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# segmentHomotopy -- a segment homotopy

- 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:
  - H, an instance of the type GateHomotopy,

## 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   9 2    3   2   7 3      7 2    7       7 2
    o2 = {-x  + -x y + -x*y  + -y  - 1, -x  + --x*y + --y  - 2}
          2     4      4       4        9     10      10

    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 a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NumericalAlgebraicGeometry/doc.m2:1422:0.