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:
- 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   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 a method function with options.