The blackbox solver works reasonably well to approximate all isolated solutions of a polynomial system. On the system we saved earlier in the file multilin, we invoke the blackbox solver as
/tmp/phc -b multilin multilin.phc
The output of the solver will be sent to the file multilin.phc.
In case the input file did not yet contain any solutions, the solution
list will be appended to the input file.
We now explain the format of the solutions, for example, the last solution in the list occurs in the following format:
solution 44 : start residual : 1.887E-14 #iterations : 1 success t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : a : 5.50304308029581E+00 -6.13068078142107E-44 b : 8.32523889626848E+00 -5.18918337570284E-45 c : 1.01021324864917E+01 -1.29182202179944E-45 d : 1.42724963260133E+01 1.38159270467025E-44 f : 4.34451307203401E+01 -6.26380413553193E-43 == err : 3.829E-12 = rco : 3.749E-03 = res : 2.730E-14 = real regular ==This is the actual output of the root refiner. As the residual at the end of the solution path and at the start of the root refinement is already 1.887E-14 = 1.887 10-14, one iteration of Newton's method suffices to confirm the quality of the root.
The next line in the output indicates that we reached the end of the path, at t=1, properly. The multiplicity of the root is one, as indicated by m = 1. Then we see the values for the five variables, as pairs of two floating-point numbers: the real and imaginary part of each value. The last line summarizes the numerical quality of the root. The value for err is the magnitude of the last correction term used in Newton's method. The number for rco is an estimate for the inverse condition number of the root. Here this means that we are guaranteed to have all decimal places correct, except for the last three decimal places. The last number represents the residual, the magnitude of the vector evaluated at the root.