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1 Electronic Transactions on Numerical Analysis. Volume 2, pp , 25. Copyright 25,. ISSN CROUT VERSIONS OF ILU FACTORIZATION WITH PIVOTING FOR SPARSE SYMMETRIC MATRICES NA LI AND YOUSEF SAAD Abstract. The Crout variant of ILU preconitioner (ILUC) evelope recently has been shown to be generally avantageous over ILU with Threshol (ILUT), a conventional row-base ILU preconitioner. This paper explores pivoting strategies for sparse symmetric matrices to improve the robustness of ILUC. We integrate two symmetrypreserving pivoting strategies, the iagonal pivoting an the Bunch-Kaufman pivoting, into ILUC without significant overheas. The performances of the pivoting methos are compare with ILUC an ILUTP ([2]) on a set of problems, incluing a few arising from sale-point (KKT) problems. Key wors. incomplete LU factorization, ILU, ILUC, sparse Gaussian elimination, crout factorization, preconitioning, iagonal pivoting, Bunch-Kaufman pivoting, ILU with threshol, iterative methos, sparse symmetric matrices AMS subject classifications. 65F1, 65F5 1. Introuction. Incomplete LU (ILU) factorization base preconitioners combine with Krylov subspace projection processes are often consiere the best general purpose iterative solvers available. Such ILU factorizations normally perform a Gaussian elimination algorithm while ropping some fill-ins. However, some variants of the Gaussian elimination algorithm may be more suitable than others for solving a given problem. For example, when ealing with sparse matrices, the traitional KIJ version [12, p 99] is impractical since all remaining rows are moifie at each step. In this case, another variant of Gaussian elimination namely the IKJ version, which implements a elaye-upate version for the matrix elements, is preferre. Although the IKJ version of Gaussian elimination is wiely use for implementing ILU factorizations (see, e.g., ILUT in SPARSKIT [2]), it has an inherent isavantage: its requirement to access to the entries of a row of a matrix in a topological orer uring the factorization. Due to the fill-ins introuce uring the factorization of a sparse matrix, a search is neee at each step in orer to locate a pivot with the smallest inex [21]. These searches often result in high computational costs, especially when the number of nonzero entries in an/or the factors is large. A strategy to reuce the cost of these searches is to construct a binary tree for the current row an utilize binary searches [22]. Recently, a more efficient alternative, terme ILUC an base on the Crout version of Gaussian elimination, has been evelope [17]. In the Crout version of Gaussian elimination, the -th column of ( ) an the - th row of ( ) are calculate at the -th step. Unlike the IKJ version, all elements that will be use to upate an at the -th step in the Crout version have alreay been calculate, i.e., the fill-ins will not interfere with the row upates at the -th step. Therefore, searching for the next pivot in the Crout version is avoie. A special ( bi-inex ) ata structure was evelope for ILUC to aress two implementation ifficulties in sparse matrix operations (see Section 3 for etails), following earlier work by Eisenstat et al. [11] in the context of the Yale Sparse Matrix Package (YSMP), an Jones an Plassmann [16]. Besies efficiency, another avantage of ILUC is that it also enables some more rigorous ropping strategies (e.g. [3, 2]) which result in improve robustness. However, there are still Receive April 6, 24. Accepte for publication January 19, 25. Recommene by A. Frommer. This work was supporte by the National Science Founation uner grant ACI-3512, an by the Minnesota Supercomputer Institute. Department of Computer Science an Engineering, University of Minnesota, 2 Union Street S.E., Minneapolis, MN

2 76 N. LI AND Y. SAAD many situations where sparse linear systems are ifficult to solve by iterative methos with the ILUC preconitioning, especially when the coefficient matrices are very ill-conitione an/or highly inefinite. In such situations, pivoting techniques can be use to further improve robustness. Nevertheless, a pivoting metho that is both efficient an effective must be carefully esigne to fit within the ata structure use by the incomplete factorization algorithm, which may not be a trivial task. In this paper, symmetry-preserving pivoting strategies that are suitable for the ata structure use in ILUC are explore an implemente for symmetric matrices. We begin our iscussion with a review of relate pivoting techniques in Section 2. We then iscuss in etail ILUC with pivoting methos in Section 3. Finally, we compare the performances of the pivoting methos with ILUC an ILUTP ([2]) on some general symmetric matrices in Section 4 an some KKT matrices in Section Relate Pivoting Techniques. For many linear systems that arise from real applications, the ILU factorization may encounter either a zero or a small pivot. In case a zero pivot occurs, one strategy to avoi a break-own of the factorization process is to replace the zero pivot with a very small element, which leas to the secon case. However, this remey works only in very limite cases. Inee, small pivots can cause an exponential growth of the entries in the factors an the resulting ILU process can eventualy break own ue to an overflow or unerflow conition. Even when this oes not happen, the prouce L an U factors are generally of poor quality an will not lea to convergence in the iteration phase. A common solution to ensure a moerate growth in the factors is to use pivoting techniques. However, pivoting techniques are often in conflict with the unerlying ata structures of the factorizations. For example, for a symmetric matrix, a column partial pivoting metho such as the ILUTP algorithm of SPARSKIT [2], will estroy symmetry. Pivoting methos that preserve symmetry are esirable. The elimination in ILUC is symmetric, i.e., at step, the th row of an the th column of are compute. Moreover, the bi-inex ata structure use to implement ILUC is also symmetric. Thus, the symmetric elimination process an the symmetric ata structure are ieal to incorporate symmetry-preserving pivoting methos into ILUC. In this paper, we explore symmetry-preserving pivoting methos that can be integrate into the existing ILUC process without significant overheas. Our goal is to improve the robustness of ILUC for symmetric systems while preserving symmetry. For this reason, we use a revise version of ILUC (terme ILDUC) to compute the factorization instea of. In ILDUC, an are unit lower triangular matrices, an is iagonal. One simple pivoting metho that preserves symmetry is to select the largest iagonal entry as the next pivot an interchange the corresponing row an column simultaneously. This metho, referre to as iagonal pivoting in the rest of the paper, works well for many symmetric matrices accoring to our tests. However, it fails for a matrix as simple as!" In 1971, Bunch an Parlett propose a pivoting metho base on Kahan s generalize pivot to inclue &'(& principal submatrices [4]. They also prove that the boun of this metho, in terms of the growth of elements in the reuce matrices, is almost as goo as that of the Gaussian elimination with complete pivoting. However, this metho requires searching the entire reuce matrix at each step uring the factorization, which makes it impractical for large sparse matrices. In 1977, Bunch an Kaufman propose a partial pivoting metho, now known as the Bunch-Kaufman pivoting metho, where a )* or &+& pivot can be "#%$

3 " W W " " $ CROUT VERSIONS OF ILU FACTORIZATION WITH PIVOTING 77 etermine by searching at most two columns of the reuce matrix at each step [6]. In 1998, Ashcraft et al. propose two alternatives of the Bunch-Kaufman algorithm, proviing better accuracy by bouning the triangular factors [1]. In 2, Higham prove the stability of the Bunch-Kaufman pivoting metho [15]. Because of the efficiency an effectiveness of the Bunch-Kaufman pivoting metho, it has been use in LINPACK an LAPACK to solve ense symmetric inefinite linear systems. In contrast to these methos, which ynamically select pivots, strategies such as weighte matching have been introuce to approximate a static pivoting orer in irect solvers [9, 1], an have been recently also explore for iterative methos [13]. In this paper, we show that the Bunch-Kaufman pivoting metho can be integrate into ILDUC to solve sparse symmetric inefinite linear systems. For the sake of completeness, we provie some etails of the Bunch-Kaufman pivoting algorithm. Let -, /. enote the matrix at step. To ecie the next pivot, 1. Determine 6587:9<;=9<?> 1'243, /. ;@ >BA which is the largest off-iagonal element in absolute value in the -th column. If, then go to step C. Otherwise let D (DEF ) be the smallest integer such that > G, /. >. 2. If >, /. 6 >H*I where I KJLMC*N PO4QSR4T, then use, /. 6 as a UV pivot. Otherwise, 3. Determine W XZY[\Y] 1'243 [^ _` > a, 6. G > cb 4. If I >, /. 6, then use >, /. 6 as a e pivot. 5. Else if > GfG, /., then interchange the >MHgI -th an the D -th rows an columns of, 6., use GfG, /. as a V pivot. 6. Else interchange the Jh icjpq -th an the D -th rows an columns of -, /. so that >, /. /587/ > >, /. /587 >, use k, /. 6, 6. as a &o& pivot. /5l7: /5l7nm In the next section, we present the methos that integrate the iagonal pivoting an the Bunch-Kaufman pivoting algorithms into the existing ILDUC process without significant overhea. 3. ILDUC with Pivoting for Sparse Symmetric Matrices. Algorithm 3.1 shows the ILDUC process to calculate the factorization pq Mr, moifie from the ILUC algorithm propose in [17]. ALGORITHM 3.1. ILDUC - Crout version of Incomplete LDU factorization 1. s ut A 'vwyx 2. For 'Fwzx Do : 3. Initialize row { : {z7f % A {4 /5l7f " \ t} /5l7f 4. For ~ > tƒxzsu = 6; Do : 5. {4 /587S )u{4 /587S ˆ = /;c sš; MŒ<;= 6587f 6. EnDo 7. Initialize column : Ž7f A /587S "} qt} /587S } 8. For ~ > tƒxzsžœ<;@ \ Do :

4 " b 78 N. LI AND Y. SAAD 9. /587S /587S VŒ ;@ s ; /5l7f ; 1. EnDo 11. Apply a ropping rule to row { 12. Apply a ropping rule to column 13. Œ u{rysš A Œ 6 F 14. B Uq URysŠ A = %K "} 15. For ~ > %Cq x A ;@ tšxzs?œ /; Do: 16. sš;8usƒ; ˆ ;@ sš MŒ /; 17. EnDo 18. EnDo One property of the Crout version of LU is that only previously calculate elements are use to upate the -th column of an the -th row of at step. For sparse matrices, this means new fill-ins will not interfere with the upates of the -th column of an the -th row of. Therefore, the upates to the -th row of (resp., the -th column of ) can be mae in any orer, i.e., the variable can be chosen in any orer in Line 4 (resp., in Line 8). This avois the expensive searches in the stanar ILUT. However, as pointe out in [17], two implementation ifficulties regaring sparse matrix operations have to be aresse. The first ifficulty lies in Lines 5 an 9 in Algorithm 3.1. At step, only the section Jh?C %wyx Q of the -th row of is neee to calculate the -th row of. Similarly, only the section Jh?C*wyx Q of the -th column of is neee to calculate the -th column of. Accessing entire rows of or columns of an then extracting the esire part is an expensive option. The secon ifficulty lies in lines 4 an 8 in Algorithm 3.1. is store column-wise, but the nonzeros in the -th row of must be accesse. Similarly, is store row-wise, but the nonzeros in the -th column of must be accesse. A carefully esigne bi-inex ata structure was use in [17] to aress these ifficulties, inspire from early work by Eisenstat et al. [11] an Jones an Plassmann [16]. The bi-inex ata structure consists of four arrays of size x : hdy, M =L, šdy, an % =L. At step, M šdy 6J=LQ, u, points to the first entry with the row inex greater than in the -th column of. In this way, the section /5l7f ; can be efficiently accesse, which aresses the first ifficulty. It is worth pointing out that the elements in a column of nees to be sorte by their inices to achieve this. However, in contrast with the searches neee in the stanar ILUT, this sorting is much more efficient for two reasons. First, the sort is performe only after a large number of elements in a column are roppe, unlike in ILUT, where searching is performe on all of the elements. Secon, a fast sorting algorithm such as Quick Sort can be easily applie, unlike in ILUT where some overhea in performing the search (e.g., when builing the binary search trees). To aress the secon ifficulty, the array is use to maintain linke lists of elements in the -th row of, where H. The linke lists are upate in such a way that the linke list of the elements in the -th row of is guarantee to be reay at step. This linke list will be use to upate the -th row of. šdy an % =L are forme in a similarly way. In the following, we iscuss how we integrate pivoting methos to the bi-inex ata structure use in ILDUC for sparse symmetric matrices. One critical issue is that the new methos with pivoting shoul have a similar cost an complexity as that of ILDUC The œ Diagonal Pivoting. For the œ iagonal pivoting, we nee to locate the largest iagonal element in absolute value at each step. For sparse matrices, a straightforwar linear search woul lea to a computational cost of )J=x Q, which is not acceptable especially when x is large. Alternatively, we buil a binary heap for the iagonal elements in orer to locate the largest iagonal entry efficiently. The binary heap is forme an maintaine so that each noe is greater than or equal to its chilren in absolute value. Therefore, the root noe

5 W W " W CROUT VERSIONS OF ILU FACTORIZATION WITH PIVOTING 79 is always the largest iagonal entry in absolute value. Algorithm 3.2 summarizes ILDUC with )ž iagonal pivoting (terme ILDUC-DP). Note that for a symmetric matrix the factorization is Ÿ ŽŸ(q \ M, so only nees to be calculate in the algorithm. ALGORITHM 3.2. ILDUC-DP - Incomplete M\ with V iagonal pivoting 1. s ut A 'vwyx 2. Initialize a binary heap for s} A 'K Ä 6A x 3. For 'Fwzx Do : 4. Locate the largest iagonal s G in absolute value from the root noe of the heap 5. Interchange column D an column if Dr u 6. Remove the root noe from the heap an reorer the heap 7. Initialize column : 7f A " /587S qt /587S } 8. For ~ > tƒxzsu 6; Do : 9. /587S 'q /587S ) e = /; sƒ;c = 6587f ; 1. EnDo 11. Apply a ropping rule to column 12. B Uq UR4s A = 6 K "} 13. For ~ > Cq x A ;@ Do: 14. sš;8qsš; ˆ ; s ; 15. Reorer the heap for s; 16. EnDo 17. EnDo In the above algorithm, the cost of initializing the binary heap is )J x Q (a bottom-up approach). The total cost of removing the root noe an reorering the heap is )J=x% ƒ x Q. The total cost of maintaining the heap is at most )J rx% z x Q (where ª x is the ropping parameter efining the maximum number of fill-ins allowe in each column) since whenever a iagonal element is moifie (line 15) the heap has to be reorere. Therefore, the total cost is boune by )J= x% ƒ x Q The Bunch-Kaufman Diagonal Pivoting. In the Bunch-Kaufman pivoting metho, to etermine the next pivot at each step only requires searching for at most two columns in the reuce matrix. This is feasible for a sparse symmetric matrix as the number of nonzero entries in each column is very small. Algorithm 3.3 escribes ILDUC with the Bunch-Kaufman pivoting (terme ILDUC-BKP). Since ILDUC uses a elaye-upate strategy, notice that the two columns must be upate before proceeing with the search in the algorithm. ALGORITHM 3.3. ILDUC-BKP - Incomplete with the Bunch-Kaufman pivoting metho 1. For 'Fwzx Do : 2. Loa an upate t 6587f. Let > > t} /587S > > «an > t G >. 3. If > t >H I Then 4. Let ŽK. 5. Else b 6. Loa an upate t /587S } G. Let q1)2y3 a G > t ag >. 7. If > t 6 > H I Then 8. ŽK 9. Else If t GfG Then H I 1. ŽK ; interchange the -th an the D -th rows an columns. 11. Else 12. Ž & ; interchange the Jh C PQ -th an the D -th rows an columns. 13. En If 14. En If

6 8 N. LI AND Y. SAAD 15. Perform ILDUC elimination process using the i pivot 16. EnDo 3.3. Implementation. Similarly to ILUTP in SPARSKIT [2], ILDUC-DP an ILDUC- BKP use a permutation array along with a reverse permutation array to hol the new orering of the variables. This strategy ensures that uring the elimination the matrix elements are kept in their original labeling. For ILDUC with pivoting, new strategies are neee to aress the two implementation ifficulties mentione earlier in this section ue to pivoting. First, we nee to efficiently access /587S ; for any to calculate the -th column of. In the implementation of ILUC, recall that an x -size array M šdy is maintaine so that hdy 6J=LQ always points to the first element with a row inex greater than in column. This also requires that the elements in a column of be store in the orer of increasing row inices. However, with pivoting enable, at the time when the -th column is calculate, the orer of the elements is unknown as they may be repositione later ue to pivoting. Thus, this metho will not work with pivoting. Our new strategy to hanle this issue is as follows. Observe that the positions of elements in section M; ± 7/ ;, M, will not be change after step n. We use M šdy 6J LQ to recor the number of nonzero elements in section ; P± 7/ ;. Since the nonzero elements in ; ; are store compactly in a linear buffer, we always store the nonzero elements in section ; ± 7/ ; in the first M šdy 6J=LQ positions in the buffer. In this way, the nonzero elements in } ; are continuously store (although they may be in any orer) starting from position M šdy 6J=LQ C² in the linear buffer, which can be efficiently accesse. After ; is use to calculate the -th column of, we nee to ensure the above property for the next step. Specifically, we nee to scan the linear buffer starting from position M šdy 6J=LQ Cq to access the nonzero elements in ;. If ; is zero, we o nothing. Otherwise, it can be locate uring the scan. We then swap ; with the elements store at position M šdy 6J=LQ an let M šdy 6J=LQUwBK hdy 6J=LQlC. Thus, at step -C, the nonzero elements in section /587S } ; are guarantee to be store continuously starting at position M šdy 6J=LQ as well. Secon, we nee to access some rows of but it is store column-wise. In ILUC, recall that a x -size array M =L is carefully esigne an maintaine such that all elements in the -th row of are guarantee to form a linke list when neee at the -th step. This linke list is embee in M =L. However, when pivoting is allowe, the availability of only the -th row of is not enough. For any ³E, the ³ -th row may be interchange with the -th row at step ue to pivoting. Therefore, we nee to maintain a linke list for each row of with a row inex greater than or equal to. We can also use strategies such as preorering an equilibration to further improve the stability of our methos. We apply the Reverse Cuthill-McKee (RCM) algorithm to preorer the matrices [7] in our implementation. A matrix is equilibrate if all its rows an columns have the same length in some norm. We use the metho propose by Bunch in [5] to equilibrate a symmetric matrix so that its rows/columns are normalize uner the max-norm. It is worth pointing out that for sparse symmetric matrices the two ifficulties in ILDUC exist in the equilibration metho as well. We aress them by using the same bi-inex ata structure in our implementation. 4. Experiments. In this section, we compare the performances of ILDUC, ILDUC with e pivoting (ILDUC-DP), an ILDUC with the Bunch-Kaufman pivoting (ILDUC-BKP). The computational coes were written in C, an the experiments were conucte on a 1.7GHz Pentium 4 PC with 1GB of main memory. All coes were compile with the -O3 optimization option. We teste ILDUC, ILDUC-DP, an ILDUC-BKP on 11 symmetric inefinite matrices selecte from the Davis collection [8]. Some generic information on these matrices is given

7 " $ CROUT VERSIONS OF ILU FACTORIZATION WITH PIVOTING 81 in Table 4.1, where x is the matrix size an x xz{ is the total number of nonzeros in a full matrix. In the tests, artificial right-han sies were generate, an GMRES(6) was use to solve the systems using a zero initial guess. The iterations were stoppe when the resiual norm was reuce by 8 orers of magnitue or when the maximum iteration count of 3 was reache. The ual criterion ropping strategy was use for all preconitioners, i.e., any element of column whose magnitue was less than a tolerance < > > /587S > > 7 was roppe; an only Lfil largest elements were kept. The parameter Lfil was selecte as a multiple of the ratio y zµ, the average number of nonzero elements per column in the original matrix. We use a parameter to etermine the ropping tolerance an a parameter to set the value of Lfil : M y yµ. Nevertheless, even uner the same control parameters, the number of fill-ins may be very ifferent from metho to metho. To better compare the preconitioners, we use an inicator calle a fill-factor, i.e., the value of x xz{cj C gcu 8QfR4x xz{cj ŽQ, to represent the sparsity of the ILU factors. A goo metho shoul yiel convergence for a small fill-factor. TABLE 4.1 Symmetric Inefinite Matrices from the Davis Collection x Matrix x xz{ Source aug3cqp D PDE bloweya Cahn-Hilliar problem bratu D Bratu problem ixmaanl Dixon-Maany optimization example mario Discretization mario Discretization olesnik Straz po Ralskem mine moel sit Straz po Ralskem mine moel stokes Stokes equation tuma Mine moel tuma Mine moel Table 4.2 shows the results of the three preconitioners on the 11 matrices, where the RCM orering an equilibration were applie. "z" To ensure the preconitioners were comparable, we fixe the ropping tolerance to ó an artificially selecte a fill-in parameter for each preconitioner such that the resulting fill-factors were similar for each matrix. For references, we also teste the linear systems with ILUTP uner similar fill-in parameters. Since ILUTP oes not take avantage of symmetry an the ILUTP coe we use was written in FORTRAN, we only compare the convergence an ignore the execution time for ILUTP. In the table, the values in the fill fiel are the fill-factors. The symbol - in the fill fiel inicates that the preconitioner faile ue to a zero pivot encountere uring ILU. its enotes the number of iterations for GMRES(6) to convergence. A symbol - in the its fiel inicates that convergence was not obtaine in 3 iterations. sec. enotes the total time in secons use for each metho (Preconitioning time +GMRES(6) time). From the table, it is clear that ILUTP was not robust an ILDUC faile in most cases ue to zero pivots encountere. ILDUC with the Bunch-Kaufman is better than ILDUC with iagonal pivoting in general. Next we offer a few aitional comments on these performances. 1. aug3cqp, stokes64: For these two matrices, with a similar amount of fill-ins al- x xz{ for aug3cqp respectively), all three ILDUC-base methos converge. The number of iterations require for conver- lowe (e.g. $ x xz{, $ ¹ Tyx xz{, an $º "

8 82 N. LI AND Y. SAAD TABLE 4.2 Performances on Symmetric Inefinite Matrices:» ¼o½¾ ½/½ Matrix ILDUC ILDUC-DP ILDUC-BKP ILUTP Name fill its sec. fill its sec. fill its sec. fill its aug3cqp bloweya bratu ixmaanl mario mario olesnik sit stokes tuma tuma gence was similar for all methos (ILDUC-DP require slightly fewer iterations to converge for matrix stokes64). As expecte, the cost of ILDUC with pivoting is of the same orer as that of ILDUC. Even with larger fill-in factors, ILUTP i not converge. 2. bloweya, bratu3: For these two matrices, ILDUC-DP, ILDUC-BKP an ILUTP converge, but ILDUC faile ue to zero pivots encountere. Nevertheless, ILDUC- BKP an ILUTP ha better performances than ILDUC-DP. 3. mario1, mario2, tuma1: For these matrices, ILDUC-DP an ILDUC-BKP both converge, but ILDUC an ILUTP faile. ILDUC-BKP ha better performances than ILDUC-DP. With a similar or even less amount of fill-ins, ILDUC-BKP require much fewer iterations to converge. For example, for matrix tuma1, ILDUC- DP require 23 iterations to converge with a fill-factor of $ OyÀ, while ILDUC-BKP only require 71 iterations with a fill-factor of $ TŠ&. Table 4.3 compares ILDUC- DP an ILDUC-BKP with various fill-in factors on matrix tuma1. From the table, we see that ILDUC-BKP is more efficient than ILDUC-DP. Figure 4.1 (a) an (b) compare the preconitioning cost an the total cost of the two methos respectively. Preconitioning Time (s) ILDUC DP ILDUC BKP Total Time (s) ILDUC DP ILDUC BKP Fill in Factor (a) Fill in Factor (b) FIG Comparison on the preconitioning time an the total time for matrix tuma1

9 Â CROUT VERSIONS OF ILU FACTORIZATION WITH PIVOTING 83 TABLE 4.3 Matrix tuma1.» ¼o½¾ ½/½. Preconitioner GMRES(6) Total Metho fill sec. its sec. sec. ILDUC-DP ILDUC-BKP ILDUC-DP ILDUC-BKP ILDUC-DP ILDUC-BKP ILDUC-DP ILDUC-BKP ILDUC-DP ILDUC-BKP olesnik, sit1: ILDUC an ILUTP faile on these two matrices, but ILDUC-DP an ILDUC-BKP converge an ha similar performances. 5. ixmaanl: ILDUC-DP an ILUTP faile on this matrix. ILDUC-BKP ha a slightly better performance than ILDUC. 6. tuma2: ILDUC-BKP was the only preconitioning metho that converge for matrix tuma2. 5. Experiments with KKT Matrices. In this section, we test the preconitioners on 14 KKT matrices (in reference to the Karush-Kuhn-Tucker first-orer necessary optimality conitions for the solution of general nonlinear programming problems [14]) as shown in Table 5.1. KKT matrices have the form: u! Á Â Ã "Ä# an are often generate from equality an inequality constraine nonlinear programming, sparse optimal control, an mixe finite-element iscretizations of partial ifferential equations [14]. The 14 KKT matrices were provie by Little [18] an Haws [14]. Matrices P288 an P1448 were permute Á from their original form [19] in orer to obtain a more narrow banwith for the part. The experiments were conucte on a 866MHz Pentium III PC with 1GB of main memory. Table 5.2 summaries the results of ILDUC, ILDUC-DP, ILDUC-BKP, an ILUTP on these matrices. Equilibration was applie. From the table, we make the following observations. First, ILDUC-BKP has the best overall performance on these matrices. ILUTP solve 6 problems, ILDUC solve 7 problems, ILDUC-DP solve 8 problems, an ILDUC-BKP solve 12 problems. Secon, as expecte, the cost of ILDUC-BKP is of the same orer as that of ILDUC. This is evience in matrices CHOI-L, LCAV-S1, STIFF5, MASS4, MASS5 an MASS6, where the number of iterations for the two methos were ientical or very close. Thir, ILDUC-DP is the best metho for matrices P288 an P1488. Especially for P1488, ILDUC-DP is the only preconitioner that converge. However, for matrices such as CHOI-L, LCAV-S1, STIFF4 an MASS5, ILDUC-DP i not converge or require significantly more iterations to converge. Finally, although ILUTP ha the worst performance on these matrices an it i not take avantage of symmetry, it was the only metho that converge for matrix TRAJ33.

10 84 N. LI AND Y. SAAD TABLE 5.1 KKT Matrices x Matrix x xz{ Source P Magnetostatic problem (2D coarse iscretization) P Magnetostatic problem (2D fine iscretization) CHOI Particles in flui simulator (5 escening particles) CHOI-L Particles in flui simulator (5 escening particles) LCAV-S Full Navier-Stokes equations in L shape cavity OPT Optimization problem STIFF Stiffness problem STIFF Stiffness problem MASS Mass problem MASS Mass problem MASS Mass problem TRAJ Sparse optimal control problem TRAJ Sparse optimal control problem LNTS Sparse optimal control problem TABLE 5.2 Performances on KKT Matrices: ÅŽ¼oƾ Ç,» ¼½¾ ½ Matrix ILDUC ILDUC-DP ILDUC-BKP ILUTP Name fill its sec. fill its sec. fill its sec. fill its P P CHOI CHOI-L LCAV-S OPT STIFF STIFF MASS MASS MASS TRAJ TRAJ LNTS Conclusion. We have explore two symmetry-preserving pivoting methos an shown how to integrate them into a Crout version of the ILU factorization. As expecte, this implementation results in better quality symmetric incomplete factorization for symmetric matrices. The overhea associate with this implementation is not significant. In aition, the pivoting methos have been emonstrate to fit the unerlying ata structure use in ILUC. Our experiments show that the Bunch-Kaufman pivoting metho can be efficiently an effectively integrate with a sparse symmetric iterative solver. 7. Acknowlegements. We woul like to thank Leigh Little, John Haws, Masha Sosonkina an Ilaria Perugia for proviing many of the test matrices use in the numerical tests Section 4 an 5. We also thank the anonymous referees for their helpful comments on an earlier

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