### Table 1: Backward recursive algorithm

2000

"... In PAGE 6: ... Since the cost functional is continuously di eren- tiable and strictly convex, Q M (k;;n) is also a convex problem with linear constraints and has a unique solu- tion at a nite point. The backward recursive algorithm for solving the optimal control problem for multiple fur- nace system, P M , can be described as in ALGO 1 and ALGO 2 in Table1 . We omit the detailed algorithm here.... ..."

Cited by 2

### Table 2. Recursive algorithms - schedule duration

2003

Cited by 5

### Table 3. Recursive algorithms - simulation runtime

2003

Cited by 5

### Table 3: The Recursive Doubling Algorithm

1995

"... In PAGE 11: ...where ; means unde ned. Table3 shows the outline of the recursive dou- bling in our implementation. After condensation of parameters, the Jacobian matrix J in Figure 1 becomes the one shown in Figure 5.... In PAGE 49: ...10% Table 29: With Pivoting 1: NDIM=96, NTST=32, NCOL=4, NMX=10 Number of Nodes Execution time Relative Speed-up Relative E ciency 64 0.10121E+04 1 100% Table3 0: With Pivoting 1: NDIM=96, NTST=64, NCOL=4, NMX=10... In PAGE 50: ...47 13.47 Table3 1: With Pivoting: Average I/O Time... In PAGE 51: ...77 23.07% Table3 2: With Pivoting 2: NDIM=12, NTST=64, NCOL=4, NMX=10 Number of Nodes Execution time Speed-up E ciency 1 0.29209E+03 1 100% 2 0.... In PAGE 51: ...25 26.96% Table3 3: With Pivoting 2: NDIM=12, NTST=128, NCOL=4, NMX=10... In PAGE 52: ...56 29.87% Table3 4: With Pivoting 2: NDIM=12, NTST=256, NCOL=4, NMX=10 Number of Nodes Execution time Speed-up E ciency 1 0.86536E+03 1 100% 2 0.... In PAGE 52: ...30 44.22% Table3 5: With Pivoting 2: NDIM=24, NTST=64, NCOL=4, NMX=10 Number of Nodes Execution time Relative Speed-up Relative E ciency 4 0.44720E+03 1 100% 16 0.... In PAGE 52: ...59 53.67% Table3 6: With Pivoting 2: NDIM=24, NTST=128, NCOL=4, NMX=10... In PAGE 53: ...63 57.90% Table3 7: With Pivoting 2: NDIM=24, NTST=256, NCOL=4, NMX=10 Number of Nodes Execution time Relative Speed-up Relative E ciency 2 0.14757E+04 1 100% 4 0.... In PAGE 53: ...76 73.52% Table3 8: With Pivoting 2: NDIM=48, NTST=32, NCOL=4, NMX=10 Number of Nodes Execution time Relative Speed-up Relative E ciency 8 0.76609E+03 1 100% 16 0.... In PAGE 53: ...40 67.44% Table3 9: With Pivoting 2: NDIM=48, NTST=64, NCOL=4, NMX=10 Number of Nodes Execution time Relative Speed-up Relative E ciency 16 0.78489E+03 1 100% 32 0.... ..."

Cited by 1

### Table 3: The Recursive Doubling Algorithm

1996

"... In PAGE 12: ...where ; means unde ned. Table3 shows the outline of the recursive dou- bling in our implementation. After condensation of parameters, the Jacobian matrix J in Figure 1 becomes the one shown in Figure 5.... In PAGE 50: ...10% Table 29: With Pivoting 1: NDIM=96, NTST=32, NCOL=4, NMX=10 Number of Nodes Execution time Relative Speed-up Relative E ciency 64 0.10121E+04 1 100% Table3 0: With Pivoting 1: NDIM=96, NTST=64, NCOL=4, NMX=10... In PAGE 51: ...47 13.47 Table3 1: With Pivoting: Average I/O Time... In PAGE 52: ...77 23.07% Table3 2: With Pivoting 2: NDIM=12, NTST=64, NCOL=4, NMX=10 Number of Nodes Execution time Speed-up E ciency 1 0.29209E+03 1 100% 2 0.... In PAGE 52: ...25 26.96% Table3 3: With Pivoting 2: NDIM=12, NTST=128, NCOL=4, NMX=10... In PAGE 53: ...56 29.87% Table3 4: With Pivoting 2: NDIM=12, NTST=256, NCOL=4, NMX=10 Number of Nodes Execution time Speed-up E ciency 1 0.86536E+03 1 100% 2 0.... In PAGE 53: ...30 44.22% Table3 5: With Pivoting 2: NDIM=24, NTST=64, NCOL=4, NMX=10 Number of Nodes Execution time Relative Speed-up Relative E ciency 4 0.44720E+03 1 100% 16 0.... In PAGE 53: ...59 53.67% Table3 6: With Pivoting 2: NDIM=24, NTST=128, NCOL=4, NMX=10... In PAGE 54: ...63 57.90% Table3 7: With Pivoting 2: NDIM=24, NTST=256, NCOL=4, NMX=10 Number of Nodes Execution time Relative Speed-up Relative E ciency 2 0.14757E+04 1 100% 4 0.... In PAGE 54: ...76 73.52% Table3 8: With Pivoting 2: NDIM=48, NTST=32, NCOL=4, NMX=10 Number of Nodes Execution time Relative Speed-up Relative E ciency 8 0.76609E+03 1 100% 16 0.... In PAGE 54: ...40 67.44% Table3 9: With Pivoting 2: NDIM=48, NTST=64, NCOL=4, NMX=10 Number of Nodes Execution time Relative Speed-up Relative E ciency 16 0.78489E+03 1 100% 32 0.... ..."

### Table 3. The recursive algorithm to compute optimal max- imal paths.

1996

"... In PAGE 18: ... A cycle of this kind will be called a looplet, or irregular cycle. The procedure optimize(p) in Table3 can be modi ed slightly so that cycles are detected and handled appropriately. The modi ed procedure optimize0 works as follows.... ..."

### Table 3. Blocked Recursive FW Algorithm (FWR)

"... In PAGE 2: ... Therefore, it is possible to perform FWI recursively, and the recursive version will be referred to as FWR for the rest of the paper. The code for FWR is shown in Table3 . There is also a tiled version of FW, which is simply a recursion by only one level.... ..."

### Table 1. Modified recursive graph bisection algorithm

1995

Cited by 6