### Table 2: Condition numbers of the three di erent Schur complement approximations for Cases 1{5.

"... In PAGE 15: ...ia ILUT. The results are shown in Figure 9. Method M1 is far superior to methods M2 and M3. The overall solution time for method M1 is not much greater than for Case 3 despite the fact the the condition number of ~ S3 (see Table2 ) is ve orders of magnitude higher. The solution time for method M3 is the nearly same as for method M1, but the preconditioning time is so great that M3 is unusable in practice.... ..."

### Table 2: Condition numbers of the three di erent Schur complement approximations for Cases 1{5.

"... In PAGE 15: ... Method M1 is far superior to methods M2 and M3. The overall solution time for method M1 is not much greater than for Case 3 despite the fact the the condition number of ~ S3 (see Table2 ) is ve orders of magnitude higher. The solution time for method M3 is the nearly same as for method M1, but the preconditioning time is so great that M3 is unusable in practice.... ..."

### Table 1 Rate of convergence of the Schur complement method.

94

### Table 2. Preconditioned Schur complement, M3b, the 2-cyclic iteration and the fast Schur complement

1994

"... In PAGE 12: ...n Section (3.3.1). In Table2 , preconditioner 3a is the Schur complement preconditioned with the matrix D from Section 3.... ..."

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### Table 4 Spectral properties of Schur complement matrices spectral properties of Schur complement matrices NE ?A=A

### Table 3: Number of zero diagonals in the original matrix and in the Schur complement matrices at each level. The parameters used are the same as those in Table 2.

1999

"... In PAGE 8: ...05 Table 2: Numerical results from solving the coating matrices by the multi-level block ILU preconditioning technique. Table3 shows the number of the rows of the matrix (Schur complements) with zero diagonal values at each level, corresponding to the parameters listed in Table 2. We see that the original matrices contain a substantial number of rows with zero diagonal elements.... In PAGE 9: ... It is shown in [22] that a row with a zero diagonal element is to remain such a status if it is not excluded as a node with links to the nodes of the independent set. However, the situation seems to be changed with the multi-level block ILU factorization, since the data in Table3 indicates that the number of zero diagonals decreases rapidly as the number of levels increases. According to our discussions in Section 3, the results in Table 3 indicate that the nodes (rows) that were excluded from the block independent set are modi ed on the coarse levels or the nature of the (relative) diagonal threshold tolerance is changed on the coarse levels.... In PAGE 9: ... However, the situation seems to be changed with the multi-level block ILU factorization, since the data in Table 3 indicates that the number of zero diagonals decreases rapidly as the number of levels increases. According to our discussions in Section 3, the results in Table3 indicate that the nodes (rows) that were excluded from the block independent set are modi ed on the coarse levels or the nature of the (relative) diagonal threshold tolerance is changed on the coarse levels. Such modi cations or changes result in the inclusion of those nodes in the coarse level independent set that were excluded from the ne level independent sets.... In PAGE 9: ... Such modi cations or changes result in the inclusion of those nodes in the coarse level independent set that were excluded from the ne level independent sets. Table3 also shows that the higher the sparsity ratio of the preconditioner, the faster the zero diagonals disappear. This information implies that the multi-level block ILU factorization procedure has the ability to stabilize the ILU factorization process by modifying small diagonal values in the approximate Schur complement.... ..."

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### Table 2. Number of GMRES iterations for the related system (6) using an ILU(0) or AMG splitting of the (1,1) block and struc- tured probing to approximate the Schur complement, for various levels of h-refinement on a single non-linear iteration in the metal deformation problem. Dashes indicate insufficient memory to run that particular combination.

2006

"... In PAGE 16: ... For each of these problems, we use structured probing with the prime divisor, balanced and greedy colorings and an ILU(0) factorization of the approximate Schur complement. Table2 shows the number of GMRES iterations necessary to solve the related system (6) to a tolerance... In PAGE 17: ... If the splitting is poor, a better approximation to the Schur complement is unlikely to yield a significant improvement in convergence. We see this effect in Table2 . The difference between the prime divisor coloring, which uses more vectors than the balanced or greedy colorings for a given stencil, is most pronounced when we use the exact splitting, F = A.... ..."

### Table 1. Eigenvalues of the Schur complement, for streamline-upwind discretization.

1996

"... In PAGE 11: ... We rst consider the bounds of Theorem 1. Table1 shows the extreme real parts and max- imum imaginary parts of the generalized eigenvalues (2.7) of the Schur complement operator, for = 1=10 and 1=100 with the streamline-upwind discretization, on three meshes.... In PAGE 11: ... The analysis also shows that the real parts and largest imaginary parts of the eigenvalues are bounded independently of ; the bound for the smallest real part is proportional to 2. The data of Table1 are in agreement with the upper bounds. Figure 5 plots the smallest real parts on a logarithmic scale, for the streamline-upwind discretization on a 64 64 grid and = 1=20, 1=40, 1=80, and 1=160.... ..."

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### Table 2: e ciency with Gauss-Seidel or dual Schur complement

"... In PAGE 12: ... But the granularity of the tasks is much larger, because the local task con- sists of a forward/backward substitution using the dense skyline factoriza- tion instead of the sparse initial matrix. Table2 presents the results with the implementation of a V-cycle, with 10 smoothing iterations at the ne grid level, and either 200 Gauss-Seidel iterations or 20 dual Schur complement iterations for solving the coarse grid problem, on a 128-processor iPSC-860 machine. Table 2: e ciency with Gauss-Seidel or dual Schur complement... ..."

### Table 2: Schur complement system (2.25)

"... In PAGE 22: ... Only the numbers for the preconditioned systems are treated. The numbers of itera- tions di er slightly from those of Table2 , as a slightly di erent preconditioner C 1 V was used for the inversion of the local single layer potentials e Vi;h. The computational times are a little bit higher than in the example of Table 2 as the computations have been executed on a di erent cluster.... ..."

Cited by 1