### Table 4. The table describes a test case involving the two traces that may be observed for the test condition stated above as th1. Note that threads p1 and p2 started at times 3 and 4 respectively may terminate in either order, and that the outputs finally returned will occur in the corresponding order.

2005

"... In PAGE 9: ... Table4... ..."

Cited by 3

### Table 1 presents the results for continuous Asian call option values computed at the starting date of the contract at an interest rate of 9% per annum with the initial stock price at 100, for various maturities, three volatility levels of .2, .25 and .3 and ve strike values of 90; 95; 100; 105, and 110. The values rise in maturity and volatility and fall with increases in the strike price as expected. We were not able to successfully invert the Geman and Yor transform for maturities below .4. The inversions were conducted using Mathematica on a Sun Sparc 10 workstation. The time taken for the inversions in CPU seconds was between 20 to 45 seconds. We also used the Abate and Whitt (1995) formulation of the Post-Widder inversion procedure with comparable results, but the computation time was considerably larger, ranging from 475 to 650 CPU seconds.

1997

"... In PAGE 13: ... For the generation of the path we used a straight update of the stock price given by St+h = Sterh+ ph t+h? 2h=2; where the random variables t+h were drawn from a standard normal distribution.3 Figures 1 through 5 and Table 3 present the results for the case of r = :09; T = :4; = :2 and the ve strikes reported in Table1 . The notation dr1ks1k refers to 1000 (\1k quot;) readings per day (\dr quot;) with 1000 (\1k quot;) path replications or simulations (\s quot;): These results employ linked replications, by which we mean that the random number seeds used for a succeeding replication begin where the random number streams used for the preceding replication nish.... ..."

Cited by 4

### Table 3: The probabilities of reaching different values of n, when p=.6, and starting at n/2.

### Table 6: Execution time table; T: period, S: start time (release time), D: deadline, A: available time P1 P2 P3 P4

"... In PAGE 5: ... The Table 7 shows the execution time of each task on each processor. The tasks have associated timing constraints as shown in the Table6 . Each processor has an associated cost given in the Table 7.... In PAGE 6: ... The tasks are either randomly generated or industrial DSP, communication, and control examples adopted from [Pot95]. Table6... ..."

### Table 3.2: Results for problem P1 for different starting points using the

### Table 3. Starting values Double turning points (p = 2) h x(n+1)=2

"... In PAGE 16: ... The thermal explosion of solid explosives is described by the discrete model xi?1 ? 2xi + xi+1 + h2( i?1 + 10 i + i+1)=12 = 0 ; i = 1; 2; : : : ; n; x0 = xn+1 = 0 ; (n ? 1)h = 1 ; i := exp xi 1 + xi ; i = 0; 1; : : : ; n + 1; see [12], [16]. Table3 gives the results for the computations of the double turning points (x h; h; h) 2 Rn+2 R1 R1 of the discretized model in dependence of the stepsize h. In order to ful l the termination criterionjjH(xh; h; h)jj lt; 10?15 Algorithm 2 performed kh iterations steps.... ..."

### Table 1: Factorizations of the numbers p 1, p 2 S.

"... In PAGE 8: ....3.2 Building sF-psp apos;s Since an sF-psp is a Carmichael number, we start by building a Carmichael number as in Section 3. We put = 27 33 52 72 11 13 17 19 and build S = fp prime; p ? 1 j ; 2(p + 1) j ; p - g: The set S has 40 elements which are given in Table1 . We now apply the method of Section 3 and nd a squarefree product of elements, call it N, such that N 1 mod .... ..."

### Table 4: higher order operators over Q(p2)(x) The IT columns of Table 3 illustrate the cuto when the fraction{free method starts becoming better than the fractional one in a system with canonical expanded forms for polynomials and fractions such as IT or axi.om [7]: F36 s m and G168 s m have orders smaller than the generic

1997

"... In PAGE 7: ...7 24.4 Table 1: second order operators over Q(x) Finally an example with algebraic numbers in its (small) coe cients: Lp2 = (x2 + 1)D3 ? p2D + x: Table 3 contains the times needed for computing symmet- ric powers by the fraction{free (Z[x]) and fractional (Q(x)) kernel methods, while Table4 contains the times needed for IT MAPLE Z[x] Q(x) Z[x] Q(x) diffop F36 s 5 8.3 10.... In PAGE 7: ... This cuto is further away on MAPLE be- cause of two characteristics of the arithmetic in MAPLE: the preprocessing step of Section 3 consists of a simplex on a matrix of machine-integers. Machine integers are however not accessible in the MAPLE user language, causing the preprocessing step to take a more signi - cant portion of the computing time than in compiled languages, thereby pushing the cuto further away; the normal function in MAPLE keeps the numerators and denominators of elements of Q(x) factorized, and this allows an appropriately coded Gaussian elimina- tion to take advantage of the many cancellations that happens on the matrices that arise in the symmetric powers computations, thereby giving an extra advan- tage to the fractional method; At the same time, the intermediate results tend to become expanded during the fraction-free elimination, giving it an additional dis- advantage; Both of the above points disappear when the constant eld is a proper extension of Q, as illustrated by Table4 , where the fraction{free method is signi cantly faster than the frac- tional one on all systems. Conclusions For second order operators, the fraction{free iteration method of Theorem 2 is the most e cient way of comput- ing symmetric powers, and it turns the algorithm of [16] into a very e cient second order linear ordinary equation solver.... ..."

Cited by 9

### Table 4: higher order operators over Q(p2)(x) The IT columns of Table 3 illustrate the cuto when the fraction{free method starts becoming better than the fractional one in a system with canonical expanded forms for polynomials and fractions such as IT or axi.om [7]: F36 s m and G168 s m have orders smaller than the generic

1997

"... In PAGE 7: ...7 24.4 Table 1: second order operators over Q(x) Finally an example with algebraic numbers in its (small) coe cients: Lp2 = (x2 + 1)D3 ? p2D + x: Table 3 contains the times needed for computing symmet- ric powers by the fraction{free (Z[x]) and fractional (Q(x)) kernel methods, while Table4 contains the times needed for IT MAPLE Z[x] Q(x) Z[x] Q(x) diffop F36 s 5 8.3 10.... In PAGE 7: ... This cuto is further away on MAPLE be- cause of two characteristics of the arithmetic in MAPLE: the preprocessing step of Section 3 consists of a simplex on a matrix of machine-integers. Machine integers are however not accessible in the MAPLE user language, causing the preprocessing step to take a more signi - cant portion of the computing time than in compiled languages, thereby pushing the cuto further away; the normal function in MAPLE keeps the numerators and denominators of elements of Q(x) factorized, and this allows an appropriately coded Gaussian elimina- tion to take advantage of the many cancellations that happens on the matrices that arise in the symmetric powers computations, thereby giving an extra advan- tage to the fractional method; At the same time, the intermediate results tend to become expanded during the fraction-free elimination, giving it an additional dis- advantage; Both of the above points disappear when the constant eld is a proper extension of Q, as illustrated by Table4 , where the fraction{free method is signi cantly faster than the frac- tional one on all systems. Conclusions For second order operators, the fraction{free iteration method of Theorem 2 is the most e cient way of comput- ing symmetric powers, and it turns the algorithm of [16] into a very e cient second order linear ordinary equation solver.... ..."

Cited by 9

### Table 1 Table 2 We consider the embedding P (t) with the strictly convex quadratic objective and the starting point (x0 1; x0 2) = (0; 0)

1997

"... In PAGE 19: ... There is not a curve quot;connecting t = 0 and t = 1 quot;. Table1 shows the singularities. We are not successful with PATH III but with JUMP I we achieve t = 1.... ..."

Cited by 1