### Table 1. CPU-Time(msec) for the computation of rational normal forms for Data1 (Each of rational normal forms consists of one companion block.) size(n) straight(t1) frac-free(t2) t2=t1

"... In PAGE 9: ... We show the CPU-times (in milli-seconds) and the ratios between Algorithm 1 and Algorithm 2. Table1 shows the results for Data1 and Table 2 for Data2. As we men- tioned in section 2, we can compute the transformation matrices simultaneously.... In PAGE 10: ...2. discussion Table1 and Table 2 show that all the ratios of the CPU-times are less than 1 except the case for n = 3; 4. For small-sized matrices, the overhead to avoid rational expressions is considered to overcome the e ect of fraction-free computation.... In PAGE 10: ... However, for larger-sized matrices, the fraction-free algorithm turns out to be more e cient than the straightfor- ward one. In Table1 , the ratio gradually decreases down to 0:27 in inverse proportion to the size of matrices. For the sizes 19 and 20, the ratio slightly shifts to the increase.... ..."

### Table 2 Di erences in utilities between rational and boundedly rational agents

in in

"... In PAGE 18: ...Table2 around here] The overall ranking of the rational agent for the same rates of experimentation, are presented in table 3. For the given rate of experimentation, the frequencies are computed by ranking utilities of the rational and N boundedly rational agents from the lowest to the highest at the end of each t for T max periods in 5 simulations.... ..."

### Table 2. CPU-Time(msec) for the computation of rational normal forms for Data2 (Each of rational normal forms consists of three companion blocks.) size(n) straight(t1) frac-free(t2) t2=t1 size of the biggest block C1

"... In PAGE 9: ... We show the CPU-times (in milli-seconds) and the ratios between Algorithm 1 and Algorithm 2. Table 1 shows the results for Data1 and Table2 for Data2. As we men- tioned in section 2, we can compute the transformation matrices simultaneously.... In PAGE 10: ...Table 1 and Table2 show that all the ratios of the CPU-times are less than 1 except the case for n = 3; 4. For small-sized matrices, the overhead to avoid rational expressions is considered to overcome the e ect of fraction-free computation.... In PAGE 10: ... Although the e ciency depends on the elements of input matrices and their arrangement, we consider that the e ect of fraction-free computation is con rmed. Table2 shows that the size of the biggest companion block C1 is an important factor in the e ciency. When the size of C1 increases at n = 4; 7; 10; 13;16; 19, the ratio increases.... ..."

### Table 1: Error in rational interpolant.

2006

"... In PAGE 14: ... Figure 1 shows plots of the rational interpolant with d = 3 for respectively n = 10, 20, 40, 80. The second column of Table1 shows the numerically computed errors in this example, for n up to 640, and they confirm the fourth order approximation predicted by Theorem 2. Figure 2 shows plots of the rational interpolant of the function f(x) = sin(x) at the same equally spaced points as in the previous example, but this time with d = 4.... In PAGE 14: ... Figure 2 shows plots of the rational interpolant of the function f(x) = sin(x) at the same equally spaced points as in the previous example, but this time with d = 4. The third column of Table1 shows the computed errors, which confirm the fifth order approximation predicted by Theorem 2. One advantage of the rational interpolants is the ease with which we can change the degree d of the blended polynomials.... ..."

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### Table 4: Comparison of floating point and rational arithmetics for small problems. Floating point computations usually fail on very large problems due to rounding errors.

### TABLE 2 Experimental results from the use of IPF( ) representations to evaluate scalar rational functions. An asterisk * means that no correct significant digits were computed.

1995

### Table 9. Abbreviated Paramedic Method for

1992

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### Table 3: Approximation errors using polynomial and rational methods reported by Matlab for log (x).

"... In PAGE 6: ... The range reduction step of the evaluated functions is exact where no quantization is involved as shown in Figure 3. We summarize the approximation errors reported by Matlab in Table3 . We can see that the approximation error generated by using the rational method is much lower than the poly- nomial method.... In PAGE 6: ... It means that if the nal computed error is smaller than these two values, the library/design can assure the accuracy requirement of either IEEE single or double precision. Esingle = 2 23 = 1:192 10 7 Edouble = 2 52 = 2:220 10 16 Now that the quantization error of the above example is lower than Esingle, however the approximation error of this example is Eapprox = 0:02983 as shown in Table3 which is much higher than Esingle. We apply this static analysis in our code generation to cover all the possible errors in the calculation for determining the best possible embedded C code.... In PAGE 8: ... Figure 18 shows that the rational method achieves much lower error than the polynomial method for small de- gree. Note that the maximum error values shown in this graph are much higher than those in Table3... ..."

### Table 1 shows that all of the leading principal submatrices of both ^ H49 and ^ H049 are fairly well-conditioned in the average. Afterwards, we have used rational interpolation at Chebyshev points for approximating even functions in the interval [?2; 2]. We have considered 3 functions fj(x), j = 1; 2; 3, (j is here a counter for the approximated functions):

1995

"... In PAGE 13: ... The absolute error j is de ned as the maximum of the absolute errors jrm;n(x(49) i ) ? fi;jj; 1 i 50; over all of the computed rational interpolating functions rm;n(x). For each test set, we consider the average errors = P100 j=1 j 100 ; 0 = P100 j=1 0j 100 ; = P100 j=1 j 100 : Table1... ..."

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### Table 4. This table gives the upper entries ij of the rational weakly-eutactic matrices in the corresponding class.

"... In PAGE 14: ... The computation times are very short, except for the determination of the eld of de nition of the eutactic form when a high degree is involved. We conclude this paper with Table 3 giving the list of the well-rounded minimal classes in dimension 5 and their invariants, and Table4 displaying the matrices of the rational weakly eutactic forms. The algebraic matrices and the automorphism groups can be requested from the author.... In PAGE 22: ...CHRISTIAN BATUT Table4 (continued) class 11 12 13 14 15 22 23 24 25 33 34 35 44 45 55 a8 21100 2100 200 20 2 b8 42000 4000 42-1 41 4 c8 84-210 82-10 820 80 8 d8 42-1-10 4110 410 40 4 f8 18 9 9 9 9 12 6 10 8 12 8 10 20 14 20 g8 12 6 8 10 9 1210 8 9 201417 2017 26 h8 12 6 9 7 8 12 9 1110 181215 2217 24 a7 42-100 4100 400 40 4 b7 21000 2000 210 20 2 c7 63000 6000 62-2 62 6 d7 63-110 61-10 620 60 6 f7 84756 8576 1481 141 16 g7 93645 9687 12810 161217 i7 24 0 9 11 11 24 12 12 12 24 15 15 30 4 30 a6 21000 2000 200 20 2 b6 31-100 3100 300 30 3 c6 41-110 41-10 410 40 4 d6 5 -1-1-1-1 5-1-1-1 5 -1-1 5-1 5 e6 20 1 9 10 10 20 12 11 11 24 15 15 26 5 26 a5 5 3 -2 -2 -2 5 -2 -2 -2 4 0 0 4 0 4 b5... ..."