuted matrix has a zero free diagonal [6]. Though the symbolic factorization does work on a matrix that contains zero entries in the diagonal, it is not preferable because it makes the overestimation too generous. The symbolic factorization process will iterate n steps and at step k, for each row i 2 P k , its structure will be updated ....

....4 and 5. This symbolic factorization is applied after an ordering is performed on the matrix A to reduce fill ins. The ordering we are currently using is the multiple minimum degree ordering for A T A. We also permute the rows of the matrix using a transversal obtained from Duff s algorithm [6] to make A have a zero free diagonal. The transversal can often help reduce fill ins [7] In the SuperLU, symbolic factorization is conducted dynamically according to the actual pivoting choice so that overestimation is not an issue. But the symbolic factorization contributes average 20 Gamma ....

I. S. Duff. On Algorithms for Obtaining a Maximum Transversal. ACM Transactions on Mathematical Software, 7(3):315--330, September 1981.

....k as follows: P k = fiji k; and a k Gamma1 ik is structurally nonzerog: We assume that a kk is always a nonzero. For any nonsingular matrix which does not have a zero free diagonal, it is always possible to permute the rows of the matrix so that the permuted matrix has a zero free diagonal [6]. Though the symbolic factorization does work on a matrix that contains zero entries in the diagonal, it is not preferable because it makes the overestimation too generous. The symbolic factorization process will iterate n steps and at step k, for each row i 2 P k , its structure will be updated ....

....4 and 5. This symbolic factorization is applied after an ordering is performed on the matrix A to reduce fill ins. The ordering we are currently using is the multiple minimum degree ordering for A T A. We also permute the rows of the matrix using a transversal obtained from Duff s algorithm [6] to make A have a zero free diagonal. The transversal can often help reduce fill ins [7] In the SuperLU, symbolic factorization is conducted dynamically according to the actual pivoting choice so that overestimation is not an issue. But the symbolic factorization contributes average 20 45 to ....

I. S. Duff. On Algorithms for Obtaining a Maximum Transversal. ACM Transactions on Mathematical Software, 7(3):315--330, September 1981.

....this collection. We also included Saad s SPARSKIT2 collection (at ftp.cs. umn.edu) and the collection at the University of Florida (available from ftp.cis.ufl.edu in the directory pub umfpack matrices) For the unsymmetric matrices in the test set, we first used the maximum transversal algorithm [4] to reorder the matrix so that the permuted matrix has a zero free diagonal. We then formed the symmetric pattern of the permuted matrix plus its transpose. This is how a minimum degree ordering algorithm is used in MUPS [1] a successor to MA37 [5, 6, 12] Table 1 lists the matrices in our test ....

I. S. Duff, On algorithms for obtaining a maximum transversal, ACM Transactions on Mathematical Software, 7 (1981), pp. 315--330.

....factorization to compute the entire LU factors. When the matrix becomes dense enough near the end of factorization (default of 50 dense) MA48 switches to a dense factorization code. MA48 can preorder a matrix to block upper triangular form (always preceded by finding a maximum transversal [8]) and then factorize each block on the diagonal [12] Off diagonal blocks do not suffer fill in. MA48 can restrict UNSYMMETRIC PATTERN MULTIFRONTAL METHOD 15 Table 6.1 Input parameters for each method. method option UMFPACK MA48 MUPS SSGETRF GPLU scaling of A yes no yes no yes no yes no ....

.... other methods we are comparing perform this test by columns, we factorize A T with MUPS and then use the factors of A T to solve the original system (Ax = b) MUPS optionally preorders a matrix to maximize the modulus of the smallest entry on the diagonal (using a maximum transversal algorithm [8]) MUPS always attempts to preserve symmetry. It does not permute the matrix to block upper triangular form. SSGETRF is a classical multifrontal method in the Cray Research, Inc. library (version 1.1) installed on the CRAY YMP. It uses Liu s multiple minimum degree (MMD) algorithm [28] on the ....

I. S. Duff, On algorithms for obtaining a maximum transversal, ACM Transactions on Mathematical Software, 7 (1981), pp. 315--330.

....fronts) The remaining 44.4 million operations are performed in only 116 frontal matrices, a typical one of which is 69 by 65 with 31 pivots) The largest frontal matrix constructed by AFstack is 216 by 281, with 171 pivots. For this matrix, Mups is directed to first find a maximum transversal [8], otherwise excessive fill in is obtained. MA28 and D2 both find a poor pivot ordering for rdist1, when compared with Mups and the AF algorithms. MA28 and D2 use only a global pivot search, and do not consider the overlap of the fill in of a previous pivot with the current pivot. Using only a ....

I. S. Duff. On algorithms for obtaining a maximum transversal. ACM Transactions on Mathematical Software, 7:315--330, 1981.

....added later in the form of an additional subroutine. We have taken the opportunity in MA48 to build iterative refinement into the solve subroutine as an option. We also provide options for calculating estimates of the relative backward error and of the error in the solution (Arioli, Demmel, and Duff 1989) A separate HSL package, MA30, was provided with MA28 for users willing to order their matrix entries by rows. It was called by MA28 to perform the fundamental tasks of matrix factorization and actual solution. We have followed the same model for the new code, with MA50 providing the fundamental facilities. One significant ....

Duff, I. S. (1981a). On algorithms for obtaining a maximum transversal. ACM Trans. Math. Softw. 7, 315-330.

....of simple cases (surprisingly common) where a row or column of A is zero or identical to another row or column. One of the main ways that we achieve high performance, particular on vector or super scalar machines, is to switch to full matrix processing using Level 3 BLAS (Dongarra, Du Croz, Duff, and Hammarling 1990) once the matrix is sufficiently dense. This has led us to using a column oriented representation internally because of the column major ordering used by Fortran. It also means that the inner loops of the solve phase will vectorize more readily because they involve adding a multiple of one vector ....

Duff, I. S. (1981a). On algorithms for obtaining a maximum transversal. ACM Trans. Math. Softw. 7, 315-330.

....parameters that control its behavior. We used the recommended defaults for most of these, with a few exceptions that we now indicate. By default, three of the five methods (MA38, UMFPACK V1.1, and MA48) preorder a matrix to block triangular form (always preceded by finding a maximum transversal [14]) and then factorize each block on the diagonal [16] This can reduce the work for unsymmetric matrices. We did not perform the preordering, since MA42 and MUPS do not provide these options. One matrix (lhr71) was so ill conditioned that it required scaling prior to its factorization. The scale ....

I. S. Duff, On algorithms for obtaining a maximum transversal, ACM Transactions on Mathematical Software, 7 (1981), pp. 315--330.

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I. Duff. On algorithms for obtaining a maximum transversal. ACM Trans. Math. Software, 7:315--330, 1981.

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I. S. Duff. On Algorithms for Obtaining a Maximum Transversal. ACM Transactions on Mathematical Software, 7(3):315--330, September 1981.

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I. S. Duff. On algorithms for obtaining a maximum transversal. ACM Transactions on Mathematical Software, 7:315--330, 1981.

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