J. R. Gilbert. Predicting structure in sparse matrix computations. SIAM Journal of Matrix Analysis and Applications, 15(1):62--79, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....to each other in the lled pattern. Unlike the Cholesky factor whose minimum graph representation is a tree (called the elimination tree, or etree for short) 48] the minimum graph representations of the L and U factors are directed acyclic graphs (called elimination DAGs, or edags for short) [31, 32]. Despite these diculties, researchers have been addressing these issues successfully for sequential and shared memory machines; available codes include MA41 [6, 5] PARDISO [57] SPOOLES [9] SuperLU [19] SuperLU MT [20] UMFPACK MA38 [15] and WSMP [34] In our earlier codes SuperLU (serial) ....

John R. Gilbert. Predicting structures in sparse matrix computations. SIAM J. Matrix Analysis and Applications, 15(1):62-79, January 1994.

....is a sequence of distinct nodes i 1 , i k such that i 1 i 2 , i 2 i 3 , and i k,1 i k , often written i 1 ### i k ,orsimplyi 1 ; i k . The transitive closure G of a graph G is one constructed on the same nodes but having i j whenever i in G. For a fuller treatment, see [20, 21]. As is shown in [21] assuming here and for the rest of this section that there is no felicitous 38 cancellation, the structure of B is given by the transitive closure of the graph of GB.Aswas mentioned before, when the forward transform M # is multiplied out (with no update steps) the ....

....nodes i 1 , i k such that i 1 i 2 , i 2 i 3 , and i k,1 i k , often written i 1 ### i k ,orsimplyi 1 ; i k . The transitive closure G of a graph G is one constructed on the same nodes but having i j whenever i in G. For a fuller treatment, see [20, 21] As is shown in [21], assuming here and for the rest of this section that there is no felicitous 38 cancellation, the structure of B is given by the transitive closure of the graph of GB.Aswas mentioned before, when the forward transform M # is multiplied out (with no update steps) the off diagonal ,P s are ....

J. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 62--79.

....the columns to reduce fill. They have shown that the fill of the LU factors of PA is essentially contained in the fill of the Cholesky factor of A T A for every row permutation P . P is a permutation matrix that permutes the rows of A and represents the actions of partial pivoting. Gilbert [10] later showed that this upper bound on the fill of the LU factors is not too loose, in the sense that for a large class of matrices, for every fill element in the Cholesky factor of A T A there is a pivoting sequence P that causes the element to fill in the LU factors of A. Thus, nonsymmetric ....

John R. Gilbert. Predicting structure in sparse matrix computations. SIAM Journal on Matrix Analysis and Applications, 15:62--79, 1994.

....used Gauss Seidel with damping for T 3, selecting values for that minimize the number of iterations. Next, we have the results for the Newton method where the linear system of size 45T is solved with the sparse LU factorization implemented in MATLAB (see Gilbert, Moler and Schreiber (1992) and Gilbert (1994)) The last two columns show the results where the linear system in the Newton step is solved with the multiple block diagonal LU factorization described in Figures 7 and 8. The linear systems of size 45 to be solved within this method are factorized with the sparse LU in MATLAB. Using the Mflops ....

Gilbert, J.R., 1994, Predicting Structure in Sparse Matrix Computations, SIAM Journal on Matrix Analysis and Applications, 15, 62--79.

....distinct nodes i 1 , i k such that i 1 # i 2 , i 2 # i 3 , and i k 1 # i k , often written i 1 # # i k , or simply i 1 ; i k . The transitive closure G # of a graph G is one constructed on the same nodes but having i # j whenever i ; j in G. For a fuller treatment, see [20, 21]. As is shown in [21] assuming here and for the rest of this section that there is no felicitous 38 cancellation, the structure of B 1 is given by the transitive closure of the graph of GB . As was mentioned before, when the forward transform M # is multiplied out (with no update steps) the ....

..... i k such that i 1 # i 2 , i 2 # i 3 , and i k 1 # i k , often written i 1 # # i k , or simply i 1 ; i k . The transitive closure G # of a graph G is one constructed on the same nodes but having i # j whenever i ; j in G. For a fuller treatment, see [20, 21] As is shown in [21], assuming here and for the rest of this section that there is no felicitous 38 cancellation, the structure of B 1 is given by the transitive closure of the graph of GB . As was mentioned before, when the forward transform M # is multiplied out (with no update steps) the off diagonal P s ....

J. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 62--79.

....of distinct nodes i 1 , i k such that i 1 # i 2 , i 2 # i 3 , and i k 1 # i k , often written i 1 # # i k , or simply i 1 i k . The transitive closure G # of a graph G is one constructed on the same nodes but having i # j whenever i j in G. For a fuller treatment, see [25, 26]. As is shown in [26] assuming here and for the rest of this section that there is no felicitous cancellation, the structure of B 1 is given by the transitive closure of the graph of GB . As shown before, the forward transform M# can be simply expressed as a triangular matrix (when there are no ....

...., i k such that i 1 # i 2 , i 2 # i 3 , and i k 1 # i k , often written i 1 # # i k , or simply i 1 i k . The transitive closure G # of a graph G is one constructed on the same nodes but having i # j whenever i j in G. For a fuller treatment, see [25, 26] As is shown in [26], assuming here and for the rest of this section that there is no felicitous cancellation, the structure of B 1 is given by the transitive closure of the graph of GB . As shown before, the forward transform M# can be simply expressed as a triangular matrix (when there are no update steps) Then ....

J. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix. Anal. Appl., 15 (1994), pp. 62--79.

....be applied to the symmetric part of any matrix. Definition 4.1. Let A be a square matrix with a triangular factorization A = LU . The I F fill of A is defined to be the total number of nonzeros in the inverses of L and U, assuming no cancellation in the forming of those inverses. From Gilbert[10] and Liu[13] we have the following graph theoretic characterization of the structure of the inverse Cholesky factor: Theorem 4.2. Let A be a SPD matrix with Cholesky factor L. Then assuming no cancellation the structure of Z = L T corresponds to the transitive closure of the graph of L T , ....

J. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix. Anal. Appl., 15 (1994), pp. 62--79.

....is the pair (V; E) where V = f1; ng and E = f(i; j)ji 6= j and R(i; j) 6= 0g. For (i; j) 2 E, i is called a predecessor of j and j a successor of i. The transitive closure of a directed graph G = V; E) is the graph G 0 = V; E 0 ) where E 0 = f(i; j)j9 path i j in Gg. Theorem 7 (Gilbert 1994) Let R be a nonsingular upper triangular matrix. Then G(R Gamma1 ) G(R) 0 : This theorem allows us to extend Lemma 6 by showing that the restriction to using elementary matrices as factors is not necessary. Instead, we can consider blocks of rows of R where the corresponding ....

Gilbert, J. R. (1994), `Predicting structure in sparse matrix computations', SIAM J. Matrix Analysis and Applications 15, 62--79.

....the columns to reduce fill. They have shown that the fill of the LU factors of PA is essentially contained in the fill of the Cholesky factor of A T A for every row permutation P . P is a permutation matrix that permutes the rows of A and represents the actions of partial pivoting. Gilbert [8] later showed that this upper bound on the fill of the LU factors is not too loose, in the sense that for a large class of matrices, for every fill element in the Cholesky factor of A T A there is a pivoting sequence P that causes the element to fill in the LU factors of A. Thus, nonsymmetric ....

John R. Gilbert. Predicting structure in sparse matrix computations. SIAM Journal on Matrix Analysis and Applications, 15:62--79, 1994.

....the columns to reduce fill. They have shown that the fill of the LU factors of PA is essentially contained in the fill of the Cholesky factor of A T A for every row permutation P . P is a permutation matrix that permutes the rows of A and represents the actions of partial pivoting. Gilbert [9] later showed that this upper bound # This research was supported by Israel Science Foundation founded by the Israel Academy of Sciences and Humanities (grant number 572 00 and grant number 9060 99) and by the University Research Fund of Tel Aviv University. School of Computer Science, Tel Aviv ....

John R. Gilbert. Predicting structure in sparse matrix computations. SIAM Journal on Matrix Analysis and Applications, 15:62--79, 1994.

....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, Predicting structure in sparse matrix computations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 62--79.

....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, Predicting Structure in Sparse Matrix Computations, Tech. report CS-86-750, Computer Science Dept., Cornell Univ., Ithaca, NY, 1986.

....the nonzero structure of an n Theta n matrix M corresponds to a graph GM with vertices labelled 1; n and directed edge i j if and only if M ij 6= 0. See [9] for example, for more discussion of graph theory and sparse matrix computations. REFINING AN APPROXIMATE INVERSE 6 As proven in [11], the inverse of a matrix M has the structure of the transitive closure G M of GM , that is a graph G M with a directed edge i j whenever there is a path from i to j in GM . The simplest characterization of the structure of the true inverse factors W T = L Gamma1 and Z = U Gamma1 is ....

J. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 62--79.

....arc i # j if and only if A ij #= 0. A directed path, or dipath, is an ordered set #u 1 , u 2 , u k # such that the arcs u 1 # u 2 , u k 1 # u k all exist often this is written as u 1 # # u k . See chapter 3 of [17] for example, for further explanation. From Gilbert [19] and Liu [23] we have the following graph theoretic characterization of the structure of the inverse Cholesky factor. ORDERING, ANISOTROPY, AND APPROXIMATE INVERSES 869 Theorem 2.2. Let A be an SPD matrix with Cholesky factor L. Then assuming no cancellation, the structure of Z = L T ....

J. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix. Anal. Appl., 15 (1994), pp. 62--79.

....increasing column number, the elementary matrices corresponding to columns in P . The condition that the nonzero structure of a factor P should be the same as the structure of its inverse corresponds in the graph model to the requirement that the column subgraph of P should be transitively closed [8]. A DAG G is transitively closed if and only if for every pair of vertices j and i such that there is a directed path in G from j to i, the edge (j; i) is present in G. Hence the graph model of (Pr1) is as follows: Pr1 0 ) Find an ordered partition P 1 OE P 2 OE 1 1 1 OE Pm of the ....

J. R. Gilbert, Predicting structure in sparse matrix computations, Tech. Report 86-750, Computer Science, Cornell University, 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. R. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 62--79.

....of a matrix shows how each unknown in a linear system depends on the other unknowns. The structure of the matrix A shows only the immediate dependencies. This suggests that in the structure of A 1 there is an edge (i j) whenever there is a directed path from vertex i to vertex j in G(A) [21] (if A is nonsingular, and ignoring coincidental cancellation) This structure is called the transitive closure of G(A) and is denoted G (A) For an irreducible matrix, this result says that the inverse is a full matrix, but it does suggest the possibility of truncating the transitive ....

J. R. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 62-79.

....3. Structural considerations. In this section we review the effect of orderings on the amount of fill in occurring in the inverse triangular factors of a sparse matrix A. See also [6] for a treatment along similar lines. The inverse of a sparse irreducible matrix is structurally full [11] [17], and this property is obviously invariant under permutations. However, A Gamma1 may be representable as the product of two sparse triangular matrices. We are interested in finding permutations of A so that the inverse triangular factors Z; W of A preserve a good deal of sparsity. For example, ....

....is the set of vertices of G(A) from which there are paths to x. The structure of a vector v is defined as Struct(v) fijv i 6= 0g. In the following we will state results for the factor L Gamma1 . Similar results hold, of course, also for the factor U Gamma1 . These results were given in [17], 18] The usual no cancellation assumption is made throughout. We denote by L Gamma1 ( i) the ith column of L Gamma1 . Proposition 3.1. Struct(L Gamma1 ( i) cl G(L) i) Let G ffi (L) denote the transitive reduction of the directed acyclic graph (dag) G(L) This is a graph with a ....

J. R. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 62--79.

....edges in the graph. This ability to identify independent rows will be needed in x 4. Also, the rigidity matrix is quite sparse, having only 2d nonzeros in each row. To save time and space, sparse techniques could be used for large problems. There are sparse QR algorithms, but none for SVD [8, 16, 17]. There are also efficient parallel algorithms for finding the rank of a matrix. Ibarra, Moran and Rosier [23] discovered an algorithm that runs in O(log 2 m) time on O(m 4 ) processors. This means that rigidity testing is in random NC for any dimension. The class NC is the set of problems ....

J. Gilbert, Predicting structure in sparse matrix computations, Tech. Rep. TR 86-750, Dept. of Computer Science, Cornell University, Ithaca, NY, 1986.

....complete. Theorem 5.2 During symbolic downdate AA T = AA T Gamma ww T (where w is a column of A) the nonzero pattern of v = L Gamma1 w is equal to the path P(k) in the (old) elimination tree of L where k = min fi : w i 6= 0g: 6) Proof. Let W = fi : w i 6= 0g. Theorem 5. 1 of Gilbert [10, 11, 14] states that the nonzero pattern of v is the set of nodes reachable from the nodes in W by paths 16 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 reachable ....

J. R. Gilbert, Predicting structure in sparse matrix computations, Tech. Report CS-86-750, Computer Science Dept., Cornell Univ., 1986. 29

....as a guideline to predict the structure of the factorized approximate inverse, and have an impact on certain aspects of the implementation. It is well known that structural nonzeros in the inverse matrix A Gamma1 can be characterized by the paths in the graph of the original matrix A (see [24] [29]) The following lemma states necessary and sufficient conditions for a new entry (fill in) to be added in one of the z vectors at the ith step of the biconjugation algorithm. A similar result holds for the w vectors. We make use of the standard no cancellation assumption. Lemma 5.1. Let 1 i ....

J. R. Gilbert. Predicting structure in sparse matrix computations. SIAM J. Matrix Anal. Appl., 15:62--79, 1994.

....3. Structural considerations. In this section we review the effect of orderings on the amount of fill occurring in the inverse triangular factors of a sparse matrix A. See also [7] for a treatment along similar lines. The inverse of a sparse irreducible matrix is structurally full [12] [19], and this property is obviously invariant under permutations. However, A Gamma1 may be representable as the product of two sparse triangular matrices. We are interested in finding permutations of A such that the inverse triangular factors Z; W of A preserve a good deal of sparsity. For ....

....is the set of vertices of G(A) from which there are paths to x. The structure of a vector v is defined as Struct(v) fijv i 6= 0g. In the following we will state results for the factor L Gamma1 . Similar results hold, of course, also for the factor U Gamma1 . These results were given in [19], 20] The usual no cancellation assumption is made throughout. We denote by L Gamma1 ( i) the ith column of L Gamma1 . Proposition 3.1. Struct(L Gamma1 ( i) cl G(L) i) Let G ffi (L) denote the transitive reduction of the directed acyclic graph (dag) G(L) This is a graph with ....

J. R. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 62--79.

....of a graph G = V; E) is a graph G = V; E) with E = f(i; j)jthere is a path in G from i to jg. If a graph G is equal to its own transitive closure, G is called transitively closed. The graph associated with L Gamma1 is equal to the transitive closure of the graph associated with L [8]. The induced subgraph of a subset of vertices V is the graph G = V; E) with E = f(i; j)j(i; j) 2 E i 2 V g. Others [1, 2, 3, 5, 11] have considered the following node partitioning problem, which is closely related to problem 1: Problem 2 Given a DAG G, find an ordered ....

....edge e ij , ergo it is transitively closed. Because of condition 2 W k 1 remains transitively closed. 1 the element (4; 1) of the inverse is not a structural nonzero but a numerical nonzero, only structural nonzeros are considered in the relation between the inverse and the transitive closure in [8]. 4 Optimal Edge Partitioning Algorithm RPOPT forall v 2 V do count(v) indegree(v) enddo F fe vw 2 Ejv; w 2 V , count(v) 0g k 1 while F 6= do W k ; H ; forall e vw 2 F do W k W k [ fe vw g count(w) count(w) Gamma 1 if count(w) 0 then H H [ fwg; endif enddo F ....

J.R. Gilbert. Predicting structure in sparse matrix computations. SIAM J. Matrix Anal. Appl., 15(1):62--79, Jan. 1994.

....The orderings for which the generator matrix has Property R [5] are semiconvergent, and we use this as our starting point. The task is di#cult because one needs to know the smallest # to say something about the worth of an ordering at hand for a given problem. The results that appear in [3] related to forecasting the nonzero structure of the inverse of an unsymmetric matrix have helped us considerably. 2. Background material. In this section, an overview of some concepts discussed in [5] and other remarks are given. Wherever something has been taken from [5] the appropriate ....

J. R. GILBERT, Predicting structure in sparse matrix computations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 62--79.

....pivoting is tremendously difficulty to implement mainly due to the following reason. The nonzero patterns in L and U depend on the row interchanges and cannot be predetermined precisely based solely on the structure of A. This can be best illustrated by the following example given by Gilbert [66]. Let the structure of A be 0 B 1 ffl 2 ffl 3 1 C A : Depending on the relative magnitudes of the nonzero entries, pivoting could cause the structure of U to be any of the four outcomes: 0 B 1 2 3 1 C A ; 0 B 1 ffl 2 ffl 3 1 C A ; 0 B 1 ffl 2 3 1 C A ; 0 B 1 ffl 2 ....

John R. Gilbert. Predicting structures in sparse matrix computations. SIAM J. Matrix Analysis and Applications, 15(1):62--79, January 1994.

....directed graph of M , that is, m ij 6= 0. Edges in the directed graph of M are directed from rows to columns. The notation i M = j means that there is a directed path from i to j in the directed graph of M . Such a path may have length zero; that is, i M = i holds if m ii 6= 0. Theorem 3 [60] f ij is nonzero (equivalently, i F j) if and only if i L( J) k A j for some k i. This result implies that the symbolic factorization of column j can be obtained as follows. Consider the nonzeros in A( j) as a subset of the vertices of the directed graph G = G(L( J) T ) ....

J. R. Gilbert. Predicting structure in sparse matrix computations. SIAM J. Matrix Analysis and Applications, 15:62--79, 1994.

....be applied to the symmetric part of any matrix. Definition 4.1. Let A be a square matrix with a triangular factorization A = LU . The I F fill of A is defined to be the total number of nonzeros in the inverses of L and U, assuming no cancellation in the forming of those inverses. From Gilbert[10] and Liu[13] we have the following graph theoretic characterization of the structure of the inverse Cholesky factor: Theorem 4.2. Let A be a SPD matrix with Cholesky factor L. Then assuming no cancellation the structure of Z = L GammaT corresponds to the transitive closure of the graph of L ....

J. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix. Anal. Appl., 15 (1994), pp. 62--79.

....for short [2] Figure 1 is an example of a lower triangular matrix and its corresponding directed acyclic graph. 2.3. Structural characterization of triangular solution. Consider the solution of the lower triangular system Lx = b, where both L and the right hand side vector b are sparse. Gilbert [7] provides a simple characterization of the sparse structure of the solution vector x in terms of that of L and b. We introduce the following notation: For any row or column vector w = w 1 ; wn ) or w = w 1 ; wn ) T , define its vector structure as the vertex subset Struct(w) ....

....notation: For any row or column vector w = w 1 ; wn ) or w = w 1 ; wn ) T , define its vector structure as the vertex subset Struct(w) fi 2 V j w i 6= 0g: Note that in this definition, i is a vertex in the graph G(L) and w i is an entry in the vector w. Theorem 2.1. [7] The structure Struct(x) of the solution vector x to Lx = b is given by the set of vertices reachable from vertices of the right hand side structure Struct(b) by paths in the directed graph G(L T ) Strictly speaking, this result gives only an upper bound on the structure of x, because ....

[Article contains additional citation context not shown here]

John R. Gilbert. Predicting structure in sparse matrix computations. Technical Report 86--750, Cornell University, 1986. To appear in SIAM Journal on Matrix Analysis and Applications.

....elements. Sections 3 through 6 contain the results of the paper: Roughly speaking, the results in Section 3 are immediate; those in Section 4 are known; most of those in Section 5 are consequences of known results; and those in Section 6 are new. This paper is based on an earlier technical report [19]. 2. Definitions. We assume that the reader is familiar with such basic graph theoretic terms as directed graph, undirected graph, and path. Harary [26] is a good general reference. 2.1. Directed graphs and matrix structures. Let A be an n by n matrix. The structure of A is its directed graph ....

John R. Gilbert. Predicting structure in sparse matrix computations. Technical Report 86--750, Cornell University, 1986.

....data structure for A. Each nonzero index of b corresponds to a vertex of the graph. The set of nonzero indices of x corresponds to the set of all vertices of b, plus all vertices that can be reached from vertices of b via directed paths in the graph of A. This is true even if A is not triangular [15]. Any graph searching algorithm could be used to identify those vertices and find the nonzero indices of x. A depth first search has the advantage that a topological order for the list can be generated during the search. We add each vertex to the list at the time the depth first search backtracks ....

John R. Gilbert. Predicting structure in sparse matrix computations. Technical Report 86--750, Cornell University, 1986. To appear in SIAM Journal on Matrix Analysis and Applications.

....model of LU factorization without row or column interchanges. The undirected version of this model is due to Parter [27] and was developed extensively by Rose [30] the directed version was developed by Rose and Tarjan [31] George and Liu [11] is a good source for the undirected model. Gilbert [14] surveys these and related results. If G = G(A) is a directed or undirected graph, we define the deficiency of a vertex v of G as the set of edges fhr; ci : v 2 AdjG (r) c 2 AdjG (v) and c = 2 AdjG (r)g : The deficiency of v corresponds to the fill that occurs in A when the (v; v) element is ....

John R. Gilbert. Predicting structure in sparse matrix computations. Technical Report 86--750, Cornell University, 1986. To appear in SIAM Journal on Matrix Analysis and Applications.

....memory. The symbolic analysis phase computes the column elimination tree (c etree) of the matrix, a postordering of this tree, and upper bounds on the number of nonzeros in each column of the factors L and U . For more information on the algorithms that are used in the symbolic analysis phase, see [4, 5]. The scheduling phase uses this information to compute an e#cient out of core schedule for the factorization. The numerical factorization phase determines which column of L updates which column of A and computes the columns of U and L. As in other pivoting LU factorization algorithms, the ....

John R. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix Analysis and Applications, 15 (1994), pp. 62--79.

....data structure for A. Each nonzero index of b corresponds to a vertex of the graph. The set of nonzero indices of x corresponds to the set of all vertices of b, plus all vertices that can be reached from vertices of b via directed paths in the graph of A. This is true even if A is not triangular [12]. Any graph searching algorithm could be used to identify those vertices and find the nonzero indices of x. A depth first search has the advantage that a topological order for the list can be generated during the search. We add each vertex to the list at the time the depth first search backtracks ....

J. R. Gilbert, Predicting structure in sparse matrix computations, Tech. Report 86--750, Cornell University, 1986. To appear in SIAM Journal on Matrix Analysis and Applications.

....the directed graph of M , that is, m ij 6= 0. Edges in the directed graph of M are directed from rows to columns. The notation i M = j means that there is a directed path from i to j in the directed graph of M . Such a path may have length zero; that is, i M = i always holds. Theorem 4.1. [24] f ij is nonzero (equivalently, i F j) if and only if i L( J) k A j for some k i. This result implies that the symbolic factorization of column j can be obtained as follows. Consider the nonzeros in A( j) as a subset of the vertices of the directed graph G = G(L( J) T ) ....

J. R. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix Analysis and Applications, 15 (1994), pp. 62--79.

....M , the notation i M j means that there is an edge from i to j in the directed graph of M , that is, m ij 6= 0. The notation i M = j means that there is a directed path from i to j in the directed graph of M . Such a path may have length zero; that is, i M = i always holds. Theorem 3 [16] f ij is nonzero (equivalently, i F j) if and only if i L( J) k A j for some k i. This result implies that the symbolic factorization of column j can be obtained as follows. Consider the nonzeros in A( j) as a subset of the vertices of the directed graph G = G(L( J) T ) ....

J. R. Gilbert. Predicting structure in sparse matrix computations. SIAM J. Matrix Analysis and Applications, 15:62--79, 1994.

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J. R. Gilbert. Predicting structure in sparse matrix computations. SIAM Journal of Matrix Analysis and Applications, 15(1):62--79, 1994.

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John Gilbert, "Predicting structure in sparse matrix computations," SIAM Journal of Matrix Analysis and Applications, vol. 15, no. 1, pp. 62--79, 1994.

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J. R. Gilbert, Predicting structure in sparse matrix computations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 62--79.

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J. R. Gilbert, Predicting structure in sparse matrix computations, Tech. Report CS-86-750, Computer Science Dept., Cornell Univ., 1986.

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