### Table 4: CPU Times (seconds) for Uzawa-like method with BBT preconditioner solved by: multifrontal (MF) or IC conjugate gradient (PCG).

1996

"... In PAGE 6: ... Incomplete cholesky preconditioned conjugate gradient. In Table4 we analyze the solution of problem (7) for the constant permeability case ( quot; = 0) with the two proposed solvers for BBT . We give the number of outer iterations and the total solution times.... ..."

Cited by 2

### Table 1: The Preconditioned Conjugate Gradient (PCG) algorithm in solving Ax = b with preconditioner M and initial guess x0.

2006

"... In PAGE 7: ....2.1 CG-based Inner-Outer Iteration Methods We suggest the PCG-PCG inner-outer algorithm given in Table 3. The algorithm is given as a modification to the reference PCG algorithm from Table1 , and therefore only the lines that change are written out. The inner PCG is in SP where SP data and arithmetic are colored in blue.... In PAGE 8: ..... 10 [check convergence and break if done] 11 end 12 x = xi Table 3: PCG-PCG algorithm with inner PCG in SP (SP data and arithmetic colored in blue). Only the lines that change from the reference PCG algorithm on Table1 are given. in convergence that is characteristic of the steepest descent algorithm.... In PAGE 16: ...930 119.23% Table1 0: Time to solution of sequential SuperLU in single and double precision for selected sparse matrices on the Intel Woodcrest with reference and optimized BLAS. The performance of the mixed precision solver is usually very close to that of the sin- gle precision mainly because the cost of each iteration is often negligible if compared to the cost of the matrix factorization.... ..."

### Table 2: Parallel performance for various steps of the preconditioned conjugate gradient implementations (35 35 35 subdomain grid). # proc 27 64 125 216

"... In PAGE 4: ... These experiments were performed on an Apple Xserve Power PC G5 platform. We depict in Table2 the preconditioner setup time for both Md and... ..."

### Table 6: Number of preconditioned conjugate gradient iterations for the heterogeneous and anisotropic problems when the number of subdomains and the subdomain mesh size is varied.

### Table 3: MbytesL1 for a single iteration of preconditioned conjugate gradient solver in POP using the test grid. The values of W SLP and W SLM are in Kbytes.

2007

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### Table 1 shows the number of iterations required by our algorithm to reduce the 2-norm of the residual by a factor of 106 when solving these two sequences of problems. Since the original matrix, K, is always symmetric and positive- de nite, the \no preconditioning quot; column shows the number of iterations re- quired by a conjugate gradient solver for each problem. The iteration counts in the other columns are obtained using GMRES with the preconditioner outlined over the previous sections. (See, for example, [1, 12, 20] for a description of conjugate gradients, GMRES and similar iterative methods.)

in A New Parallel Domain Decomposition Preconditioner II: Generalization to a Mesh-Free Parallel Solver

2000

"... In PAGE 12: ... Table1 : Iteration counts, using the proposed algebraic preconditioner, for the sequences of matrices obtained from piecewise linear discretizations of Problems 1 and 2. Note that without preconditioning the number of iterations doubles when- ever n is quadrupled | implying that the condition numbers for these two se- quences of matrices are approximately proportional to n (as expected since, here, n = O(h?2)).... ..."

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### Table 8. Performance of preconditioned conjugate gradient (PCG, old algorithm) and wavefront recursion (WFR, new algorithm) for POP barotropic solve. Times are elapsed seconds used to update the barotropic equations for four timesteps for a 100x116 problem, omitting the cost of the initial timestep.

"... In PAGE 42: ... Performance results reported here were obtained on the seaborg IBM SP machine at the NERSC facility in California. Performance results for the PCG and WFR solvers are presented in Table8 , and their respective initialization costs in Table 9. Because of a variety of difficulties, complete results have been obtained only for a relatively coarse resolution problem with 100 longitude x 116 latitude zones.... ..."

### Table 9. Initialization performance of preconditioned conjugate gradient (PCG, old algorithm) and wavefront recursion (WFR, new algorithm) for POP barotropic solve. Times are elapsed seconds used to update the barotropic equations for the initial timestep for a 100x116 problem.

"... In PAGE 42: ... Performance results reported here were obtained on the seaborg IBM SP machine at the NERSC facility in California. Performance results for the PCG and WFR solvers are presented in Table 8, and their respective initialization costs in Table9 . Because of a variety of difficulties, complete results have been obtained only for a relatively coarse resolution problem with 100 longitude x 116 latitude zones.... ..."

### Table 4.6: Jacobi preconditioned conjugate gradient iterations required for

1998

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