REP-1994-160 - AN APPROXIMATE MINIMUM DEGREE ORDERING...

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Unformatted text preview: AN APPROXIMATE MINIMUM DEGREE ORDERING ALGORITHM TIMOTHY A. DAVIS , PATRICK AMESTOYy , AND IAIN S. DUFFz Computer and Information Sciences Dept., University of Florida, Technical Report TR-94-039, December, 1994 (revised July 1995). Abstract. An Approximate Minimum Degree ordering algorithm (AMD) for preordering a symmetric sparse matrix prior to numerical factorization is presented. We use techniques based on the quotient graph for matrix factorization that allow us to obtain computationally cheap bounds for the minimum degree. We show that these bounds are often equal to the actual degree. The resulting algorithm is typically much faster than previous minimum degree ordering algorithms, and produces results that are comparable in quality with the best orderings from other minimum degree algorithms. Keywords: approximate minimum degree ordering algorithm, quotient graph, sparse matrices, graph algorithms, ordering algorithms AMS classi cations: 65F50, 65F05. 1. Introduction. When solving large sparse symmetric linear systems of the form Ax = b, it is common to precede the numerical factorization by a symmetric reordering. This reordering is chosen so that pivoting down the diagonal in order on the resulting permuted matrix PAPT = LLT produces much less ll-in and work than computing the factors of A by pivoting down the diagonal in the original order. This reordering is computed using only information on the matrix structure without taking account of numerical values and so may not be stable for general matrices. However, if the matrix A is positive-de nite 21], a Cholesky factorization can safely be used. This technique of preceding the numerical factorization with a symbolic analysis can also be extended to unsymmetric systems although the numerical factorization phase must allow for subsequent numerical pivoting 1, 2, 16]. The goal of the preordering is to nd a permutation matrix P so that the subsequent factorization has the least ll-in. Unfortunately, this problem is NP-complete 31], so heuristics are used. The minimum degree ordering algorithm is one of the most widely used heuristics, since it produces factors with relatively low ll-in on a wide range of matrices. Because of this, the algorithm has received much attention over the past three decades. The algorithm is a symmetric analogue of Markowitz' method 26] and was rst proposed by Tinney and Walker 30] as algorithm S2. Rose 27, 28] developed a graph theoretical model of Tinney and Walker's algorithm and renamed it the minimum degree algorithm, since it performs its pivot selection by selecting from a graph a node of minimum degree. Later implementations have dramatically improved the time and memory requirements of Tinney and Walker's method, while maintaining the basic idea of selecting a node or set of nodes of minimum degree. These improvements have reduced the memory complexity so that the algorithm can operate within the storage of the original matrix, and have reduced the amount of work needed to keep track of the degrees of nodes in the graph (which is the most computationally intensive part Computer and Information Sciences Department University of Florida, Gainesville, Florida, USA. phone: (904) 392-1481, email: davis@cis.u .edu. Support for this project was provided by the National Science Foundation (ASC-9111263 and DMS-9223088). Portions of this work were supported by a post-doctoral grant from CERFACS. y ENSEEIHT-IRIT, Toulouse, France. email: amestoy@enseeiht.fr. z Rutherford Appleton Laboratory, Chilton, Didcot, Oxon. 0X11 0QX England, and European Center for Research and Advanced Training in Scienti c Computation (CERFACS), Toulouse, France. email: isd@letterbox.rl.ac.uk. 1 2 P. AMESTOY, T. A. DAVIS, , AND I. S. DUFF of the algorithm). This work includes that of Du and Reid 10, 13, 14, 15]; George and McIntyre 23]; Eisenstat, Gursky, Schultz, and Sherman 17, 18]; George and Liu 19, 20, 21, 22]; and Liu 25]. More recently, several researchers have relaxed this heuristic by computing upper bounds on the degrees, rather than the exact degrees, and selecting a node of minimum upper bound on the degree. This work includes that of Gilbert, Moler, and Schreiber 24], and Davis and Du 7, 8]. Davis and Du use degree bounds in the unsymmetric-pattern multifrontal method (UMFPACK), an unsymmetric Markowitz-style algorithm. In this paper, we describe an approximate minimum degree ordering algorithm based on the symmetric analogue of the degree bounds used in UMFPACK. Section 2 presents the original minimum degree algorithm of Tinney and Walker in the context of the graph model of Rose. Section 3 discusses the quotient graph (or element graph) model and the use of that model to reduce the time taken by the algorithm. In this context, we present our notation for the quotient graph, and present a small example matrix and its graphs. We then use the notation to describe our approximate degree bounds in Section 4. The Approximate Minimum Degree (AMD) algorithm and its time complexity is presented in Section 5. In Section 6, we rst analyse the performance and accuracy of our approximate degree bounds on a set of test matrices from a wide range of disciplines. The AMD algorithm is then compared with other established codes that compute minimum degree orderings. 2. Elimination graphs. The nonzero pattern of a symmetric n-by-n matrix, A, can be represented by a graph G0 = (V 0; E0), with nodes V 0 = f1; :::; ng and edges E 0. An edge (i; j ) is in E 0 if and only if aij 6= 0. Since A is symmetric, G0 is undirected. The elimination graph, Gk = (V k ; E k ), describes the nonzero pattern of the submatrix still to be factorized after the rst k pivots have been chosen. It is undirected, since the matrix remains symmetric as it is factorized. At step k, the graph Gk depends on Gk 1 and the selection of the kth pivot. To nd Gk , the kth pivot node p is selected from V k 1. Edges are added to E k 1 to make the nodes adjacent to p in Gk 1 a clique (a fully connected subgraph). This addition of edges ( ll-in) means that we cannot know the storage requirements in advance. The edges added correspond to ll-in caused by the kth step of factorization. A ll-in is a nonzero entry Lij , where (PAPT )ij is zero. The pivot node p and its incident edges are then removed from the graph Gk 1 to yield the graph Gk . Let AdjG (i) denote the set of nodes adjacent to i in the graph Gk . Throughout this paper, we will use the superscript k to denote a graph, set, or other structure obtained after the rst k pivots have been chosen. For simplicity, we will drop the superscript when the context is clear. The minimum degree algorithm selects node p as the kth pivot such that the degree of p, tp jAdjG 1 (p)j, is minimized (where j:::j denotes the size of a set or the number of nonzeros in a matrix). The minimum degree algorithm is a non-optimal greedy heuristic for reducing the number of new edges ( ll-ins) introduced during the factorization. We have already noted that the optimal solution is NP-complete 31]. By minimizing the degree, the algorithm minimizes the upper bound on the ll-in caused by the kth pivot. Selecting p as pivot creates at most (t2 tp )=2 new edges in p G. 3. Quotient graphs. In contrast to the elimination graph, the quotient graph models the factorization of A using an amount of storage that never exceeds the storage for the original graph, G0 21]. The quotient graph is also referred to as the k k APPROXIMATE MINIMUM DEGREE 3 generalized element model 13, 14, 15, 29]. An important component of a quotient graph is a clique. It is a particularly economic structure since a clique is represented by a list of its members rather than by a list of all the edges in the clique. Following the generalized element model, we refer to nodes removed from the elimination graph as elements (George and Liu refer to them as eliminated nodes). We use the term variable to refer to uneliminated nodes. The quotient graph, G k = (V k ; V k ; E k ; Ek ), implicitly represents the elimination graph Gk , where G 0 = G0. For clarity, we drop the superscript k in the following. The nodes in G consist of variables (the set V ), and elements (the set V ). The edges are divided into two sets: edges between variables E V V , and between variables and elements E V V . There are no edges between elements since they are removed by element absorption. The sets V 0 and E 0 are empty. We use the following set notation (A, E , and L) to describe the quotient graph model and our approximate degree bounds. Let Ai be the set of variables adjacent to variable i in G , and let Ei be the set of elements adjacent to variable i in G (we refer to Ei as element list i). That is, if i is a variable in V , then Ai fj : (i; j ) 2 E g V ; Ei fe : (i; e) 2 E g V ; and AdjG (i) Ai Ei V V : The set Ai refers to a subset of the nonzero entries in row i of the original matrix A (thus the notation A). That is, A0 fj : aij 6= 0g, and Ak Ak 1, for 1 k n. i i i Let Le denote the set of variables adjacent to element e in G . That is, if e is an element in V , then we de ne Le AdjG (e) = fi : (i; e) 2 E g V : The edges E and E in the quotient graph are represented explicitly as the sets Ai and Ei for each variable in G , and the sets Le for each element in G . We will use A, E , and L to denote three sets containing all Ai , Ei , and Le, respectively, for all variables i and all elements e. George and Liu 21] show that the quotient graph takes no more storage than the original graph (jAk j + jE k j + jLkj jA0 j for all k). The quotient graph G and the elimination graph G are closely related. If i is a variable in G, it is also a variable in G , and (3.1) AdjG (i) = Ai ! e2Ei Le n fig; where the \n" is the standard set subtraction operator. When variable p is selected as the kth pivot, element p is formed (variable p is removed from V and added to V ). The set Lp = AdjG (p) is found using Equation (3.1). The set Lp represents a permuted nonzero pattern of the kth column of L (thus the notation L). If i 2 Lp, where p is the kth pivot, and variable i will become the mth pivot (for some m > k), then the entry Lmk will be nonzero. Equation (3.1) implies that Le n fpg Lp for all elements e adjacent to variable p. This means that all variables adjacent to an element e 2 Ep are adjacent to the element p and these elements 4 P. AMESTOY, T. A. DAVIS, , AND I. S. DUFF Fig. 3.1. Elimination graph, quotient graph, and matrix for rst three steps. G 0 G 2 1 G 2 5 2 G 5 5 9 7 3 6 9 7 10 8 4 1 10 8 3 9 7 3 10 3 8 4 6 4 5 9 7 10 6 8 6 4 (a) Elimination graph 0 1 2 2 2 5 3 2 5 2 5 5 9 7 3 6 9 7 3 6 9 7 3 6 9 7 3 6 10 8 4 1 10 8 4 1 10 8 4 1 10 8 4 1 (b) Quotient graph 1 1 2 1 2 3 3 4 2 3 4 5 1 2 3 4 5 6 6 7 7 8 9 10 6 7 8 9 5 6 7 8 4 5 8 9 10 9 10 10 (c) Factors and active submatrix e 2 Ep are no longer needed. They are absorbed into the new element p and deleted 15], and reference to them is replaced by reference to the new element p. The new element p is added to the element lists, Ei, for all variables i adjacent to element p. Absorbed elements, e 2 Ep , are removed from all element lists. The sets Ap and Ep , and Le for all e in Ep , are deleted. Finally, any entry j in Ai , where both i and j are in Lp , is redundant and is deleted. The set Ai is thus disjoint with any set Le for e 2 Ei . In other words, Ak is the pattern of those entries in row i of A that are not i modi ed by steps 1 through k of the Cholesky factorization of PAPT . The net result is that the new graph G takes the same, or less, storage than before the kth pivot was selected. 3.1. Quotient graph example. We illustrate the sequence of elimination graphs and quotient graphs of a 10-by-10 sparse matrix in Figures 3.1 and 3.2. The example is ordered so that a minimum degree algorithm recommends pivoting down the diagonal in the natural order (that is, the permutation matrix is the identity). In Figures 3.1 and 3.2, variables and elements are shown as thin-lined and heavy-lined circles, respectively. In the matrices in these gures, diagonal entries are numbered, unmodi ed original nonzero entries (entries in A) are shown as a solid squares. The solid squares in row i form the set Ai . The variables in current unabsorbed elements (sets Le) are indicated by solid circles in the columns of L corresponding to the unabsorbed 5 APPROXIMATE MINIMUM DEGREE Fig. 3.2. Elimination graph, quotient graph, and matrix for steps 4 to 7. G 4 G 5 G 6 G7 5 9 7 10 8 9 7 10 6 8 9 7 9 10 6 8 10 8 (a) Elimination graph 2 4 5 9 7 8 5 3 10 9 6 5 9 4 8 9 6 4 7 10 7 10 6 8 9 7 10 6 8 (b) Quotient graph 1 1 2 1 2 3 3 4 2 3 4 5 1 2 3 4 5 6 6 7 7 8 9 10 6 7 8 9 5 6 7 8 4 5 8 9 10 9 10 10 (c) Factors and active submatrix elements. The solid circles in row i form the set Ei . Entries that do not correspond to edges in the quotient graph are shown as an . Figure 3.1 shows the elimination graph, quotient graph, and the matrix prior to elimination (in the left column) and after the rst three steps (from left to right). Figure 3.2 continues the example for the next four steps. Consider the transformation of the graph G 2 to the graph G 3 . Variable 3 is selected as pivot. We have L3 = A3 = f5; 6; 7g (a simple case of Equation (3.1)). The new element 3 represents the pairwise adjacency of variables 5, 6, and 7. The explicit edge (5,7) is now redundant, and is deleted from A5 and A7 . Also consider the transformation of the graph G 4 to the graph G 5 . Variable 5 is selected as pivot. The set A5 is empty and E5 = f2; 3g. Following Equation (3.1), L5 = (A5 L2 L3 ) n f5g = (; f5; 6; 9g f5; 6; 7g) n f5g = f6; 7; 9g; which is the pattern of column 5 of L (excluding the diagonal). Since the new element 5 implies that variables 6, 7, and, 9 are pairwise adjacent, elements 2 and 3 do not add any information to the graph. They are removed, having been \absorbed" into element 5. Additionally, the edge (7, 9) is redundant, and is removed from A7 and 6 P. AMESTOY, T. A. DAVIS, , AND I. S. DUFF A9 . In G 4, we have A6 = ; E6 = f2; 3; 4g A7 = f9; 10g E7 = f3; 4g : A9 = f7; 8; 10g E9 = f2g After these transformations, we have in G 5 , A6 = ; E6 = f4; 5g A7 = f10g E7 = f4; 5g ; A9 = f8; 10g E9 = f5g and the new element in G 5 , L5 = f6; 7; 9g: 3.2. Indistinguishable variables and external degree. Two variables i and j are indistinguishable in G if AdjG (i) fig = AdjG (j ) fj g. They will have the same degree until one is selected as pivot. If i is selected, then j can be selected next without causing any additional ll-in. Selecting i and j together is called mass elimination 23]. Variables i and j are replaced in G by a supervariable containing both i and j , labeled by its principal variable (i, say) 13, 14, 15]. Variables that are not supervariables are called simple variables. In practice, new supervariables are constructed at step k only if both i and j are in Lp (where p is the pivot selected at step k). In addition, rather than checking the graph G for indistinguishability, we use the quotient graph G so that two variables i and j are found to be indistinguishable if AdjG (i) fig = AdjG (j ) fj g. This comparison is faster than determining if two variables are indistinguishable in G, but may miss some identi cations because, although indistinguishability in G implies indistinguishability in G, the reverse is not true. We denote the set of simple variables in the supervariable with principal variable i as i, and de ne i = fig if i is a simple variable. When p is selected as pivot at the kth step, all variables in p are eliminated. The use of supervariables greatly reduces the number of degree computations performed, which is the most costly part of the algorithm. Non-principal variables and their incident edges are removed from the quotient graph data structure when they are detected. The set notation A and L refers either to a set of supervariables or to the variables represented by the supervariables, depending on the context. In degree computations and when used in representing elimination graphs, the sets refer to variables; otherwise they refer to supervariables. In Figure 3.2, detected supervariables are circled by dashed lines. Non-principal variables are left inside the dashed supervariables. These are, however, removed from the quotient graph. The last quotient graph in Figure 3.2 represents the selection of pivots 7, 8, and 9, and thus the right column of the gure depicts G7, G 9 , and the matrix after the ninth pivot step. The external degree di ti jij + 1 of a principal variable i is (3.2) di = jAdjG (i) n ij = jAi n ij + ! e2Ei Le n i ; since the set Ai is disjoint from any set Le for e 2 Ei . At most (d2 di)=2 ll-ins occur i if all variables in i are selected as pivots. We refer to ti as the true degree of variable i. Selecting the pivot with minimum external degree tends to produce a better ordering than selecting the pivot with minimum true degree 25] (also see Section 6.2). APPROXIMATE MINIMUM DEGREE 7 Algorithm 1 (Minimum degree algorithm, based on quotient graph) V = f1:::ng V =; for i = 1 to n do Ai = fj : aij = 0 and i = j g 6 6 Ei = ; di = jAi j i = fig end for k=1 while k n do mass elimination: select variable p 2 V that minimizes dp Lp = Ap Se2E Le n p for each i 2 Lp do remove redundant entries: Ai = (Ai n Lp ) n p element absorption: Ei = (Ei n Ep ) fpg compute externalSdegree: di = jAi n ij + e2E Le n i end for supervariable detection, pairs found via hash function: for each pair i and j 2 Lp do if i and j are indistinguishable then remove the supervariable j: i=i j di = di jjj V = V n fj g Aj = ; Ej = ; end if end for p i convert variable p to element p: V = (V fpg) n Ep V = V n fpg Ap = ; Ep = ; k = k + jpj end while 3.3. Quotient-graph-based minimum degree algorithm. A minimum degree algorithm based on the quotient graph is shown in Algorithm 1. It includes element absorption, mass elimination, supervariables, and external degrees. Supervariable detection is simpli ed by computing a hash function on each variable, so that not all pairs of variables need be compared 3]. Algorithm 1 does not include two 8 P. AMESTOY, T. A. DAVIS, , AND I. S. DUFF important features of Liu's Multiple Minimum Degree algorithm (MMD): incomplete update 17, 18] and multiple elimination 25]. With multiple elimination, an independent set of pivots with minimum degree is selected before any degrees are updated. If a variable is adjacent to two or more pivot elements, its degree is computed only once. A variable j is outmatched if AdjG (i) AdjG (j ). With incomplete degree update, the degree update of the outmatched variable j is avoided until variable i is selected as pivot. These two features further reduce the amount of work needed for the degree computation in MMD. We will discuss their relationship to the AMD algorithm in the next section. The time taken to compute di using Equation (3.2) by a quotient-graph-based minimum degree algorithm is (jAi j + (3.3) X e2Ei jLej); which is (jAdjG (i)j) if all variables are simple.1 This degree computation is the most costly part of the minimum degree algorithm. When supervariables are present, the time taken is in the best case proportional to the degree of the variable in the \compressed" elimination graph, where all non-principal variables and their incident edges are removed. 4. Approximate degree. Having now discussed the data structures and the standard minimum degree implementations, we now consider our approximation for the minimum degree and indicate its lower complexity. We assume that p is the kth pivot, and that we compute the bounds only for supervariables i 2 Lp. Rather than computing the exact external degree, di , our Approximate Minimum Degree algorithm (AMD) computes an upper bound 7, 8], k (4.1) 8 n k; 9 > k1 > < di + jLp n ij; = k X : di = min > jAi n ij + jLp n ij + jLe n Lp j > : ; e2E nfpg i The rst two terms (n k, the size of the active submatrix, and dk 1 + jLp n ij, i the worst case ll-in) are usually not as tight as the third term in Equation (4.1). Algorithm 2 computes jLe n Lp j for all elements e in the entire quotient graph. The set Le splits into two disjoint subsets: the external subset Le n Lp and the internal subset Le \ Lp . If Algorithm 2 scans element e, the term w(e) is initialized to jLej and then decremented once for each variable i in the internal subset Le \ Lp , and, at the end of Algorithm 2, we have w(e) = jLej jLe \ Lp j = jLe n Lpj. If Algorithm 2 does not scan element e, the term w(e) is less than zero. Combining these two cases, we obtain w if w(e) (4.2) jLe n Lp j = jL(eej) otherwise 0 ; for all e 2 V : 1 Asymptotic complexity notation is de ned in 6]. We write f (n) = (g(n)) if there exist positive constants c1 , c2 , and n0 such that 0 c1 g(n) f (n) c2 g(n) for all n > n0 . Similarly, f (n) = (g(n)) if there exist positive constants c and n0 such that 0 cg(n) f (n) for all n > n0 ; and f (n) = O(g(n)) if there exist positive constants c and n0 such that 0 f (n) cg(n) for all n > n0 . APPROXIMATE MINIMUM DEGREE 9 Algorithm 2 (Computation of jLe n Lp j for all e 2 V ) assume w(1 : : :n) < 0 for each supervariable i 2 Lp do for each element e 2 Ei do if (w(e) < 0) then w(e) = jLej w(e) = w(e) jij end for end for Algorithm 2 is followed by a second loop to compute our upper bound degree, di for each supervariable i 2 Lp , using Equations (4.1) and (4.2). The total time for Algorithm 2 is ( X i2L jEij): p The second loop to compute the upper bound degrees takes time (4.3) ( X i2L (jAij + jEij)); p which is thus equal to the total asymptotic time. Multiple elimination 25] improves the minimumdegree algorithm by updating the degree of a variable only once for each set of independent pivots. Incomplete degree update 17, 18] skips the degree update of outmatched variables. We cannot take full advantage of the incomplete degree update since it avoids the degree update for some supervariables adjacent to the pivot element. With our technique (Algorithm 2), we must scan the element lists for all supervariables i in Lp . If the degree update of one of the supervariables is to be skipped, its element list must still be scanned so that the external subset terms can be computed for the degree update of other supervariables in Lp . The only advantage of multiple elimination or incomplete degree update would be to skip the second loop that computes the upper bound degrees for outmatched variables or supervariables for which the degree has already been computed. If the total time in Equation (4.3) is amortized across the computation of all supervariables i 2 Lp, then the time taken to compute di is (jAi j + jEi j) = O(jA0j); i which is (jAdjG (i)j) if all variables are simple. Computing our bound takes time proportional to the degree of the variable in the quotient graph, G . This is much faster than the time taken to compute the exact external degree (see Equation (3.3)). 4.1. Accuracy of our approximate degrees. Gilbert, Moler, and Schreiber 24] also use approximate external degrees that they can compute in the same time b as our degree bound d. In our notation, their bound di is k b di = jAi n ij + X e2Ei jLe n ij: 10 P. AMESTOY, T. A. DAVIS, , AND I. S. DUFF Since many pivotal variables are adjacent to two or fewer elements when selected, b Ashcraft and Eisenstat 4] have suggested a combination of d and d, e d= d if jEij = 2 : b d otherwise e b Computing d takes the same time as d or d, except when jEij = 2. In this case, it e, whereas computing d or d takes (jAij) time. b takes O(jAij + jLej) time to compute d In the Yale Sparse Matrix Package 17] the jLe n Lp j term for the Ei = fe; pg case is computed by scanning Le once. It is then used to compute di for all i 2 Lp for e which Ei = fe; pg. This technique can also be used to compute d, and thus the time e to compute d is O(jAi j + jLej) and not (jAi j + jLej). Theorem 1: Relationship between external degree and the three apeb proximate degree bounds. The equality di = di = di = di holds when jEi j 1. ei di holds when jEij = 2. Finally, the inequality b The inequality di = di = d eb di di di = di holds when jEij > 2. Proof: b The bound di is equal to the exact degree when variable i is adjacent to at most one element (jEij 1). The accuracy of their bound is una ected by the size of Ai , since entries are removed from A that fall within the pattern L of an element. Thus, b if there is just one element (the current element p, say), the bound di is tight. If jEij is two (the current element, p, and a prior element e, say), we have b di = jAi n ij + jLp n ij + jLe n ij = di + j(Le \ Lp ) n ij: b b The bound di counts entries in the set (Le \ Lp ) n i twice, and so di will be an overestimate in the possible (even likely) case that a variable j 6= i exists that is e eb adjacent to both e and p. Combined with the de nition of d, we have di = di = di ei di when jEij = 2, and di di = di when jEij > 2. b eb when jEi j 1, di = d b If jEij 1 our bound di is exact for the same reason that di is exact. If jEij is two we have di = jAi n ij + jLp n ij + jLe n Lpj = di : No entry is in both Ai and any element L, since these redundant entries are removed from Ai . Any entry in Lp does not appear in the external subset (Le n Lp ). Thus, no b entry is counted twice, and di = di when jEij 2. Finally, consider both di and di when jEi j > 2. We have di = jAi n ij + jLp n ij + and X e2Ei nfpg b di = jAi n ij + jLp n ij + jLe n Lp j X e2Ei nfpg jLe n ij: Since these degree bounds are only used when computing the degree of a supervariable b i 2 Lp , we have i Lp. Thus, di di when jEij > 2. 2 APPROXIMATE MINIMUM DEGREE 11 eb Combining the three inequalities in Theorem 1, the inequality di di di di holds for all values of jEij. Note that, if a variable i is adjacent to two elements or less then our bound is equal to the exact external degree. This is very important, since most variables of minimum degree are adjacent to two elements or less. Additionally, our degree bounds take advantage of element absorption, since the bound depends on jEij after elements are absorbed. 4.2. Degree computation example. We illustrate the computation of our approximate external degree bound in Figures 3.1 and 3.2. Variable 6 is adjacent to three elements in G 3 and G 4. All other variables are adjacent to two or less elements. In G 3 , the bound d6 is tight, since the two sets jL1 n L3j and jL2 n L3 j are disjoint. In graph G 4, the current pivot element is p = 4. We compute d6 = jAi n ij + jLp n ij + ( = = = = X e2Ei nfpg jLe n Lpj) j; n f6gj + jf6; 7; 8gn f6gj + (jL2 n L4j + jL3 n L4 j) jf7; 8gj + (jf5; 6; 9g n f6; 7; 8gj + jf5; 6; 7gn f6; 7; 8gj) jf7; 8gj + (jf5; 9gj + jf5gj) 5: The exact external degree of variable 6 is d6 = 4, as can be seen in the elimination graph G4 on the left of Figure 3.2(a). Our bound is one more than the exact external degree, since the variable 5 appears in both L2 n L4 and L3 n L4, but is one less than b the bound di which is equal to 6 in this case. Our bound on the degree of variable 6 is again tight after the next pivot step, since elements 2 and 3 are absorbed into element 5. 5. The approximate minimum degree algorithm. The Approximate Minimum Degree algorithm is identical to Algorithm 1, except that the external degree, di , is replaced with di, throughout. The bound on the external degree, di , is computed using Algorithm 2 and Equations (4.1) and (4.2). In addition to absorbing elements in Ep , any element with an empty external subset (jLe n Lpj = 0) is also absorbed into element p, even if e is not adjacent to p. This \aggressive" element absorption improves the degree bounds by reducing jEj. As in many other minimum degree algorithms, we use linked lists to assist the search for a variable of minimum degree. List d holds all supervariables i with degree bound di = d. Maintaining this data structure takes time proportional to the total number of degree computations, or O(jLj). Computing the pattern of each pivot element, Lp, takes a total of O(jLj) time overall, since each element is used in the computation of at most one other element, and the total sizes of all elements constructed is O(jLj). The AMD algorithm is based on the quotient graph data structure used in the MA27 minimum degree algorithm 13, 14, 15]. Initially, the sets A are stored, followed by a small amount of elbow room. When the set Lp is formed, it is placed in the elbow room (or in place of Ap if jEp j = 0). Garbage collection occurs if the elbow room is exhausted. During garbage collection, the space taken by Ai and Ei is reduced to exactly jAij + jEi j for each supervariable i (which is less than or equal to jA0j) and i the extra space is reclaimed. The space for Ae and Ee for all elements e 2 V is fully reclaimed, as is the space for Le of any absorbed elements e. Each garbage collection 12 P. AMESTOY, T. A. DAVIS, , AND I. S. DUFF takes time that is proportional to the size of the workspace (normally (jAj)). In practice, elbow room of size n is su cient. During the computation of our degree bounds, we compute the following hash function for supervariable detection 3], 80 9 < X X1 = Hash(i) = :@ j + eA mod (n 1); + 1; j2A e2E i i which increases the degree computation time by a small constant factor. We place each supervariable i in a hash bucket according to Hash(i), taking time O(jLj) overall. If two or more supervariables are placed in the same hash bucket, then each pair of supervariables i and j in the hash bucket are tested for indistinguishability. If the hash function results in no collisions then the total time taken by the comparison is O(jAj). Ashcraft 3] uses this hash function as a preprocessing step on the entire matrix (without the mod(n 1) term, and with an O(jV j log jV j) sort instead of jV j hash buckets). In contrast, we use this function during the ordering, and only hash those variables adjacent to the current pivot element. For example, variables 7, 8, and 9 are indistinguishable in G5 , in Figure 3.2(a). The AMD algorithm would not consider variable 8 at step 5, since it is not adjacent to the pivot element 5 (refer to quotient graph G 5 in Figure 3.2(b)). AMD would not construct 7 = f7; 9g at step 5, since 7 and 9 are distinguishable in G 5. It would construct 7 = f7; 8; 9g at step 6, however. The total number of times the approximate degree di of variable i is computed during elimination is no more than the number of nonzero entries in row k of L, where variable i is the kth pivot. The time taken to compute di is O(jA0j), or equivalently i O(j(PAPT )k j), the number of nonzero entries in row k of the permuted matrix. The total time taken by the entire AMD algorithm is thus bounded by the degree computation, (5.1) O n X k=1 jLk j j(PAPT )k ! j: This bound assumes no (or few) supervariable hash collisions and a constant number of garbage collections. In practice these assumptions seem to hold, but the asymptotic time would be higher if they did not. In many problem domains, the number of nonzeros per row of A is a constant, independent of n. For matrices in these domains, our AMD algorithm takes time O(jLj) (with the same assumptions). 6. Performance results. In this section, we present the results of our experiments with AMD on a wide range of test matrices. We rst compare the degree e b computations discussed above (t, d, d, d, and d), as well as an upper bound on the true degree, t d + jij 1. We then compare the AMD algorithm with other established minimum degree codes (MMD and MA27). 6.1. Test Matrices. We tested all degree bounds and codes on all matrices in the Harwell/Boeing collection of type PUA, RUA, PSA, and RSA 11, 12] (at orion.cerfacs.fr or numerical.cc.rl.ac.uk), all non-singular matrices in Saad's SPARSKIT2 collection (at ftp.cs.umn.edu), all matrices in the University of Florida collection (available from ftp.cis.ufl.edu in the directory pub/umfpack/matrices), APPROXIMATE MINIMUM DEGREE 13 Table 6.1 Selected matrices in test set Matrix RAEFSKY3 VENKAT01 BCSSTK32 EX19 BCSSTK30 CT20STIF NASASRB OLAF RAEFSKY1 CRYSTK03 RAEFSKY4 CRYSTK02 BCSSTK33 BCSSTK31 EX11 FINAN512 RIM BBMAT EX40 WANG4 LHR34 WANG3 LHR71 ORANI678 PSMIGR1 APPU n nz 21,200 62,424 44,609 12,005 28,924 52,329 54,870 16,146 3,242 24,696 19,779 13,965 8,738 35,588 16,614 74,752 22,560 38,744 7,740 26,068 35,152 26,064 70,304 2,529 3,140 14,000 733,784 827,684 985,046 123,937 1,007,284 1,323,067 1,311,227 499,505 145,517 863,241 654,416 477,309 291,583 572,914 540,167 261,120 862,411 1,274,141 225,136 75,564 608,830 75,552 1,199,704 85,426 410,781 1,789,392 Percentage of Description 0.00 0.71 0.20 1.57 0.66 0.77 0.06 0.41 0.00 0.00 0.00 0.00 0.00 0.60 0.04 1.32 2.34 5.81 17.45 15.32 7.69 15.29 8.47 6.68 6.65 15.64 uid/structure interaction, turbulence unstructured 2D Euler solver structural eng., automobile chassis 2D developing pipe ow (turbulent) structural eng., o -shore platform structural eng., CT20 engine block shuttle rocket booster NASA test problem incompressible ow, pressure-driven pipe structural eng., crystal vibration buckling problem for container model structural eng., crystal vibration structural eng., auto steering mech. structural eng., automobile component CFD, 3D cylinder & at plate heat exch. economics, portfolio optimization chemical eng., uid mechanics problem CFD, 2D airfoil with turbulence CFD, 3D die swell problem on square die 3D MOSFET semicond. (30x30x30 grid) chemical eng., light hydrocarbon recovery 3D diode semiconductor (30x30x30 grid) chemical eng., light hydrocarbon recovery Australian economic model US county-by-county migration NASA test problem (random matrix) jEp j > 2 jEi j > 2 13.4 15.7 27.3 29.4 31.8 33.2 35.0 35.2 38.9 40.9 41.4 42.0 42.6 43.1 43.3 46.6 63.2 64.4 64.7 78.3 78.7 79.2 81.1 86.9 91.0 94.4 and several other matrices from NASA and Boeing. Of those 378 matrices, we present results below on those matrices requiring 500 million or more oating-point operations for the Cholesky factorization, as well as the ORANI678 matrix in the Harwell/Boeing collection and the EX19 in Saad's collection (a total of 26 matrices). The latter two are best-case and worst-case examples from the set of smaller matrices. For the unsymmetric matrices in the test set, we rst used the maximumtransversal algorithm MC21 from the Harwell Subroutine Library 9] 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, 2]. For these matrices, Table 6.1 lists the statistics for the symmetrized pattern. Table 6.1 lists the matrix name, the order, the number of nonzeros in lower triangular part, two statistics obtained with an exact minimumdegree ordering (using d), and a description. In column 4, we report the percentage of pivots p such that jEp j > 2. Column 4 shows that there is only a small percentage of pivots selected using an exact minimum degree ordering that have more than two elements in their adjacency list. Therefore, we can expect a good quality ordering with an algorithm based on our approximate degree bound. In column 5, we indicate how often a degree di is computed when jEij > 2 (as a percentage of the total number of degree updates). Table 6.1 is sorted according to this degree update percentage. Column 5 thus reports the percentage of \costly" degree updates performed by a minimum degree algorithm based on the exact degree. For matrices with relatively large values 14 P. AMESTOY, T. A. DAVIS, , AND I. S. DUFF in column 5, signi cant time reductions can be expected with an approximate degree based algorithm. Since any minimum degree algorithm is sensitive to tie-breaking issues, we randomly permuted all matrices and their adjacency lists 21 times (except for the random APPU matrix, which we ran only once). All methods were given the same set of 21 randomized matrices. We also ran each method on the original matrix. On some matrices, the original matrix gives better ordering time and ll-in results for all methods than the best result obtained with the randomized matrices. The overall comparisons are not however dependent on whether original or randomized matrices are used. We thus report only the median ordering time and ll-in obtained for the randomized matrices. The APPU matrix is a random matrix used in a NASA benchmark, and is thus not representative of sparse matrices from real problems. We include it in our test set as a pathological case that demonstrates how well AMD handles a very irregular problem. Its factors are about 90% dense. It was not practical to run the APPU matrix 21 times because the exact degree update algorithms took too much time. 6.2. Comparing the exact and approximate degrees. To make a valid comparison between degree update methods, we modi ed our code for the AMD algorithm so that we could compute the exact external degree (d), our bound (d), Ashcraft and e b Eisenstat's bound (d), Gilbert, Moler, and Schreiber's bound (d), the exact true dee gree (t), and our upper bound on the true degree (t). The six codes based on d, d, d, b d, t, and t (columns 3 to 8 of Table 2) di er only in how they compute the degree. Since aggressive absorption is more di cult when using some bounds than others, we switched o aggressive absorption for these six codes. The actual AMD code (in column 2 of Table 2) uses d with aggressive absorption. Table 6.2 lists the median number of nonzeros below the diagonal in L (in thousands) for each method. Results 20% higher than the lowest median jLj in the table (or higher) are underlined. Our upper bound on the true degree (t) and the exact true degree (t) give nearly identical results. As expected, using minimum degree algorithms based on external degree noticeably improves the quality of the ordering (compare eb columns 3 and 7, or columns 4 and 8). From the inequality d d d d, we would expect a similar ranking in the quality of ordering produced by these methods. Table 6.2 con rms this. The bound d and the exact external degree d produce nearly identical results. Comparing the AMD results and the d column, aggressive absorption tends to result in slightly lower ll-in, since it reduces jEj and thus improves the e accuracy of our bound. The d bound is often accurate enough to produce good results, but can fail catastrophically for matrices with a high percentage of approximate b pivots (see column 4 in Table 6.1). The less accurate d bound produces notably worse results for many matrices. Comparing all 378 matrices, the median jLj when using d is never more than 9% higher than the median ll-in obtained when using the exact external degree, d (with the exception of the FINAN512 matrix). The ll-in results for d and d are identical for nearly half of the 378 matrices. The approximate degree bound d thus gives a very reliable estimation of the degree in the context of a minimum degree algorithm. The FINAN512 matrix is highly sensitive to tie-breaking variations. Its graph consists of two types of nodes: \constraint" nodes and \linking" nodes 5]. The constraint nodes form independent sparse subgraphs, connected together via a tree of linking nodes. This matrix is a pathological worst-case matrix for any minimumdegree APPROXIMATE MINIMUM DEGREE 15 Table 6.2 Median ll-in results of the degree update methods Number of nonzeros below diagonal in L, in thousands e b AMD d d d d t t RAEFSKY3 4709 4709 4709 4709 5114 4992 4992 5798 6399 6245 6261 VENKAT01 5789 5771 5789 5080 5081 5079 5083 5721 5693 5665 BCSSTK32 EX19 319 319 319 318 366 343 343 BCSSTK30 3752 3751 3753 3759 4332 4483 4502 CT20STIF 10858 10758 10801 11057 13367 12877 12846 NASASRB 12282 12306 12284 12676 14909 14348 14227 OLAF 2860 2858 2860 2860 3271 3089 3090 RAEFSKY1 1151 1151 1151 1151 1318 1262 1262 CRYSTK03 13836 13836 13836 13836 17550 15507 15507 7685 9294 8196 8196 RAEFSKY4 7685 7685 7685 CRYSTK02 6007 6007 6007 6007 7366 6449 6449 BCSSTK33 2624 2624 2624 2640 3236 2788 2787 5115 5096 5132 5225 6194 6079 6057 BCSSTK31 6014 6016 6014 6014 7619 6673 6721 EX11 4778 4036 6042 11418 11505 8235 8486 FINAN512 RIM 3948 3898 3952 3955 4645 4268 4210 BBMAT 19673 19880 19673 21422 37820 21197 21445 1418 1386 1417 1687 1966 1526 1530 EX40 WANG4 6547 6808 6548 6566 7871 7779 7598 LHR34 3618 3743 3879 11909 27125 4383 4435 WANG3 6545 6697 6545 6497 7896 7555 7358 7933 8127 8499 28241 60175 9437 9623 LHR71 ORANI678 147 147 146 150 150 147 146 3020 3025 3011 3031 3176 2966 2975 PSMIGR1 APPU 87648 87613 87648 87566 87562 87605 87631 Matrix method. All constraint nodes should be ordered rst, but linking nodes have low degree and tend to be selected rst, which causes high ll-in. Using a tree dissection algorithm, Berger, Mulvey, Rothberg, and Vanderbei 5] obtain an ordering with only 1.83 million nonzeros in L. Table 6.3 lists the median ordering time (in seconds on a SUN SPARCstation 10) for each method. Ordering time twice that of the minimum median ordering b time listed in the table (or higher) is underlined. Computing the d bound is often the fastest, since it requires a single pass over the element lists instead of the two passes required for the d bound. It is, however, sometimes slower than d because it can generate more ll-in, which increases the ordering time (see Equation 5.1). The ordering time of the two exact degree updates (d and t) increases dramatically as the percentage of \costly" degree updates increases (those for which jEij > 2). Garbage collection has little e ect on the ordering time obtained. In the above runs, we gave each method elbow room of size n. Usually a single garbage collection occurred. At most two garbage collections occurred for AMD, and at most three for the other methods (aggressive absorption reduces the memory requirements). 6.3. Comparing algorithms. In this section, we compare AMD with two other established minimum degree codes: Liu's Multiple Minimum Degree (MMD) code 25] and Du and Reid's MA27 code 15]. MMD stores the element patterns L in a fragmented manner and requires no elbow room 20, 21]. It uses the exact external degree, d. MMD creates supervariables only when two variables i and j have no 16 P. AMESTOY, T. A. DAVIS, , AND I. S. DUFF Table 6.3 Median ordering time of the degree update methods Matrix AMD RAEFSKY3 1.05 4.07 VENKAT01 4.67 BCSSTK32 EX19 0.87 BCSSTK30 3.51 CT20STIF 6.62 7.69 NASASRB OLAF 1.83 RAEFSKY1 0.27 CRYSTK03 3.30 RAEFSKY4 2.32 CRYSTK02 1.49 BCSSTK33 0.91 4.55 BCSSTK31 2.70 EX11 15.03 FINAN512 RIM 5.74 BBMAT 27.80 1.04 EX40 WANG4 5.45 LHR34 19.56 WANG3 5.02 46.03 LHR71 ORANI678 5.49 10.61 PSMIGR1 APPU 41.75 d 1.10 4.95 5.64 1.12 5.30 8.66 11.03 2.56 0.34 4.84 2.90 2.34 1.36 7.53 4.06 34.11 10.38 115.75 1.56 11.45 109.10 10.45 349.58 196.01 334.27 2970.54 Ordering time, in seconds d 1.09 4.11 4.54 0.89 3.55 6.54 7.73 1.90 0.28 3.08 2.18 1.55 1.05 4.92 2.77 14.45 5.69 27.44 1.10 5.56 25.62 5.49 58.25 8.13 10.07 39.83 e d 1.05 4.47 4.91 1.01 3.65 7.07 9.23 2.16 0.32 3.68 2.45 1.64 0.99 5.68 3.00 17.79 6.12 42.17 1.09 6.98 45.36 6.52 129.85 6.97 14.20 43.20 b d 1.02 3.88 4.35 0.86 3.51 6.31 7.78 1.83 0.25 3.14 2.08 1.45 0.85 4.56 2.60 15.84 5.72 23.02 0.95 5.21 43.70 4.81 121.96 7.23 8.16 40.64 t 1.15 4.32 5.55 1.09 4.38 8.63 11.78 2.33 0.35 5.23 3.12 2.04 1.62 7.41 4.23 46.49 10.01 129.32 1.46 11.59 125.41 11.02 389.70 199.01 339.28 3074.44 t 1.09 3.85 4.48 0.87 3.38 6.45 7.99 1.78 0.28 3.30 2.07 1.52 0.91 4.92 2.89 18.58 5.58 28.33 1.12 5.88 24.73 5.02 60.40 8.45 9.94 38.93 adjacent variables and exactly two adjacent elements (Ei = Ej = fe; pg, and Ai = Aj = ;, where p is the current pivot element). A hash function is not required. MMD takes advantage of multiple elimination and incomplete update. MA27 uses the true degree, t, and the same data structures as AMD. It detects supervariables whenever two variables are adjacent to the current pivot element and have the same structure in the quotient graph (as does AMD). MA27 uses the true degree as the hash function for supervariable detection, and does aggressive absorption. Neither AMD nor MA27 take advantage of multiple elimination or incomplete update. Structural engineering matrices tend to have many rows of identical nonzero pattern. Ashcraft has found that the total ordering time of MMD can be signi cantly improved by detecting these initial supervariables before starting the elimination 3]. We implemented Ashcraft's pre-compression algorithm, and modi ed MMD to allow for initial supervariables. We call the resulting code CMMD (\compressed" MMD). Pre-compression has little e ect on AMD, since it nds these supervariables when their degrees are rst updated. The AMD algorithm on compressed matrices together with the cost of pre-compression was never faster than AMD. Table 6.4 lists the median number of nonzeros below the diagonal in L (in thousands) for each code. Results 20% higher than the lowest median jLj in the table (or higher) are underlined. AMD, MMD, and CMMD nd orderings of about the same quality. MA27 is slightly worse because it uses the true degree (t) instead of the external degree (d). APPROXIMATE MINIMUM DEGREE 17 Table 6.4 Median ll-in results of the four codes Matrix Number of nonzeros below diagonal in L, in thousands AMD MMD CMMD MA27 RAEFSKY3 4709 4779 4724 5041 VENKAT01 5789 5768 5811 6303 BCSSTK32 5080 5157 5154 5710 EX19 319 322 324 345 3752 3788 3712 4529 BCSSTK30 CT20STIF 10858 11212 10833 12760 NASASRB 12282 12490 12483 14068 2860 2876 2872 3063 OLAF 1177 1255 RAEFSKY1 1151 1165 14066 15496 CRYSTK03 13836 13812 RAEFSKY4 7685 7539 7582 8245 CRYSTK02 6007 5980 6155 6507 2624 2599 2604 2766 BCSSTK33 5115 5231 5216 6056 BCSSTK31 EX11 6014 5947 6022 6619 FINAN512 4778 8180 8180 8159 RIM 3948 3947 3914 4283 19673 19876 19876 21139 BBMAT 1418 1408 1401 1521 EX40 WANG4 6547 6619 6619 7598 LHR34 3618 4162 4162 4384 WANG3 6545 6657 6657 7707 7933 9190 9190 9438 LHR71 147 147 147 147 ORANI678 PSMIGR1 3020 2974 2974 2966 APPU 87648 87647 87647 87605 Considering the entire set of 378 matrices, AMD produces a better median ll-in than MMD, CMMD, and MA27 for 62%, 61%, and 81% of the matrices, respectively. AMD never generates more than 7%, 7%, and 4% more nonzeros in L than MMD, CMMD, and MA27, respectively. We have shown empirically that AMD produces an ordering at least as good as these other three methods for this large test set. If the apparent slight di erence in ordering quality between AMD and MMD is statistically signi cant, we conjecture that it has more to do with earlier supervariable detection (which a ects the external degree) than with the di erences between the external degree and our upper bound. Table 6.5 lists the median ordering time (in seconds on a SUN SPARCstation 10) for each method. The ordering time for CMMD includes the time taken by the precompression algorithm. Ordering time twice that of the minimum median ordering time listed in the table (or higher) is underlined. On certain classes of matrices, typically those from structural analysis applications, CMMD is signi cantly faster than MMD. AMD is the fastest method for all but the EX19 matrix. For the other 352 matrices in our full test set, the di erences in ordering time between these various methods is typically less. If we compare the ordering time of AMD with the other methods on all matrices in our test set requiring at least a tenth of a second of ordering time, then AMD is slower than MMD, CMMD, and MA27 only for 6, 15, and 8 matrices respectively. For the full set of matrices, AMD is never more than 30% slower than these other methods. The best and worst cases for the relative run-time 18 P. AMESTOY, T. A. DAVIS, , AND I. S. DUFF Table 6.5 Median ordering time of the four codes Matrix AMD RAEFSKY3 1.05 4.07 VENKAT01 BCSSTK32 4.67 EX19 0.87 BCSSTK30 3.51 6.62 CT20STIF 7.69 NASASRB OLAF 1.83 RAEFSKY1 0.27 3.30 CRYSTK03 RAEFSKY4 2.32 CRYSTK02 1.49 BCSSTK33 0.91 4.55 BCSSTK31 2.70 EX11 FINAN512 15.03 RIM 5.74 27.80 BBMAT EX40 1.04 5.45 WANG4 LHR34 19.56 WANG3 5.02 LHR71 46.03 ORANI678 5.49 PSMIGR1 10.61 APPU 41.75 Ordering time, in seconds MMD CMMD MA27 2.79 1.18 1.23 9.01 4.50 5.08 12.47 5.51 6.21 0.69 0.83 1.03 7.78 3.71 4.40 26.00 9.59 9.81 22.47 11.28 12.75 5.67 4.41 2.64 0.82 0.28 0.40 10.63 3.86 5.27 5.24 2.36 2.91 3.89 1.53 2.37 2.24 1.32 1.31 11.60 7.76 7.92 7.45 5.05 3.90 895.23 897.15 40.31 9.09 8.11 10.13 200.86 201.03 134.58 2.13 2.04 1.30 10.79 11.60 9.86 139.49 141.16 77.83 10.37 10.62 8.23 396.03 400.40 215.01 124.99 127.10 124.66 186.07 185.74 229.51 5423.23 5339.24 2683.27 of AMD for the smaller matrices are included in Table 6.5 (the EX19 and ORANI678 matrices). 7. Summary. We have described a new upper bound for the degree of nodes in the elimination graph that can be easily computed in the context of a minimum degree algorithm. We have demonstrated that this upper-bound for the degree is more accurate than all previously used degree approximations. We have experimentally shown that we can replace an exact degree update by our approximate degree update and obtain almost identical ll-in. An Approximate Minimum Degree (AMD) based on external degree approximation has been described. We have shown that the AMD algorithm is highly competitive with other ordering algorithms. It is typically faster than other minimum degree algorithms, and produces comparable results to MMD (which is also based on external degree) in terms of ll-in. AMD typically produces better results, in terms of ll-in and computing time, than the MA27 minimum degree algorithm (based on true degrees). 8. Acknowledgments. 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