### Table 1: Algorithm for multiple sorted QR-decompositions.

2005

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### Table 5: Comparative results for computing the QR decomposition.

"... In PAGE 12: ... Table 4 gives a summary description of the applications considered in our comparison, together with the corresponding dimensions. In Table5 , some results for the QR decompositions of the associated block Toeplitz matrices are Appl. # Application k l m n 1 Glass furnace 1 9 1227 20 2 Flexible robot arm 1 2 984 40 3 CD player arm 1 4 2018 30 4 Ball and beam 1 2 960 40 5 Wall temperature 1 3 1640 40 Table 4: Summary description of applications.... ..."

### Table 4.4 Model potential fluid flow problem on a rectangular domain with a constant tensor of hydraulic permeability. Number of nonzeros of the projected matrix onto the null-space basis Z of the block CT (see Algorithm 3.1, Step 3), iteration counts and timings of the preconditioned conjugate gradient method applied to the orthogonally projected indefinite system compared to the memory requirements and iteration counts for the solution of the same system based on the sparse QR decomposition of its off-diagonal block ZTB.

2001

Cited by 1

### Table 5.6: MOPS Results using QR Decomposition.

### Table 2: Costs for QR Decomposition in Floating Point Operations (Flops).

### Table 1: Execution time of the column Householder algorithm for a 1000 1000 matrix. 4 Experimental Results and Discussion Several algorithms for QR decomposition have been implemented on the 128-cell AP1000 lo- cated at the Australian National University. Many results have been obtained through extensive tests. In the following some of these experimental results are presented together with a theo- retical discussion. In all cases the matrices are dense (our algorithms do not take advantage of sparsity) and the runtimes are independent of the input data (since the algorithms do not involve pivoting). For simplicity we only present results for the QR decomposition of square matrices and only count the time for the computation of the triangular factor R, not for accu- mulation of the orthogonal transformations. In all cases double-precision (64-bit) oating-point arithmetic is used.

"... In PAGE 6: ... In this experiment the unblocked column Householder algorithm is applied to factorise a dense 1000 1000 matrix. It can be seen from Table1 that there is a decrease in the total computational time when the aspect ratio increases from less than one to one or two. This is due to the nature of the communication pattern of the algorithm.... In PAGE 6: ...ne or two. This is due to the nature of the communication pattern of the algorithm. Since the communication required in the column Householder algorithm is column oriented, that is, the horizontal communication volume is higher than the vertical communication volume, the number of columns in the cell array should be decreased so that the overall communication cost may be reduced. However, Table1 shows that further increasing the aspect ratio (even to P, that is, using a one dimensional array to eliminate all the horizontal communication), the performance gets worse. There are two reasons for this { 1.... In PAGE 7: ... Since the column Householder algorithm has h gt; v, we expect the optimal aspect ratio to be slightly (but not too much) greater than unity. The results given in Table1 show that the optimal ratio is either one or two. 4.... ..."

### Table 7.3. Numerical results for Example 7.3. Only the QR based methods can successfully deal with both contraction and rotation of the initial set. For these methods, the overestimation of the final flow is hardly noticeable. This agrees with the general observation that the QR decomposition is a very effective tool in fighting the wrapping effect, both for the interval method and for the preconditioned Taylor model method.

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### Table 1 Comparison of simulation effort of double insulator chain model

2007

"... In PAGE 11: ... Here symbolic elements are colored black, zero elements are blank and numerical elements are non-blank. Figure 7: Matrices of the double insulator chain example, (a) A1 (b) A2 In Table1 are shown the densities of the matrices, numerical costs of their QR- decomposition using the preprocessing module, and the numerical costs of the QR- decomposition using the dense solver. ... In PAGE 12: ...From Table1 follows that the distributed calculation of accelerations needs the decomposi- tion only of sparse matrices, which can be efficiently calculated using the preprocessing mod- ule. 6 CONCLUSION AND FUTURE WORK The sparse matrix method, implemented in VSD Software, performs the efficient compo- nent-oriented simulation of dynamics of CAD models of multibodies.... ..."