Point -- a type used to store a point in complex space

• coordinates -- get the coordinates (returns a list)
• status -- get the type of solution (e.g., Regular)
• matrix -- get the coordinates (returns a matrix)

• Regular -- the jacobian of the polynomial system is regular at the point
• Singular -- the jacobian of the polynomial system is (near)singular at the point
• Infinity -- the solution path has been deemed divergent
• MinStepFailure -- the tracker failed to stay above the minimal step increment threshold
• NumericalRankFailure -- it is likely that in a sequence of deflations numerical rank did not give the correct rank
• -- the point has not been classified

i1 : loadPackage "NumericalAlgebraicGeometry";

i2 : R = CC[x,y];

i3 : sols = solveSystem{x^2+y^2-3, x^3-y^3-7}

o3 = {{-1.7957+1.31322*ii, 1.7957+1.31322*ii}, {-1.7957-1.31322*ii,
     ------------------------------------------------------------------------
     1.7957-1.31322*ii}, {-.101284-.779159*ii, -1.89699+.041601*ii},
     ------------------------------------------------------------------------
     {1.89699+.041601*ii, .101284-.779159*ii}, {1.89699-.041601*ii,
     ------------------------------------------------------------------------
     .101284+.779159*ii}, {-.101284+.779159*ii, -1.89699-.041601*ii}}

o3 : List

i4 : pt = first sols

o4 = {-1.7957+1.31322*ii, 1.7957+1.31322*ii}

o4 : Point

i5 : peek pt

o5 = Point{ConditionNumber => 5.99383                            }
           Coordinates => {-1.7957+1.31322*ii, 1.7957+1.31322*ii}
           ErrorBoundEstimate => 2.99478e-16
           LastT => 1
           Multiplicity => 1
           SolutionStatus => Regular

i6 : coordinates pt

o6 = {-1.7957+1.31322*ii, 1.7957+1.31322*ii}

o6 : List

i7 : status pt

o7 = Regular

o7 : Symbol

For example, one may see the condition number of the Jacobian of the polynomial system, evaluated at this point (the smaller the value, the better) as follows.

i8 : pt.ConditionNumber

o8 = 5.99382650733725

o8 : RR (of precision 53)

The other keys that may be attached include

• NumberOfSteps -- the number of steps in made by the continuation procedure
• LastT -- the last value of the continuation parameter produced during tracking (equals 1 for a regular solution)
• ErrorBoundEstimate -- an estimate of the distance from the approximation to the actual solution
• MaxPrecision -- max precision used during the homotopy tracking
• WindingNumber -- the winding numeber of a singular solution determined in the end-games
• DeflationNumber -- number of first-order deflations in the regularization of a singular solution
• Tracker -- reserved for developers