### Table 11: Model problem of Gear{Saad, N = 80, rtol = atol = 10?6

### TABLES TABLE I. Orbital energies for the oxygen dimer, from Chelikowsky, Troullier, Wu, and Saad (1994). FD-12 refers to high-order FD calculations in a 12 au box. PW-12 and PW-24 refer to plane-wave calculations with supercells of 12 and 24 au on a side. Energies are in eV.

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### Table 5 shows the performance statistics of BILUTM of Saad and Zhang [41] for solving the WIGTO966 matrix with the parameters used by MDRILU in Table 2. Although BILUTM converged for the last three sets of parameters, it did not converge for the rst three sets of parameters. MDRILU is more robust than BILUTM for solving the WIGTO966 matrix. BILUTM is also more expensive than MDRILU to construct. Thus the dual reordering strategy does have advantages in multilevel factorizations.

in A Multilevel Dual Reordering Strategy for Robust Incomplete LU Factorization of Indefinite Matrices

"... In PAGE 14: ... Table5 : Test results of BILUTM for solving the WIGTO966 matrix using parameters corresponding to those in Table 2. RAEFSKY4 matrix.... ..."

### Table 4.4: Summary of the characteristics of each test problem in uences the conditioning of the matrices, and since this conditioning is relevant to the performance of our solution techniques, we have created two instances of CEGB2802 with signi cantly di ering spectra. The patterns of CEGB2802 and LOCK3491 arise from structural engineering problems; MAN5976 comes from deformation problems; MAT32 and MAT33 are nite element matri- ces generated with the SPARSKIT software (see Saad (1990)), TORSION1 and NOBNDTOR are quadratic elastic torsion problems arising from as an obstacle problem on a square, NET3 is a very ill-conditioned example which arises from the optimization of a high- pressure gas network and CBRATU3D is obtained by discretizing a complex 3D PDE problem in a cubic region. A summary of each problem characteristics is given Table 4.4. The degree of overlap is the average number of elements containing each variable, that is the sum of the element dimensions divided by the order of the matrix.

### Table 3. Scattering of an incident plane wave on a circular cylinder of radius a

"... In PAGE 12: ... Numerical results A matrix free iterative solver has been implemented by utilizing the two-dimensional version of the high-frequency integrator described in the preceding sections in con- junction with the GMRES algorithm (Saad amp; Schultz 1986). Table3 shows results produced by means of this two-dimensional solver on a 1.5 GHz PC, applied to a circular cylinder of radius a.... ..."

### Table 6: Test results of RILUMc with di erent uniform size blocks.

in RILUM: A General Framework for Robust Multilevel Recursive Incomplete LU Preconditioning Techniques

1999

"... In PAGE 15: ... This is another way of testing scalability since di erent size blocks may be needed in certain applications, such as in the implementation of RILUMc on parallel computers with di erent number of processors. The results are listed in Table6 . It can be seen that the convergence rate was not a ected by the block sizes.... In PAGE 16: ...The results of Table6 again suggest that large size blocks accelerate convergence rate of multilevel ILU preconditioning techniques, a fact that has been reported by Saad and Zhang [40]. It is interesting to note that using large size blocks did not increase the sparsity ratio very much, in some cases, it even reduced the sparsity ratio.... ..."

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### Table 2. The product of terrain and geology classi cations Terrain classes

2004

"... In PAGE 4: ...Alexandre Sorokine and Thomas Bittner Table 1. Hierarchical Arrangement of Subsections [3, Table2 , p. 16] I.... In PAGE 8: ...ig. 4(a) and Fig. 4(b) respectively. The product of these classi cations is depicted in Table2 , with terrain classes as columns and geologic classes as rows. Each cell of the table contains the number of individuals that instantiate corresponding classes of both hierarchies.... In PAGE 8: ... The class \Volcanics quot; that violates the weak supplementation principle was moved from the terrains hierarchy into the geologic classes hier- archy (Fig. 4 and Table2 ). This is a more natural place for this class because there is already a class with an identical name.... In PAGE 12: ... Most likely N will be greater than the number of classes that can be instantiated by the in- stances. In our example most of the cells of the Table2 are empty indicating that there are no individuals that instantiate classes in either classi cation tree. The reason for it is that certain higher-level classes do not demonstrate as much diversity on the studied territory as other classes do.... ..."

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### Table 3 Performance comparison of solvers in VIPRE

"... In PAGE 15: ... 1. The transient was performed using the direct, ADI, and BICGSTAB solver for the inner iteration and results are summarized in Table3 . Parametrics were performed on the convergence criteria and the best results were obtained with 10 4 on the relative infinite l1 norm of both the inner and outer iteration.... In PAGE 15: ... All cases shown here were executed on a single processor of a Sun Ultrasparc II. The total number of outer and inner iterations required during the transient simulation is shown in Table3 , as well as the computational profile for each of the primary modules. Since an iterative solution is most practical for this problem, it is most worthwhile to focus on comparisons of the ADI and BICGSTAB results.... In PAGE 15: ... In fact, the linear solution provided by BICGSTAB with a tolerance of 10 4 is su ciently accurate to make the number of outer iterations nearly the same as the machine precision DIRECT inner solution. Finally, it is worth noting in Table3 that the average number of inner iterations per outer is about 50% lower with BICGSTAB than with ADI, which is a direct result of the superior error reduction per iteration for BICGSTAB. This is a general advantage of preconditioned Krylov methods over stationary iterative methods such as ADI (Saad, 1996).... ..."

### 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.... ..."

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