1. $C^3$ Partial Factorization versus POLYLIB, 64-bit

We now compare $C^3$ convex hull implementation using partial factorization for 64-bit, denoted C3C64, and POLYLIB's convex hull implementation for 64-bit, denoted C3P64, in order to see whether the partial factorization, which can be seen as a pre-processing step, improves the convex hull computation or not.

In 6_fig:PerfectClub_convex_hull_dimension_C3C64_C3P64, the histogram shows green zones meaning the cases where C3P64 is faster than C3C64 and red zones meaning the opposite cases. The blue zones mean we cannot distinguish the difference because of the time resolution . We notice that C3C64 is faster in $67$ percent of all tests, whereas C3P64 is faster in only $7$ percent of all tests. But, the total execution time ratio for this database indicates that C3P64 is faster than C3C64 with a ratio of $0,78:1$, which means approximately $1,25$ times faster. This means in some cases of PerfectClub databases, the partial factorization does not work very well.

On the contrary, 6_fig:SPEC95_convex_hull_dimension_C3C64_C3P64 presents a much better performance of the partial factorization C3C64, with SPEC95 databases. Indeed, C3C64 is faster in $63$ percent of all tests, whereas C3P64 is faster in only $7$ percent of all tests, and most important, C3C64 is $5$ times faster than C3P64 by the total accumulated run times.

Figure 37: PerfectClub: Dimension C3C64 vs C3P64 in full database

Figure 38: SPEC95: Dimension C3C64 vs C3P64 in full database