# segmentHomotopy -- a segment homotopy

## Synopsis

• Usage:
H = segmentHomotopy(S,T)
• Inputs:
• S, an instance of the type System, start system (could be GateSystem, PolySystem, or list of polynomials)
• T, an instance of the type System, target system (could be GateSystem, PolySystem, or list of polynomials)
• Optional inputs:
• gamma => ..., default value 1, options for core functions of Numerical Algebraic Geometry
• Outputs:

## Description

This method produces a Homotopy (1-t) S+ t \gamma T, t\in[0,1].

 i1 : R = QQ[x,y] o1 = R o1 : PolynomialRing i2 : T = {random(3,R)-1, random(2,R)-2} 9 3 1 2 9 2 1 3 2 3 3 2 o2 = {-x + -x y + -x*y + -y - 1, x + -x*y + -y - 2} 2 2 4 2 4 2 o2 : List i3 : (S,solsS) = totalDegreeStartSystem T 3 2 o3 = ({x - 1, y - 1}, {{1, -1}, {-.5-.866025*ii, -1}, {-.5+.866025*ii, -1}, ------------------------------------------------------------------------ {-.5-.866025*ii, 1}, {1, 1}, {-.5+.866025*ii, 1}}) o3 : Sequence i4 : H = segmentHomotopy(S,T,gamma=>1+ii) o4 = GateHomotopy{...11...} o4 : GateHomotopy i5 : evaluateH(H,transpose matrix first solsS,0) o5 = | 0 | | -2.44929e-16ii | 2 1 o5 : Matrix CC <--- CC 53 53

## Ways to use segmentHomotopy :

• "segmentHomotopy(GateSystem,GateSystem)"
• "segmentHomotopy(List,List)"
• "segmentHomotopy(PolySystem,PolySystem)"

## For the programmer

The object segmentHomotopy is .