alphaCertified v1.2.0 (August 15, 2011)
  Jonathan D. Hauenstein and Frank Sottile
         GMP v4.3.1 & MPFR v2.3.1


Please note that all coefficients must be complex rational numbers.

alphaCertified is using the polynomial certification algorithms.

Analyzing 18 points using exact arithmetic.

Isolating 18 approximate solutions.

Classifying 18 distinct approximate solutions.

Rational certification results:

Number of points tested:           18
Certified approximate solutions:   18
Certified distinct solutions:      18
Certified real distinct solutions: 2

------------------------------------------------------------------------------------------------------------
The following files have been created:

approxSolns:          A list of points that are certified approximate solutions.
constantValues:       A list of the values of alpha, beta, and gamma for the points.
distinctSolns:        A list of points that correspond to distinct solutions.
isApproxSoln:         A list which describes if the ith point is an approximate solution.
isDistinctSoln:       A list which describes if the ith point is listed in 'distinctSolns'.
isRealSoln:           A list which describes if the ith point corresponds to a real solution.
nonrealDistinctSolns: A list of points that correspond to distinct nonreal solutions.
realDistinctSolns:    A list of points that correspond to distinct real solutions.
redundantSolns:       A list of points that correspond to the same solution as one in 'distinctSolns'.
refinedPoints:        A list of points that are the best internally computed approximation of each solution.
summary:              A human-readable summary for each point - main output file.
unknownPoints:        A list of points that which cannot be certified as approximate solutions.
------------------------------------------------------------------------------------------------------------