# endGameCauchy -- Cauchy end game for getting a better approximation of a singular solution

## Synopsis

• Usage:
endGameCauchy(H,t'end,p0)
endGameCauchy(H,t'end,points)
• Inputs:
• H, an instance of the type GateHomotopy,
• t'end, ,
• p0, ,
• points, ,
• Optional inputs:
• backtrack factor
• CorrectorTolerance => ..., default value null, options for core functions of Numerical Algebraic Geometry
• EndZoneFactor => ..., default value null, options for core functions of Numerical Algebraic Geometry
• InfinityThreshold => ..., default value null, options for core functions of Numerical Algebraic Geometry
• maxCorrSteps => ..., default value null, options for core functions of Numerical Algebraic Geometry
• number of vertices
• numberSuccessesBeforeIncrease => ..., default value null, options for core functions of Numerical Algebraic Geometry
• stepIncreaseFactor => ..., default value null, options for core functions of Numerical Algebraic Geometry
• tStep => ..., default value null, options for core functions of Numerical Algebraic Geometry
• tStepMin => ..., default value null, options for core functions of Numerical Algebraic Geometry

## Description

Refines an approximation of a (singular) solution to a polynomial system which was obtained via homotopy continuation. This method is used for posprocessing in the blackbox solver implemented in solveSystem.

 i1 : CC[x,y] o1 = CC [x..y] 53 o1 : PolynomialRing i2 : T = {(x-2)^3,y-x+x^2-x^3} 3 2 3 2 o2 = {x - 6x + 12x - 8, - x + x - x + y} o2 : List i3 : sols = solveSystem(T,PostProcess=>false); i4 : p0 = first sols; i5 : peek p0 o5 = Point{Coordinates => {2.01079-.011783*ii, 434.14+1247.66*ii}} H => GateHomotopy{...12...} LastIncrement => 5.72205e-7 LastT => .999999 NumberOfSteps => 26 SolutionStatus => MinStepFailure i6 : t'end = 1 o6 = 1 i7 : p = endGameCauchy(p0#"H",t'end,p0) o7 = p o7 : Point