### TABLE I The computation time (ms) of conjugate gradient solver on dense matrix and sparse matrix, the test case is a 479*479 sparse matrix.

### Table 14: Direct coarse grid solution, test4.i We now look at a V1;1 multigrid cycle with a direct coarse grid solver. Take some band matrix or sparse matrix data structure and apply a direct solver. Take care

1996

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### Table 3: This matrix comes from The Harwell-Boeing Sparse Matrix Collection, and it is obtained from a biharmonic operator on a rectangular plate with one side xed and the others free. Times shown in table are in microseconds. experiments with 1, 2, 4, 8, and 16 processors. We used two SPD matrices for running the experiments. The rst matrix is the result of a discretization on a 64 64 grid: Laplace apos;s equation. The matrix is sparse of order 4096, and has 20224 nonzero entries. The second matrix, LANPRO of order 960 with 8402 nonzero entries, comes from the Harwell-Boeing Sparse Matrix Collection [9]. The sequential time reported in Tables 2 and 3 is the result of running a sequential im- plementation of Block-CG without any routines for handling parallelism. The sequential implementation uses the same BLAS and LAPACK [1] routines as the parallel Block-CG

### Table 1 is an example of rating matrix in which three users have reported ratings on five different items. Some entries in the matrix are empty because users do not rate every item. The last row represents the ratings of a user for which the system will make a prediction. Typically, the rating matrix is sparse because most users do not rate most items.

2000

"... In PAGE 3: ... Table1 : Example of User Ratings in a sparse matrix I1 I2 I3 I4 I5 U1like 1 0 0 0 1 U1dislike 0 1 1 0 0 U2like 0 0 0 0 0 U2dislike 1 0 0 1 1 U3like 0 1 1 0 1 U3disike 0 0 0 0 0 Class Label Like Dislike Like Like ? Table 2. Boolean Feature transformation of Ratings Matrix Billsus and Pazzani [1] proposed transforming the format of rating matrix so that every cell has an entry.... ..."

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### Table 1 Condition numbers for the scaled prewavelet sti ness matrix on sparse grids (2), = 0 (top) and di erent values of for xed d = 3 and l = 7 (bottom).

34

### Table 2: Execution time (in ms) on IBM SP2 for the two datasets. u = A v is for a matrix-vector multiplication, and cgsolve is for running the conjugate gradient solver (shown in Figure 2) for 10 iterations. The table on the left is for the 800 800 dataset, and the table on the right is for the 479 479 dataset. but on other domains of interests (i.e., non-zero ratios, compression/distribution schemes, cost functions, etc.), is known as abstract interpretation in the functional programming literature. Several recent works have used statistical information about the applications or the target machines to help select implementation strategies [4, 5, 11]. These methods seem to concentrate on speci c applications, while ours center around high-level matrix classes and dynamic selections of specialized classes for faster execution. Runtime compilation techniques and their applications to sparse computation can be found at [12, 16]. For the emerging roles of Java and Web technology in parallel computing, see [2, 7, 9, 14].

"... In PAGE 7: ... Each node runs a copy of the Java virtual machine with the JavaPVM interface to the underlining PVM library. Table2 shows the performance results. Note that Table 2 shows only the results of xed selections of class implementations.... In PAGE 7: ... Table 2 shows the performance results. Note that Table2 shows only the results of xed selections of class implementations. At the beginning of an execution, we may not know that the matrix is sparse or dense, and an automatic selection of speci c class implementations is needed at run-time.... In PAGE 7: ... From our measurements, the dense-to- sparse matrix conversion cost can be amortized over a long execution. Also, in Table2 , the sparse matrix class being selected uses the (Block, *) scheme for distribution and the CRS scheme for compression. These two schemes are most suitable for the two test programs.... ..."

### Table 2: Time consumption for seq, peeling-off and sparse- ruling-set for different problem sizes using P = 16 PUs for the parallel algorithms.

"... In PAGE 3: ... For the second test the number of PUs for the parallel al- gorithms was fixed on P = 16, for sparse-ruling-set we in- creased S up to 2,000. The results in Table2 show that the parallel algorithms achieve some speed-up and can be used for problems that are too large for the main memory of a sin- gle processor. For small N peeling-off is better, because it performs fewer start-ups, but for larger values of N sparse- ruling-set is winning because of its work efficiency and its... ..."

### Table 1: Left: Time comsumption in seconds, speedup, efficiency, local time and routing volume divided by C6, for a list with C6 BP BEBEBC on up to 16 PUs. For the peeling-off algorithm CS BP BI with CUD8B4CSB5 BP B4BDBMBIB7BDBMBCBHA1CSA0D8B5BPB4BIB7CSA0D8B5. For sparse-ruling-set CS BP BE and CB BP BKBCBC. Right: Time consumption for seq, peeling-off and sparse-ruling-set for different problem sizes using C8 BP BDBI PUs for the parallel algorithms. Now CB BP 2,000 for sparse-ruling-set.

1999

"... In PAGE 25: ... We implemented the sequential list ranking algorithm, seq, the parallel peeling-off algorithm, pof and the parallel sparse-ruling-set algorithm, srs. The results in Table1 show that the parallel algorithms achieve some speed-up and can be used for problems that are too large for the main memory of a single processor. For small C6 peeling-off is better, because it performs fewer start-ups, but for larger values of C6 sparse-ruling-set is winning because of its work efficiency and its smaller routing volume.... ..."

### Table 2. Performance of sparse greedy approximation vs. explicit solution of the full learning problem. In these experiments, the Abalone dataset was split into 3000 training and 1177 test samples. To obtain more reliable estimates, the algorithm was run over 10 random splits of the whole dataset.

2002

"... In PAGE 34: ... The same applies for the log posterior. See Table2 for details. Consequently for all practical purposes, full inversion of the covariance matrix and the sparse greedy approximation have comparable generalization performance.... ..."

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### Table 1 Comparison of several list ranking algorithms, where n is the length of the list, p is the number of processors, and m is a parameter of our algorithm (m lt; n= logn, and for the CRAY C-90 m O((logn)3)).

1994

"... In PAGE 4: ... Other work efficient parallel PRAM list ranking algorithms have very large constants, which has inhibited their implementation. Table1 gives a comparison of list ranking algorithms, and Figure 1 compares the running times of five list ranking algorithms on one processor of the CRAY C-90. The Miller/Reif and Anderson/Miller algorithms use random pointer jumping, and the Belloch/Reid-Miller algorithm is the one on which we report here.... ..."

Cited by 34