### Table 4 Complexity Rules for Sequential Composition, Conditional Statements and Loops

"... In PAGE 9: ... Without loss of generality we assume all expressions use basic operations of the domain D, only. Table4 contains rules de ning sequential composition, conditional statements, and loops. For brevity, we omit the rules for static and cumulative measures.... ..."

### Table 1: Generalization accuracies obtained with the variational Kalman filter (vkf) and sequential variational inference (svi).

2003

"... In PAGE 6: ... We also use the pima diabetes data set from [16]3. Table1 compares the generalization accuracies (in fractions) obtained with the variational Kalman filter with generalization accuracies obtained with sequential variational inference. The probability of the null hypothesis, C8D2D9D0D0, that both classifiers are equal suggests that only the differences for the Balance scale and the Pima Indian data sets are significant, with either method being better in one case.... ..."

Cited by 2

### Table 1: Generalization accuracies obtained with the variational Kalman filter (vkf) and sequential variational inference (svi).

1997

"... In PAGE 6: ... We also use the pima diabetes data set from [16]3. Table1 compares the generalization accuracies (in fractions) obtained with the variational Kalman filter with generalization accuracies obtained with sequential variational inference. The probability of the null hypothesis, Pnull, that both classifiers are equal suggests that only the differences for the Balance scale and the Pima Indian data sets are significant, with either method being better in one case.... ..."

Cited by 2

### Table 1: Generalization accuracies obtained with the variational Kalman filter (vkf) and sequential variational inference (svi).

1997

"... In PAGE 6: ... We also use the pima diabetes data set from [16]3. Table1 compares the generalization accuracies (in fractions) obtained with the variational Kalman filter with generalization accuracies obtained with sequential variational inference. The probability of the null hypothesis, a4 a5 a9a8a11a10a12a10 , that both classifiers are equal suggests that only the differences for the Balance scale and the Pima Indian data sets are significant, with either method being better in one case.... ..."

Cited by 2

### Table 1 Iteration scaling with grid size. Nit = CNb e

"... In PAGE 6: ... Similar data is also shown by Burns [2] using diagonal preconditioning for a three dimensional problem with spatial domain decomposition. Table1 summarizes the iteration scaling for three solvers - generalized minimal residual (GMRES), stabilized biconjugate gradient (BiCGStab), and conjugate gradient squared (CGS) - using diagonal, Neumann, and least squares preconditioning 2. Assuming that the number of iterations scales as Nit = CNb e, Table 1 gives values for C and b for each solver/preconditioner combination.... In PAGE 6: ... Table 1 summarizes the iteration scaling for three solvers - generalized minimal residual (GMRES), stabilized biconjugate gradient (BiCGStab), and conjugate gradient squared (CGS) - using diagonal, Neumann, and least squares preconditioning 2. Assuming that the number of iterations scales as Nit = CNb e, Table1 gives values for C and b for each solver/preconditioner combination. All of the data shown in Table 1 were obtained using the AZTEC Krylov matrix solver library developed at Sandia National Laboratories in Albuquerque, New Mexico.... In PAGE 6: ... Assuming that the number of iterations scales as Nit = CNb e, Table 1 gives values for C and b for each solver/preconditioner combination. All of the data shown in Table1 were obtained using the AZTEC Krylov matrix solver library developed at Sandia National Laboratories in Albuquerque, New Mexico. The data in Table 1 shows that, of the three solvers considered , the BiCGStab solver requires the fewest number of iterations for a given preconditioner.... In PAGE 6: ... All of the data shown in Table 1 were obtained using the AZTEC Krylov matrix solver library developed at Sandia National Laboratories in Albuquerque, New Mexico. The data in Table1 shows that, of the three solvers considered , the BiCGStab solver requires the fewest number of iterations for a given preconditioner. BiCGStab also provides the slowest increase in the number of iterations, i.... In PAGE 6: ... A similar result was obtained by Burns [2]. The data in Table1 also shows that preconditioning has a bene cial e ect on the BiCGStab algorithm in terms of the iteration count. The additional cost of the Neumann and least squares preconditioners, however, tend to negate this bene t and 2Using a Krylov subspace size of 30 for GMRES and a polynomial order of 2 for Neumann and least... ..."

### Table 1. Average speedup due to the sequential improvements.

"... In PAGE 2: ... In addition, the master checks the stop condition and sends a stop message to all workers if the stopping rule applies. 3 RESULTS The performance of the sequential improvement and parallelization relative to the accelerated sequential version was tested on four data- sets ( Table1 ) on a homogeneous Linux cluster of 15 nodes with 2 CPUs of 2.0GHz each, connected via Gigabit Ethernet.... In PAGE 2: ..., 1985) of evolution was used and the WAG model (Whelan and Goldman, 2001) for pro- tein sequences. The number of iterations in the OS was fixed (see Table1 ) to avoid runtime fluctuations caused by the stopping rule. Table 1 and Fig.... In PAGE 2: ... The number of iterations in the OS was fixed (see Table 1) to avoid runtime fluctuations caused by the stopping rule. Table1 and Fig. 2 display the average speedup from 10 independent runs for each dataset.... ..."

### Table 2. Parallel vs Sequential Time for the

"... In PAGE 5: ... For each of the sample problems, the lower and upper bounds of the measured pairwise distances were initialized from the input data and the bounds of unmeasured distances were initialized to 0 #09 A for the lower bounds and 99 #09 A for the upper bounds. In Table2 , we list the number of iterations and the average time taken for each iteration of the tetrangle inequality algorithm. All these problems were run on the Intel Paragon X#2FPS machine con#02gured for 32 processors.... In PAGE 5: ... All these problems were run on the Intel Paragon X#2FPS machine con#02gured for 32 processors. It can be observed from Table2 that while it took a parallel time of about 11 hours for one iteration of the tetrangle-inequality bound-smoothing algorithm for the MIP-2 molecule, it would take an estimated time of about 10 days for the same on a state-of-the-art sequential ma- chine like the Sun Sparc Ultra-Enterprise. Since our par- allel algorithm scales well with the number of atoms, with more processors, it should be possible to compute bounds for molecules of fairly large sizes in a reasonable amount of time.... ..."

### Table 1: Algorithmic framework for a general probabilistic population-based EA. Initially, construct a probability density p0(x) that represents the model of the search space (or where it is believed that good solutions are distributed in IRn). This initial density may be quite general, or it may be tailored to the speci cs of a given problem, through the incorporation of any available prior knowledge. At each iteration (generation) t of the algorithm, a set D of k solutions is generated for the optimization problem by sampling k times from the probability density pt(x) D(k) t

### Table 1: Iteration count scaling of Schwarz-preconditioned Krylov methods Iteration Count

1999

"... In PAGE 4: ...o nonsymmetric problems, e.g., GMRES. Krylov-Schwarz iterative methods typically converge in a number of iterations that scales as the square-root of the condition number of the Schwarz-preconditioned system. Table1 lists the expected number of iterations to achieve a given reduction ratio in the resid- ual norm. (Here we gloss over unresolved issues in 2-norm and operator-norm convergence de nitions, but see [4].... In PAGE 5: ... global reduction time of CP1=d. Assume no coarse-grid solve. For simplicity, we neglect the cost of neighbor-only communication relative to arithmetic and global reductions. The rst line of Table 2 shows the estimated execution time per iteration in the left column and the overall execution time (factoring in the number of iterations for 1-level additive Schwarz from Table1 ) in the right column. All of the work terms (matrix-vector multiplies, subdomain preconditioner sweeps or incomplete factorizations, DAXPYs, and local summations of inner product computations) are contained in A, and, since it is given in units of time, A also re ects per-processor oating-point performance, including local memory system e ects.... ..."

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### Table 9: Performance comparison between single and double precision arithmetics on a fixed number of iterations of Conjugate Gradient (100 iterations) and Generalized Minimal RESidual (2 cycles of GMRES(20)) methods both with and without diago- nal scaling preconditioner. The runs were performed on Intel Woodcrest (3GHz on a 1333MHz bus).

2006

"... In PAGE 11: ... The BLAS libraries used are capable of exploiting the vector units where available and, thus, the speedups shown in Table 8 are due to a combination of higher number of floating point operations completed at each clock cycle, reduced memory traffic on the bus and higher cache hit rate. Table9 shows the difference in performance for the single and double precision implementation of the two sparse iterative solvers Conjugate Gradient and General- ized Minimum Residual. Columns 2 and 3 report the ratio between the performance of single and double precision CG for a fixed number (100) of iterations in both pre- conditioned and unpreconditioned cases.... In PAGE 11: ... Columns 4 and 5 report the same information for the GMRES(20) method where the number of cycles has been fixed to 2. Since the sparse matrix kernels involved in these computations have not been vectorized, the speedup shown in Table9 is exclusively due to reduced data traffic on the bus and higher cache hit rate.... ..."