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

Description

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

• 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 is 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
• RefinementFailure -- a solution refinement function failed
• Origin -- the solution path approaches the origin (impossible to give a relative error estimate)
• IncreasePrecision -- the current precision is deemed inadequate for robust computation
• DecreasePrecision -- the current precision is deemed excessive (more than the double of sufficient precision)
• null -- 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}, {-.101284+.779159*ii, -1.89699-.041601*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}}

o3 : List

i4 : pt = first sols

o4 = pt

o4 : Point

i5 : peek pt

o5 = Point{Coordinates => {-1.7957-1.31322*ii, 1.7957-1.31322*ii}}
           H => GateHomotopy{...12...}
           LastIncrement => .15
           LastT => 1
           NumberOfSteps => 9
           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

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
• Multiplicity -- the multiplicity of an isolated solution
• 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

Basic service functions:

• areEqual -- determine if solutions are equal
• sortSolutions -- sort the list of solutions
• isRealPoint -- determine whether a point is real
• realPoints -- select real points
• solutionsWithMultiplicity -- replaces clusters of approximately equal points by single points with multiplicity
• norm(Thing,Point) -- p-norm of the point
• toAffineChart -- coordinates of a point in the projective space in an affine chart