### Table 11. Some rational solutions of (28) for k = 3 and k = 30

### Table 1: Relative error for solutions of (7) based on rational approximants.

2002

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### Table 2: Relative errors for solutions of (2) and (4) based on rational approximants.

2002

Cited by 3

### Table 1 shows the times that were needed to nd the rational solutions of the Legendre equation Lm(y) = (1 ? x2)y00 ? 2xy0 + m(m + 1) y = 0 where n is a xed positive integer. This equation has a one- dimensional space of rational solutions generated by the mth Legendre polynomial which has degree m. For that equation and the basis Pn(x) = xn=n! we get r = 2, d = 2, A = 2, B = 2, p0 = m(m + 1), p1 = ?2x, p2 = 1 ? x2, ?2 = 1, ?1 = 1 = 2 = 0, 0(n) = m(m + 1) ? n(n + 1), b = 0 and N = maxf0; mg = m, so recurrence (9) is: vn+2 + (m(m + 1) ? n(n + 1))vn = 0

"... In PAGE 5: ... Table1 : Maple CPU seconds to solve Lm(y) = 0 (SPARC 10/41).Ratlode is the ratlode package of the Maple share li- brary, which uses a straightforward implementation of the method of undetermined coe cients; Series is a straight- forward implementation of the method presented here, and Orthopoly is the built-in orthopoly[P] function of Maple which computes Legendre polynomials.... ..."

### Table B.1 shows the list of rationally feasible solutions. Each solution was obtained by solving the system of FOCs where each condition corresponding to the zero-value variable is relaxed in such a way that allows it to have any non-positive value. Solutions for the monopolistic collusion and the welfare maximizing modes are given in the same table, because they are very similar. The difference between the monopoly and welfare maximizing solution is captured in a constant h (with h:=2b in the monopoly problem and h:=b in the welfare maximizing one). The FOCs for these two problems can be presented as:

### Table 1 Running times for rational solver on 170 small problems

2007

"... In PAGE 2: ... The fifth column gives the average number of bits needed to represent each nonzero entry of the optimal primal and dual solutions, and the last col- umn gives the ratio between the running time of the rational code and the time needed for the original QSopt code. Table1 shows the details for all prob- lems whose running-time ratio was above 800, and an average for all other instances. All runs were car- ried out on a Linux workstation with a 2.... ..."

Cited by 1

### Table 1 Running times for rational solver on 170 small problems

"... In PAGE 2: ... The fifth column gives the average number of bits needed to represent each nonzero entry of the optimal primal and dual solutions, and the last col- umn gives the ratio between the running time of the rational code and the time needed for the original QSopt code. Table1 shows the details for all prob- lems whose running-time ratio was above 800, and an average for all other instances. All runs were car- ried out on a Linux workstation with a 2.... ..."

### Table 3: Rational polynomial iterators, obtained by solving all the transcen- dental subexpressions rst Consider the equation sin x = x=100. Providing for all solutions of sin x yields three iterators:

"... In PAGE 13: ... From these possible values for x1 we select, as before, the closest to x0, x1 = 1:2024:::. More formally, using Maple apos;s RootOf notation, the iterator becomes xi+1 = RootOf z ? sin xi 1 ? z2=2 = tan xi ? z 1 + z2=2 ; z! Table3 shows the simulation results for this heuristic. Transcendental Iterators.... ..."

### Table 1: Relative errors of u(x; 105) calculated by various rational approximants.

2002

"... In PAGE 5: ...based on R7;8 is much more accurate. The numerical solutions at z = 105 based on various rational approximants are compared with the \exact quot; solution of (5) and the relative errors (in L2 norm) are listed in Table1 . Similar results are obtained with x = 1=60 and h = 10=3.... ..."

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### Table 6. Summary of results for CBS and NBS (* average with respect to a configu- ration that includes all relations)

"... In PAGE 14: ... These values less than the number of consistent and equivalent results by using CBS or NBS. A summary of results for CBS and NBS is presented in Table6 . In this table, one has to consider that each configuration may have more than one solution and that each configuration has a number of consistent and equivalent relations (100% of consistent relations implies that a solution is consistent).... ..."