### Table 1. Examples of Dynamic Programming recurrences

"... In PAGE 6: ...ig. 4. Evalua method generated for the Optimal Matrix Parenthesization (see Table 1 and Equation 1). However, scientists typically represent the functional equations as mathe- matical expressions, like those shown in Table1 , by using their favorite equation editor (Latex, OpenO ce, etc.).... In PAGE 8: ...Table1 . As an example to illustrate the use of DPSPEC, we will consider the functional equation for the multiplication parenthesization problem (Equation 1).... In PAGE 9: ... In terms of the e ciency of the generated code, Figure 5 shows the perfor- mance obtained with the DPSKEL-generated parallel code. Series of instances for the problems in Table1 were randomly generated for each problem. The instances of the skeleton generated use the OpenMP library [10] on an RS-6000 SP IBM platform.... ..."

### Table 2: Total b(2n) and normal pn partitions of 2n b(27) and b(28) can be veri ed by enumerating all partitions, using the program of section 1, or by reference to [13]. Readily, b(2n) must be even because, as observed, a0 must be even, so every subsequence of normal partitions is even. It is not hard to show that jPnj grows super exponentially with respect to n. Based on the expression log b(n) (log n)2=2 found in [10], Churchhouse [2] gives the asymptotic upper bound b(n) O(n1=2 log2n) or b(2n) = jPnj O((2n)n=2): (10) The nature of this super exponential growth is di cult to intuitively comprehend because, unfortunately, equation (10) is a poor approximation for small values of n. In Table 3, we compare b(2n) with two lower bounding functions, nn and (2n)n=3, and the upper bound (2n)n=2 to which it is eventually asymptotic. Besides giving some concrete feeling for the growth of the binary partition function, this table illustrates that a wealth of closure spaces exist for even small n.

"... In PAGE 8: ...break; g sum = 0; for (k=2; k lt;=max k; k += 2) f sum = sum + sigma[n][k]*k; g p[n] = sum; return sum; g With this code one can generate the following Table2 of partitions of 2n. The values of... ..."

### Table 1. Error Sum of Squares between Estimated and Observed Mean Value Functions

1996

"... In PAGE 4: ... Figure 4 gives the mean value functions for data set 1 (actual) and those estimated for the exponential and s- shaped and Weibull coverage functions. The error sum of squares between the predicted and ob- served mean value functions are summarized in Table1 . It is evident that the best approximation of the observed mean value function is achieved by different coverage functions for different data sets, in a least error sum of squares sense.... ..."

Cited by 6

### Table 1. Error Sum of Squares between Estimated and Observed Mean Value Func- tions

1996

"... In PAGE 7: ... Similarly, Figure 5 shows plots for the Weibull covearge function for various val- ues of . The error sum of squares for the various coverage functions and the data sets is sum- marized in Table1 . It is evident from Table 1 that the best approximation of the observed mean value function is achieved by di erent coverage functions for di erent data sets, in a least error sum of squares sense.... ..."

Cited by 6

### Table 1. Error Sum of Squares between Estimated and Observed Mean Value Func- tions

1996

Cited by 6

### Table 1. Error Sum of Squares between Estimated and Observed Mean Value Func- tions

1996

Cited by 6

### Table 6 shows the estimated wage equation coefficients and asymptotic standard errors

2001

### Table 5 reports the coefficient estimates and asymptotic t-statistics of equation

"... In PAGE 12: ...Table5... ..."

### Table 7 reports the coefficient estimates and asymptotic t-statistics of equation

### Table 1 lists some general classes of recurrence pKoblems that are suitable for solution by FORA. For each diass, the general companion function and particular examples are given. Class 1 in Table 1 covers associative functions such as +,. X, max, and min: It is clear that all such functions Satisfy Definition 2 directly. In these cases the FORA solutions are identical to direct modifications of the log- sum algdrithm. The second class of recurrence problems listed in

1974

"... In PAGE 4: ...Table1 Applications of the FORA algorithm. Problem class Companion function amp; apos;(a, b) Examples (D, = domuin of variable r) I.... In PAGE 5: ... In these cases the FORA solutions are identical to direct modifications of the log- sum algdrithm. The second class of recurrence problems listed in Table1 has the formf(a, x) =f(a(2), g(a( l),x)), where f and g have cei-taih functional properties. Suitable prob- lems inciude the introductory linear problem ( 1 i and several highly nonlinear ones.... In PAGE 5: ... This particular class of problems has been solxed previously by Kogge and Stone [9] who used the concept of recursive doubling. The third class of problems in Table1 represehts certain nonlinear problems with no previously known general parallel solution. Stone [6] was able to solve example 3 of this class with a parallel algorithm that is faste; than FORA.... In PAGE 5: ... The same procedure diagrammed in Fig. 5 can be ex- tended to cover any recurrence satisfying Class 2 of Table1 . Within this class one recurrence in particular that has received extensive study is the carry equation for binary addition of two binary numbers.... ..."

Cited by 7