Execution Modes and Tools

Since we have to respect a strict processing order and we may expect computationally lengthy jobs, PHC is organized as a menu-driven and file-oriented program.

The simplest way to solve systems by PHC is to type

   phc -b input output

when input is the name of the input file that contains the system. This mode is the so-called black-box mode and requires no other input than the polynomial system. Results can be found in the file output. One particular choice for a black-box solver is outlined in section nine.

The second mode is the full mode where PHC runs through all stages of the solver and asks the user to confirm the default settings while giving the opportunity to modify the settings interactively. This mode is invoked by default, just by typing phc after the prompt.

Some stages may be skipped, whereas more than one root-counting method can be invoked before the construction of a homotopy. Therefore, the tool mode has been created, see Figure 5. Another advantage of working with tools is that intermediate results, such as a mixed subdivision and a random coefficient start system, can be valuable stepping stones in the resolution of a large and difficult system.

  

Figure 5: Schematic overview of the tools offered by the package PHC.

Table 1 gives an overview of the tools and the options of PHC to invoke them.

 

stage acronym description of the tool option

1 scal coefficient scaling phc -s

redu reduction of degrees phc -d

2 roco root counts and start systems phc -r

mvc mixed-volume computation phc -m

3 poco polynomial continuation phc -p

4 vali validation of results phc -v

x enum enumerative geometry phc -e

 

The need for a separate tool for mvc comes from the amount of computational work that is not negligible for computing mixed volumes and performing polyhedral continuation. The reduction redu tool applies S-polynomials as described in [Verschelde and Cools 1992]. This technique generalizes the linear reduction on the coefficient matrix of the system [Morgan 1987].