alpha theory uses three auxiliary quantities related to the input polynomial system and point.
Beta value is defined by the length of the Newton step and gamma value is the quantity which is inversely proportional to the length between exact solution and the given point.
Alpha value is defined by the multiplication of beta and gamma. When it is smaller than $0.157671$, then the input point is an approximate solution to the system. The function certifySolution does this process.
i1 : R = QQ[x1,x2,y1,y2]; |
i2 : f = polySystem {3*y1 + 2*y2 -1, 3*x1 + 2*x2 -7/2,x1^2 + y1^2 -1, x2^2 + y2^2 - 1}; |
i3 : p = point{{.95,.32,-.30,.95}}; |
i4 : (a, b, g) = computeConstants(f,p) o4 = (.0505356, .00526407, 9.60011) o4 : Sequence |
The object computeConstants is a method function.