### Table 3 NASASRB: Run-times in milliseconds for the irregular sparse matrix-vector product (matvec), its gather part (gather) and its computation part (comp), and for the inner product (ddot).

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### Table 5 SHYY161: Run-times in milliseconds for the irregular sparse matrix-vector product (matvec), its gather part (gather) and its computation part (comp), and for the inner product (ddot).

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### Table 1 shows the results of the computation on an Alliant FX/80 of the eight eigenpairs with largest real parts of a random sparse matrix of order 1000. The nonzero o -diagonal and the full diagonal entries are in the range [-1,+1] and [0,20] respectively.

"... In PAGE 5: ... Finally, these codes have been used as a platform for the implementation of the uniprocessor version of Level 3 BLAS on the BBN TC2000 (see next Section). We show in Table1 the MFlops rates of the parallel matrix-matrix multiplication, and in Table 2 the performance of the LU factorization (we use a blocked code similar to the LAPACK one) on the ALLIANT FX/80, the CRAY-2, and the IBM 3090-600J obtained using our parallel version of the Level 3 BLAS. Note that our parallel Level 3 BLAS uses the serial manufacturer-supplied versions of GEMM on all the computers.... In PAGE 6: ... This package is available without payment and will be sent to anyone who is interested. We show in Table1 the performance of the single and double precision GEMM on di erent numbers of processors. Table 2 shows the performance of the LAPACK codes corresponding to the blocked LU factorization (GETRF, right-looking variant), and the blocked Cholesky factorization (POTRF, top-looking variant).... In PAGE 8: ... The second part concerned the performance we obtained with tuning and parallelizing these codes, and by introducing library kernels. We give in Table1 a brief summary of the results we have obtained: One of the most important points to mention here is the great impact of the use of basic linear algebra kernels (Level 3 BLAS) and the LAPACK library. The conclusion involves recommendations for a methodology for both porting and developing codes on parallel computers, performance analysis of the target computers, and some comments relating to the numerical algorithms encountered.... In PAGE 12: ... Because of the depth rst search order, the contribution blocks required to build a new frontal matrix are always at the top of the stack. The minimum size of the LU area (see column 5 of Table1 ) is computed during during the symbolic factorization step. The comparison between columns 4 and 5 of Table 1 shows that the size of the LU area is only slightly larger than the space required to store the LU factors.... In PAGE 12: ... The minimum size of the LU area (see column 5 of Table 1) is computed during during the symbolic factorization step. The comparison between columns 4 and 5 of Table1 shows that the size of the LU area is only slightly larger than the space required to store the LU factors. Frontal matrices are stored in a part of the global working space that will be referred to as the additional space.... In PAGE 12: ... In a uniprocessor environment, only one active frontal matrix need be stored at a time. Therefore, the minimum real space (see column 7 of Table1 ) to run the numerical factorization is the sum of the LU area, the space to store the largest frontal matrix and the space to store the original matrix. Matrix Order Nb of nonzeros in Min.... In PAGE 13: ... In this case the size of the LU area can be increased using a user-selectable parameter. On our largest matrix (BBMAT), by increasing the space required to run the factorization (see column 7 in Table1 ) by less than 15 percent from the minimum, we could handle the ll-in due to numerical pivoting and run e ciently in a multiprocessor environment. We reached 1149 M ops during numerical factorization with a speed-up of 4.... In PAGE 14: ...ack after computation. Interleaving and cachability are also used for all shared data. Note that, to prevent cache inconsistency problems, cache ush instructions must be inserted in the code. We show, in Table1 , timings obtained for the numerical factorization of a medium- size (3948 3948) sparse matrix from the Harwell-Boeing set [3]. The minimum degree ordering is used during analysis.... In PAGE 14: ... -in rows (1) we exploit only parallelism from the tree; -in rows (2) we combine the two levels of parallelism. As expected, we rst notice, in Table1 , that version 1 is much faster than version 2... In PAGE 15: ... Results obtained on version 3 clearly illustrate the gain coming from the modi cations of the code both in terms of speed-up and performance. Furthermore, when only parallelism from the elimination tree is used (see rows (1) in Table1 ) all frontal matrices can be allocated in the private area of memory. Most operations are then performed from the private memory and we obtain speedups comparable to those obtained on shared memory computers with the same number of processors [1].... In PAGE 15: ... Most operations are then performed from the private memory and we obtain speedups comparable to those obtained on shared memory computers with the same number of processors [1]. We nally notice, in Table1 , that although the second level of parallelism nicely supplements that from the elimination tree it does not provide all the parallelism that could be expected [1]. The second level of parallelism can even introduce a small speed down on a small number of processors as shown in column 3 of Table 1.... In PAGE 15: ... We nally notice, in Table 1, that although the second level of parallelism nicely supplements that from the elimination tree it does not provide all the parallelism that could be expected [1]. The second level of parallelism can even introduce a small speed down on a small number of processors as shown in column 3 of Table1 . The main reason is that frontal matrices must be allocated in the shared area when the second level of parallelism is enabled.... In PAGE 17: ...5 28.2 Table1 : Results in Mega ops on parallel computers. In Table 1, it can be seen that the performance of the program on the Alliant FX/80 in double precision is better than the performance of the single precision ver- sion.... In PAGE 17: ...2 Table 1: Results in Mega ops on parallel computers. In Table1 , it can be seen that the performance of the program on the Alliant FX/80 in double precision is better than the performance of the single precision ver- sion. The reason for this is that the single precision mathematical library routines are less optimized.... In PAGE 19: ... A comparison with the block preconditioned conjugate gradient is presently being investigated.In Table1 , we compare three partitioning strategies of the number of right-hand sides for solving the system of equations M?1AX = M?1B, where A is the ma- trix BCSSTK27 from Harwell-Boeing collection, B is a rectangular matrix with 16 columns, and M is the ILU(0) preconditioner. Method 1 2 3 1 block.... In PAGE 26: ...111 2000 lapack code 0.559 Table1 : Results on matrices of bandwith 9.... In PAGE 30: ... We call \global approach quot; the use of a direct solver on the entire linear system at each outer iteration, and we want to compare it with the use of our mixed solver, in the case of a simple splitting into 2 subdomains. We show the timings (in seconds) in Table1 on 1 processor and in Table 2 on 2 processors, for the following operations : construction amp; assembly : Construction and Assembly, 14% of the elapsed time, factorization : Local Factorization (Dirichlet+Neumann), 23%, substitution/pcg : Iterations of the PCG on Schur complement, 55%, total time The same code is used for the global direct solver and the local direct solvers, which takes advantage of the block-tridiagonal structure due to the privileged direction. Moreover, there has been no special e ort for parallelizing the mono-domain approach.... ..."

### Table 4 Test matrices from UF Sparse Matrix Collection.

"... In PAGE 17: .... Experimental results. In this section, we present the experimental results on some test matrices from the collection at the University of Florida1. Table4 lists the name, a description, the order, and the number of nonzeros in the lower triangular part of the test matrices. All experiments were performed on a SGI Origin 3800 system, with 32 processors.... ..."

### Table 1 Operation count for applying the operator R2, assuming a sparse LU factorization of Q( 0) is available. 4 sparse triangular solves 2 symmetric sparse matrix-vector products Mz 3 skew symmetric sparse matrix-vector products Gz 5 saxpy operations. The sparse LU factorization of Q( 0) needs to be computed only once; then R2 can be applied repeatedly. No further LU factorizations are needed unless the target 0 is changed. If 0 is real, then all operations are real and we may use R2 instead of R1.

2000

"... In PAGE 11: ... If 0 is neither real nor purely imaginary, there is a trick that halves the cost of computing R1( 0; W) [30]. In this case one easily veri es that R1( 0; W) is pro- portional to the imaginary part of R2( 0; W), so we can obtain R1( 0; W) at the costs listed in Table1 (with complex arithmetic) by simply computing R2( 0; W) and keeping the imaginary part. The symplectic operators S1 and S2 can be applied similarly.... ..."

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### Table 2: Performance of the hash-based algorithm on a test sample of 14 sparse matrices

2001

"... In PAGE 4: ... For RSA matrices, only the lower triangular part is stored and nnz represents its number of nonzero elements. Table2 shows some statistics for the hash-based algorithm. The second and third columns of the table are the vertex and edge compression rates, respectively, achieved by the algorithm.... In PAGE 5: ... However, since for most matrices the time required to nd indistinguishable nodes is negligible relative to the time it takes to solve the system, this is an issue that has not received much attention. This can be easily understood from the results of Table2 . Therefore, potential improvements to the simple choice (2) may exist but will not be explored in this paper.... In PAGE 7: ... EndDo 18. EndDo Table 3 shows statistics that are similar to those shown in Table2 for the hash-based algorithm. A grouping tolerance of = 0:8 was used for the tests.... In PAGE 9: ...00 2.08 Table 4: Performance of the hybrid algorithm on a test sample of 14 sparse matrices Table 4 shows statistics that are similar to those shown in Table2 and in Table 3. The same grouping tolerance of = 0:8 as in Table 3 was used.... ..."

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