This type is used to store a solution to a polynomial system obtained by such fuctions as
solveSystem (missing documentation),
track (missing documentation). The following methods can be used to access a
Point:
- coordinates -- get the coordinates (returns a list)
- status -- get the type of solution (e.g., Regular)
- matrix -- get the coordinates (returns a matrix)
Possible types of Points (accessed by
status):
- 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
Only coordinates are displayed (by
net); to see the rest use
peek. Different algorithms attach different information describing the point. For example, the solveSystem function with default options produces the following.
i1 : loadPackage "NumericalAlgebraicGeometry";
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i2 : R = CC[x,y];
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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},
------------------------------------------------------------------------
{-.101284+.779159*ii, -1.89699-.041601*ii}, {1.89699+.041601*ii,
------------------------------------------------------------------------
.101284-.779159*ii}, {1.89699-.041601*ii, .101284+.779159*ii}}
o3 : List
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i4 : pt = first sols
o4 = {-1.7957-1.31322*ii, 1.7957-1.31322*ii}
o4 : Point
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i5 : peek pt
o5 = Point{ConditionNumber => 5.99383 }
Coordinates => {-1.7957-1.31322*ii, 1.7957-1.31322*ii}
ErrorBoundEstimate => 2.26602e-16
LastT => 1
Multiplicity => 1
SolutionStatus => Regular
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i6 : coordinates pt
o6 = {-1.7957-1.31322*ii, 1.7957-1.31322*ii}
o6 : List
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i7 : status pt
o7 = Regular
o7 : Symbol
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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)
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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