### Table 1: Computational complexities of the fast motion estimation algorithms described.

"... In PAGE 21: ... To perform a direct comparison among the computational e ciencies of the fast search methods, the computational complexity of each method has been formulated for a motion displacement of size w. Table1 shows the computational complexities of the methods when w is a power of 2. Some logarithmic searches may not be able to search all the positions at the boundaries of the search window.... ..."

### Table 1 Comparison of computational complexities Unit Equation

1999

"... In PAGE 4: ... In fact in [7], it was shown that log(x), can be evaluated to n significant bits in O(Mu(n)log(n))) steps, where Mu(n) is units of time required to multiply n-bit numbers. In Table1 , we have calculated the order of complexity for different units of Figure 1. In many cases K lt; lt;M, therefore, the complexities in Table 1 can still be reduced by a factor of K.... In PAGE 4: ... In Table 1, we have calculated the order of complexity for different units of Figure 1. In many cases K lt; lt;M, therefore, the complexities in Table1 can still be reduced by a factor of K. Table 1 Comparison of computational complexities Unit Equation... In PAGE 4: ...As can be seen from Table1 , the tracking unit has the highest order of complexity. It is clear that the core of this unit is the sample-by-sample CG algorithm.... ..."

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### Table 2: Bounds on the number of solutions of the FKP for a robot with planar platform (9 unknowns) When there are more than 3 sensors (we always assume that the sensors are not redundant, which means that they actually give information), it is not interesting to build the dialytic matrix. Indeed it is better to solve the non-linear system by taking advantage of its structure when the linear equations have been eliminated. We obtain in this way a better bound. If 6 sensors are used and give information, we obtain a unique solution by solving the linear system corresponding to the 6 equations of type IV given by the sensors and the 3 equations of type III. The cpu times given in Table 3 are the times we needed to obtain the bound with the symbolic method of Section 3.3. They are only indicative. In fact in practice this computation is not done since we only want to compute numerically the result.

"... In PAGE 26: ... Sensors 3 2 1 Before linear elimination 28 After linear elimination 10 15 21 Table 1: Number of monomials present in the equations before and after the resolu tion of the linear equations 5.1 Planar platform Table2 gives bounds on the number of solutions, depending on the number of extra sensors that are added on the robot. They also give the number of unknowns in the initial non-linear system, the number of equations in the square system obtained by... ..."

### Table 5: The complexity of di erent Lie-group methods for linear equations. Another shortcoming in our analysis is that we have addressed ourselves solely to the linear case. Both Fer and Magnus methods can be generalized to a nonlinear setting (Zanna 1999) but detailed complexity analysis of this construct has not yet taken place. In Figure 2 we have compared the e ciency of a number of numerical methods, as applied to the Airy equation

"... In PAGE 32: ...5.9) by a 4th-order RK-MK method. 6 Conclusions In Section 4 we have derived complexity results for Fer and Magnus methods of orders up to eight. These results are summed up in Table5 , where we have also included... In PAGE 33: ... Moreover, we recall that our results are upper bounds. It is entirely possible, exploiting speci c values of coe cients and aggregating terms in an astute manner, to improve upon the estimates of Table5 . This has been recently accomplished by Blanes, Casas amp; Ros (1999) for Magnus expansions of orders six and eight.... ..."

### Table 5: The complexity of di erent Lie-group methods for linear equations. Another shortcoming in our analysis is that we have addressed ourselves solely to the linear case. Both Fer and Magnus methods can be generalized to a nonlinear setting (Zanna 1999) but detailed complexity analysis of this construct has not yet taken place. In Figure 2 we have compared the e ciency of a number of numerical methods, as applied to the Airy equation

"... In PAGE 32: ...5.9) by a 4th-order RK-MK method. 6 Conclusions In Section 4 we have derived complexity results for Fer and Magnus methods of orders up to eight. These results are summed up in Table5 , where we have also included... In PAGE 33: ... Moreover, we recall that our results are upper bounds. It is entirely possible, exploiting speci c values of coe cients and aggregating terms in an astute manner, to improve upon the estimates of Table5 . This has been recently accomplished by Blanes, Casas amp; Ros (1999) for Magnus expansions of orders six and eight.... ..."

### Table 1. Comparison of the substitution complexities for the published fast algebraic attacks.

"... In PAGE 2: ...2 The complexity was originally underestimated as only O(DE) [C03], where D is the size of the linear combination and E is the size of the second system of equations. Table1 lists the values of O(DE) for previously published attacks from [A04,C03]. However, simple substitution would require a complexity of O(DE2) (see Section 2.... ..."

### Table 1: Computational complexity of the full and aggregate methods

"... In PAGE 12: ... 5 A comparison of the full and aggregate methods This section compares both the computational complexity and the quality of the solution of the full method with those of the aggregate method. Table1 compares the complexity of the full and aggregate methods in terms of the number of action evaluations required per time period. The second last column shows that for the full method the number of action evaluations required increases exponentially with the number of reservoirs, while for the aggregate method this number only increases linearly with the number of reservoirs.... In PAGE 17: ...00 11831.30 Table1 0: Terminal value functions for H03 i i;Dry;1 i;Dry;2 i;Dry;3 i;Wet;1 i;Wet;2 i;Wet;3 1 203.86 143.... In PAGE 18: ...Table1 1: Terminal value functions for H04 i i;Dry;1 i;Dry;2 i;Dry;3 i;Wet;1 i;Wet;2 i;Wet;3 1 203.51 140.... In PAGE 18: ...56 3.01 Table1 2: Terminal value functions for L08 i i;Dry;1 i;Dry;2 i;Dry;3 i;Wet;1 i;Wet;2 i;Wet;3 1 210.36 160.... In PAGE 18: ...89 3.16 Table1 3: Terminal value functions for L17 i i;Dry;1 i;Dry;2 i;Dry;3 i;Wet;1 i;Wet;2 i;Wet;3 1 116.00 116.... ..."

### Table 3: The fast multichannel LMS algorithm. Equation # Mults.

"... In PAGE 3: ... The method for calculating e(n) in (23){(27) can be com- bined with the previously-derived algorithm to obtain a fast version of the LMS algorithm for multichannel active noise control. Table3 lists the complete algorithm. The number of multiplies at each iteration is C(f) LMS = NxNy 2L + Ne 4 + 3 Nx + 1 Nx + 2 Ny M ? 2Nx ? Ny + Ne: (28) As in the previous case, the complexity of this algorithm is also O(NxNy(2L)) if NeM is somewhat less than L Remark: The quantity H(j)T U(j)(n ? 1) is of O( (n)).... ..."

### Table 1: Patient case with a follow-up. Values of observables are shown for every step. Recom- mendations for each step are ordered according to the best cost score (the top choice is in bold). Two scores listed are computed by the incremental linear function method (method 1) and the fast informed bound method (method 2). Note that both methods suggest the same action choices.

"... In PAGE 16: ... To approximate the optimal action choices, we used one-step decision tree lookahead and the resulting value function approxima- tions. Table1 illustrates a sequence of recommendations for a single patient case with a follow-up obtained for two of the best-performing approximation methods from [10]. The first method is the incremental linear function method with 15 update cycles.... In PAGE 16: ...ach step. Note that the top choice for both methods is the same for all steps. 5. Evaluation To test the IHD model we performed an initial evaluation on a set of 10 patient cases with follow- ups (similar to the one shown in Table1 ). These cases where generated by a cardiologist with the objective to test the dynamic behavior of the model and identify its weaknesses.... ..."

### Table 1: Stability and order complexity when computing the jump probabili-

"... In PAGE 17: ... 4.4 Comparison of Methods Table1 lists the characteristics of the discussed methods for computing the jump probabilities in AU, as well as for computing the Poisson probabilities in SU by the scheme of Fox and Glynn #5B3#5D. N a , N s and N B are de#0Cned in #2814#29, #287#29 and #2843#29, i.... In PAGE 17: ... In that case N B = N s . We see from Table1 that none of the methods outperforms the computa- tion of Poisson probabilities in SU. Furthermore, the only numerically stable... In PAGE 17: ... We see from Table 1 that none of the methods outperforms the computa- tion of Poisson probabilities in SU. Furthermore, the only numerically stable method for computing the jump probabilities for AUin Table1... ..."