J. W. H. Liu, A compact row storage scheme for Cholesky factors using elimination trees, ACM Trans. Math. Software, 12 (1986), pp. 127--148.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= i Gamma first(i) Gamma parent(c j ) i size(c j ) x x x x x x x x x x x x x x 29 15 14 16 17 18 19 20 21 24 23 25 22 26 27 28 Figure 3: An example of a part of an postordered elimination tree. The lemma is a simple consequence of the basic properties of postorderings (see [1] [15]) An example is given in Figure 3, which shows a part of a postordered elimination tree. The number of leaves of root 29 of the tree from Figure 3 can be computed as no of leaves(29) 29 Gamma first(29) Gamma (size(T [22] size(T [25] 29 Gamma 14 Gamma (3 9) 3: Lemma 3.2 suggests an ....

....3 can be computed as no of leaves(29) 29 Gamma first(29) Gamma (size(T [22] size(T [25] 29 Gamma 14 Gamma (3 9) 3: Lemma 3. 2 suggests an efficient way how to implement the whole modification of K to get K: First, we obtain a postordered elimination tree by standard algorithms (see [15]) Then we reorder it so that the leaves in each node are ordered consecutively after all the other subtrees rooted in children of the node. In fact, a similar transformation that reorders children sets in the elimination tree is a component of any up to date sparse symmetric Cholesky solver. ....

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J.W.H. Liu. A compact row storage scheme for Cholesky factors using elimination trees. ACM Trans. Math. Software, 12:127--148, 1986.

....A) from Struct[A T A] The most expensive part of the algorithm in Figure 1 is the tree traversals to determine the roots. However, the eciency of the tree traversals can be improved by various kinds of path compression, as used in disjoint set union algorithms [26] We refer the reader to Liu [18, 20] for details of disjoint set union in elimination tree algorithms. If balancing and path compression are both used then the elimination tree algorithm has time complexity O(jA T Aj (jA T Aj; n) Liu [20] recommends using path compression without balancing; the resulting algorithm has time ....

....compression. 6 2. 2 The nonsymmetric algorithm We now shift our attention to how T (A T A) can be computed from Struct[A] rather than Struct[A T A] Every o diagonal nonzero entry in A T A, and hence every edge of G(A T A) corresponds to an ancestor descendant pair in T (A T A) [18, 20, 25]. In particular, if (i; j) and (j; k) are two edges in G(A T A) with i j k, then vertices i, j, and k must belong to the unique path joining i and k, with i being the descendant of both j and k. For 1 i m, let the column indices of the rst and last nonzeros in row A i be denoted by f ....

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J. W. Liu. A compact row storage scheme for Cholesky factors using elimination trees. ACM Trans. Math. Software, 12:127-148, 1986.

....the inverse of the Ackerman function. They also showed that with the elimination tree one can count fill (and get al..l the vertex degrees in the filled graph) in another O(n#(n) time. For proofs of these statements and many other fascinating facts about elimination trees and filled graphs consult [14, 15, 16, 28, 29]. Therefore, we have the following corollary. Corollary 5.1. If the graph of a symmetric sparse matrix A is a well shaped mesh, then a provably good nested dissection ordering of A, its elimination tree, and its fill information can be found in O(n log log n) time. 5.2. 3D point location. One ....

J. W. H. Liu, A compact row storage scheme for cholesky factors using elimination trees,ACM Trans. Math. Software, 12 (1986), pp. 127--148.

....G 0 . Let G 0 = X ; E [ F ; ff) be a chordal graph and let m(i) 1 i N Gamma 1 be given by m(i) minfjjx j 2 madj(x i )g: 3.1) Then the m tree T for G 0 is the tree with vertex set X and edges E 0 = fe im(i) g N Gamma1 i=1 . The m tree is also called the elimination tree by Liu [7] and Schreiber [12] See Liu [8] for a recent survey of the role of m trees in sparse Gaussian elimination. Among 4 ffifl fflfi 4 ffifl fflfi 6 ffifl fflfi 3 ffifl fflfi 7 ffifl fflfi 8 ffifl fflfi 9 ffifl fflfi 1 ffifl fflfi 5 ffifl fflfi 2 Fig. 3.1. A 3 Theta 3 grid graph ....

....Since x p 2 cadj(x i ) 2 p k Gamma 1, 6 e p i 2 E [F . Thus the path and the backedge form a cycle in G 0 ; this cycle is chorded by the edges e p i ; 2 p k Gamma 1. Let B denote the set of backedges and K the set of chords. It is easy to see E [ F = E 0 [ B [ K: Following Liu [7], the graph S = X ; E 0 [B) is called the skeleton of G 0 . The skeleton of the 3 Theta 3 grid graph is shown in Fig. 5. ffifl fflfi 9 ffifl fflfi 8 ffifl fflfi 7 ffifl fflfi 5 ffifl fflfi 6 ffifl fflfi 2 ffifl fflfi 1 ffifl fflfi 4 ffifl fflfi 3 Q Q Q j j j T T T T . ....

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J. W. H. Liu, A compact row storage scheme for cholesky factors using elimination trees, ACM TOMS, 12 (1986), pp. 127--148.

.... and such basic techniques as depth first search [26] We also assume a basic knowledge of the four steps in solving sparse systems by Cholesky factorization, and with the use of graphs in these algorithms [15] More specifically, we assume familiarity with elimination trees [19] skeleton graphs [18], postorderings, supernodes [1, 2, 16, 20, 22] and the subscript compression scheme for L [15, 27] 1.1 Applications Here we survey some of the sparse matrix settings in which it is useful to precompute the row counts, the column counts, or the total number of nonzeros in the Cholesky factor of ....

....that G(A) be processed three times, and we doubt that any multiple pass implementation will come close to realizing the practical efficiency of the single pass implementation presented in this section. Second, we must discard some edges of G(A) that do not affect the result. Recall from Liu [18] that the skeleton graph G Gamma = G Gamma (A) is obtained from G(A) by removing every edge (u; v) for which v u and the vertex v is not a leaf of T r [u] The skeleton graph is the smallest subgraph of G(A) whose filled graph is identical with that of G(A) Consequently, the new algorithm ....

J. W-H. Liu. A compact row storage scheme for Cholesky factors using elimination trees. ACM Trans. Math. Software, 12:127--148, 1986.

....but may lead to somewhat less work later due to the exact computation of the structure of L. In any case, the latter approach is now more viable on distributed memory machines as a consequence of our algorithms. Our work relies significantly on earlier algorithms to compute an elimination tree [10, 19, 20], compute the structure of L [11] perform numeric factorization using the multifrontal method [3, 12] and to perform dense matrix operations on distributed machines [4, 6] Thus, throughout this paper we include brief reviews of relevant background material interspersed with descriptions of our ....

J. W.-H. Liu, A compact row storage scheme for Cholesky factors using elimination trees, ACM Trans. Math. Software, 12 (1986), pp. 127--148.

....the definition by Tarjan [33] a topological ordering of the elimination tree is an ordering that numbers the children nodes before their parent node. Therefore, topological orderings preserve the elimination tree. Postorderings, mentioned in Section 2, are a subset of topological orderings. Liu [21] observes that any topological ordering of the elimination tree corresponds to an equivalent reordering of the matrix M . However, as we have mentioned above, the opposite is not true. All the leaves of the elimination tree are simplicial vertices in the filled graph. But there may be more ....

J. W. H. Liu, A compact row storage scheme for Cholesky factors using elimination trees, ACM Trans. Math. Software, 12 (1986), pp. 127--148.

....j de structure non vide, l indice du premier el ement non nul dans la colonne j, au dessous de la diagonale. p(j) minfi 2 Struct(L j )g si Struct(L j ) 6= j sinon: 8 Jocelyne Erhel, Mounir Hahad, Thierry Priol D efinition 6 L arbre d elimination T(A) de A est d efini comme suit[13] : ffl l ensemble de ses noeuds est l ensemble des indices des colonnes de L ; ffl i est le p ere de j (avec i j) i = p(j) 2 Notons ici que, si la matrice A est irr eductible, l arbre d elimination est effectivement un arbre dans le sens o u il a une racine. Dans le cas contraire, il est ....

....tache de base devient T(j) dans le cas du FanIn et J(k) dans le cas du Fan Out. Dans ces termes, les d ependances sont illustr ees par les figures 11 et 12. Une r eduction transitive de ces graphes de d ependances permet de retrouver l arbre d elimination de la matrice. Il a et e d emontr e dans [13] qu un noeud de l arbre ne peut d ependre que de ses descendants. Nous pouvons en d eduire que des noeuds (colonnes de L) n ayant pas de relation ascendant descendant sont ind ependants l un de l autre, et peuvent etre evalu es en parall ele. 2.2 Algorithme Fan Out 2.2.1 Distribution des ....

Joseph LIU. A compact row storage scheme for cholesky factors using elimination trees. ACM transactions on mathematical software, 1986.

....and the row merge heap used here. The elimination forest construction is a more compact representation of the information required for the factorization because the nonzero structure of the rows of the Cholesky factor can be easily computed from the elimination tree and the original matrix [9, 14]. In the row merge heap, the structure of a vertex is only a subset of the set of its ancestors. This means that the row merge heap, without the vertex structures, is insufficient to exactly determine the nonzero structures of the upper trapezoidal matrices associated with the vertices, or the ....

J. W.-H. Liu, A compact row storage scheme for Cholesky factors using elimination trees, ACM Transactions on Mathematical Software, 12 (1986), pp. 127--148.

....the definition by Tarjan [33] a topological ordering of the elimination tree is an ordering that numbers the children nodes before their parent node. Therefore, topological orderings preserve the elimination tree. Postorderings, mentioned in Section 2, are a subset of topological orderings. Liu [21] observes that any topological ordering of the elimination tree corresponds to an equivalent reordering of the matrix M . However, as we have mentioned above, the opposite is not true. All the leaves of the elimination tree are simplicial vertices in the filled graph. But there may be more ....

J. W. H. Liu, A compact row storage scheme for Cholesky factors using elimination trees, ACM Trans. Math. Software, 12 (1986), pp. 127--148.

....results indicate that the new algorithm is usually faster than earlier methods. Key words. sparse matrix algorithms, Gaussian elimination, LU factorization, elimination tree, elimination dag. AMS(MOS) subject classifications. 05C20, 05C75, 65F05, 65F50. 1. Introduction. The elimination tree [10, 14] is central to the study of symmetric factorization of sparse positive definite matrices. Liu [11] surveys the use of this tree structure in many aspects of sparse Cholesky factorization, including sparse storage schemes, matrix reordering, symbolic and numerical factorization algorithms, and ....

....Figure 4 displays the two elimination dags corresponding to this matrix example. 3.3. The symmetric case: Elimination trees. The situation is simpler if the matrix A is symmetric. Still assuming no pivoting, Gaussian elimination yields a symmetric factorization A = LL T . The elimination tree [10, 14] has been used extensively to study symmetric Gaussian elimination. Let L be the Cholesky factor of a symmetric positive definite matrix A. Formally, the elimination tree T (A) of A has n vertices f1; ng, and hj; ki is an edge if and only if k = minfr j j rj 6= 0g: This structure is a ....

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J. W. H. Liu. A compact row storage scheme for Cholesky factors using elimination trees. ACM Transactions on Mathematical Software, 12:127--148, 1986.

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J. W. H. Liu, A compact row storage scheme for Cholesky factors using elimination trees, ACM Trans. Math. Software, 12 (1986), pp. 127--148.

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J. W. H. Liu. A compact row storage scheme for Cholesky factors using elimination trees. ACM Transactions on Mathematical Software, 12:127--148, 1986.

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