### Table 2: Numerical results from solving the coating matrices by the multi-level block ILU preconditioning technique.

1999

"... In PAGE 7: ... Convergence was achieved when the 2-norm residual was reduced by 7 orders of magnitude. Table2 lists some statistics of the computations and the parameters used. In Table 2 and in all other tables followed, \iter.... In PAGE 7: ... Table 2 lists some statistics of the computations and the parameters used. In Table2 and in all other tables followed, \iter. quot; denotes the number of preconditioned iterations, \fact.... In PAGE 8: ... In some tables we use \{ quot; to indicate a lack of convergence. The results in Table2 show that our preconditioned iterative solver converged fast for solving the given matrices with the chosen parameters. We mention that, once the parameter quot; was chosen, the choices of the other parameters were not very critical in terms of achieving successful convergence, except in the case for solving the BKNIFE-01 matrix.... In PAGE 8: ...Table 2: Numerical results from solving the coating matrices by the multi-level block ILU preconditioning technique. Table 3 shows the number of the rows of the matrix (Schur complements) with zero diagonal values at each level, corresponding to the parameters listed in Table2 . We see that the original matrices contain a substantial number of rows with zero diagonal elements.... In PAGE 10: ... It can be seen that the multi-level block ILU preconditioner with a diagonal threshold strategy actually used less memory to achieve faster convergence rates than that yielded by the same preconditioner without a diagonal threshold strategy. Figure 5 shows the convergence history of the multi-level block ILU preconditioner for solving the PDF-1 matrix with di erent diagonal threshold tolerance , the other parameters were chosen the same as those used in Table2 . We see the suitable values of for this test problem are between 0:1 and 0:05.... ..."

Cited by 8

### Table 2. Statistics of the multi-level preconditioner

"... In PAGE 5: ... In the table, the 5th and 6th columns indicate the total number of Newton iterations and Krylov iterations used in the Newton loop and by the GMRES solver, respectively, before the simulation convergence is reached. The performance of the proposed multi-level preconditioner is summarized in Table2 on the same set of designs, where the total number of Krylov iterations corresponds to that is used by the top- level FGMRES solver. Different from the previous experiments, we have adopted a multi-level structure where the largest sub- problem size on the next level is approximately one fourth of that on the current level.... ..."

### Table 4. Multi-Level Threshold Results

"... In PAGE 7: ... Multi-Level Threshold Results Threshold Level 4 3 2 1 Value #28in Meters#29 30 18 8 2 Table 5. Multi-Level Threshold Values Table4 shows the number of PDUs generated and the average error in AOI and SR when our multi- level threshold dead reckoning algorithm is used. The threshold values used in di#0Berent levels are listed in Table 5.... In PAGE 7: ... The threshold values used in di#0Berent levels are listed in Table 5. It can be seen from Table4 that there is a great reduction in the average error in SR, compared to the average error in AOI. In our algorithm, if entity A is in entity B apos;s SR, a minimum threshold will be used in the dead reckoning so that B will receive A apos;s update packets most frequently.... ..."

### Table 2. Statistics of multi-level preconditioner

"... In PAGE 5: ... In the table, the 5th and 6th columns indicate the total number of Krylov iterations used in the Newton loop and by the GMRES solver, respectively, before the convergence is reached. The performance of the proposed multi-level preconditioner is summarized in Table2 . on the same set of designs, where the total number of Krylov iterations corresponds to that is used by the top- level FGMRES solver.... ..."

### Table 4: Comparison of the multi-level block ILU preconditioners for solving the TAIL-01 matrix with and without a diagonal threshold strategy.

1999

"... In PAGE 9: ... The parameters used are the same as those in Table 2. In Table4 we compare the results for solving the TAIL-01 matrix by the multi-level block ILU preconditioner with a diagonal threshold strategy ( = 0:1) and without a diagonal thresh- old strategy ( = 0). It shows that the multi-level block ILU preconditioner is not robust without a diagonal threshold strategy for solving the given matrix.... In PAGE 10: ...Saad and Zhang An interesting observation on the results in Table4 shows that, for the given parameters, using a diagonal threshold strategy entails a slightly larger sparsity ratio than not using a diagonal threshold strategy. This can be explained as the result of excluding certain nodes with small diagonal values from the independent set.... In PAGE 10: ... The consequence is an increase in the sparsity ratio and the memory cost. For example, corresponding to the rst data row in Table4 , the multi-level block ILU preconditioner without the diagonal threshold strategy has only 4 levels, the sizes of the independent sets at each levels are 1333; 634; 281 and 204. On the other hand, the use of the diagonal threshold strategy requires 6 levels, the sizes of the independent sets are 1165; 584; 346; 188; 109 and 55.... In PAGE 10: ... On the other hand, the use of the diagonal threshold strategy requires 6 levels, the sizes of the independent sets are 1165; 584; 346; 188; 109 and 55. However, the results in Table4 clearly indicates that the use of a diagonal threshold strategy ensures convergence and increases robustness. It can be seen that the multi-level block ILU preconditioner with a diagonal threshold strategy actually used less memory to achieve faster convergence rates than that yielded by the same preconditioner without a diagonal threshold strategy.... ..."

Cited by 8

### Table 4. OPC of the reordered bcsstk29 matrix when multi-level band GA is used for all levels of nested dissection. Classical multi-level FM yields an OPC of 3.43e +8in 0.74s.

2006

"... In PAGE 7: ... ParMeTiS, however, uses MPI, and runs fully in parallel. Table4 provides some results for graph bcsstk29. These results show that GA converges quite well, and that quality can be improved by increasing computation time and/or population size.... ..."

Cited by 3

### Table 1: Multi-level benchmarks: number of literals in factored form

"... In PAGE 16: ... The following tools were used for generating the layouts: the GORDIAN package for placement [14], the TIMBERWOLF global router [15], and the YACR detailed router [21]. Table1 goes here. In both cases, the technology-independent optimizations were performed using the MIS pro- gram.... ..."

### Table 4: Power consumption for multi-level logic.

1994

"... In PAGE 21: ... The encoded machines were synthesized and mapped using the sis package and a gate library from industry. Table4 summarizes the results. Columns 3 and 5 give the % power reduction of MWHD and LPSA over JEDI, respectively.... In PAGE 21: ...ackage and a gate library from industry.Table 4 summarizes the results. Columns 3 and 5 give the % power reduction of MWHD and LPSA over JEDI, respectively. Table4 shows that in general the lowpower state assignment produces better results than JEDI and the MWHD encoding. An average of 14.... ..."

Cited by 2

### Table 2 shows the performance improvement of the multi-level optimization over the usual optimization method.

"... In PAGE 10: ...5 W 0 0 100 200 300 400 CPU Time (msec) Figure 8: Convergence Curve for Bending Cow. Table2 : Performance Improvement by Multi-Level Optimization. 7, and lt; are the total CPU time (msec) without and with Multi Level Optimization respectively.... ..."

### Table 1: Specification of the multi-leveled articulatory features. The right-hand column gives the cardinality of each feature.

"... In PAGE 2: ... The right-hand column gives the cardinality of each feature. A set of 6 multi-leveled features as shown in Table1 is used in this work. The feature groups are self-explanatory other than static, which gives an indication of rate of spectral change, such as occurs during, for example, diphthongs.... ..."