A compliant mechanism

• Save the files polyExpG, polyExpGp, B, Bp, config96, config1024, configNewton96, and configNewton1024 in the same directory and copy the binary of alphaCertified into this directory.
• Execute the command
        >   ./alphaCertified   polyExpG   B   config96
  from this directory.
• Here is the output displayed to the screen. From this output, we see that, using 96-bit precision, both B₁ and B₂ are approximate solutions of G = 0 with real distinct associated solutions.
• The file summary provides soft certified upper bounds, to four digits, of 0.0166 and 0.0427 for &alpha(G,B₁) and &alpha(G,B₂), respectively.
• Execute the command
        >   ./alphaCertified   polyExpG   B   config1024
  from this directory.
• Here is the output displayed to the screen. From this output, we see that, using 1024-bit precision, both B₁ and B₂ are approximate solutions of G = 0 with real distinct associated solutions.
• The file summary provides soft certified upper bounds, to four digits, of 0.0166 and 0.0427 for &alpha(G,B₁) and &alpha(G,B₂), respectively.
• Execute the command
        >   ./alphaCertified   polyExpGp   Bp   config96
  from this directory.
• Here is the output displayed to the screen. The file summary produced by the previous command with the above summary file provides the data for the following table.
  
  
• In order to certify the solutions for G' = 0, we need to first apply a Newton iteration. To do this using 96-bit precision, execute the command
        >   ./alphaCertified   polyExpGp   Bp   configNewton96
  from this directory.
• Here is the output displayed to the screen. By looking at the final values of &alpha in the file summary, we see that the refined points are approximate solutions.
• To utilize 1024-bit precision, execute the command
        >   ./alphaCertified   polyExpGp   Bp   configNewton1024
  from this directory.
• Here is the output displayed to the screen. As above, by looking at the final values of &alpha in the file summary, we see that the refined points are approximate solutions.

• Save the files inputStart, input_top, input_bottom, inputSolvePoly, inputPolyToExp, setupCoeffParam.c, and polyExpG, and config96 in the same directory and copy the binaries of Bertini and alphaCertified into this directory.
• Execute the command
        >   ./bertini   inputStart
  from this directory.
• Here is the screen output. From this output, we see that Bertini has heuristically computed two nonsingular isolated solutions for this starting system. The file nonsingular_solutions provides the two numerical approximations computed by Bertini.
• Execute the commands
        >   cp   nonsingular_solutions   start
        >   gcc   setupCoeffParam.c
        >   ./a.out
  from this directory.
• This yields 356 points in final_solutions.
• Execute the commands
        >   cp   final_solutions   start
        >   ./bertini   inputSolvePoly
  from this directory.
• Here is the screen output. The file nonsingular_solutions contains 120 points.
• Execute the commands
        >   cp   nonsingular_solutions   start
        >   ./bertini   inputPolyToExp
  from this directory.
• Here is the screen output. The file nonsingular_solutions contains 93 points.
• Execute the command
        >   ./alphaCertified   polyExpG   nonsingular_solutions   config96
  from this directory.
• Here is the output displayed to the screen and summary. From this output, we see that, using 96-bit precision, the 93 points computed by Bertini are approximate solutions of G = 0 and that 65 of them have real associated solutions of G = 0.