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NAGtypes :: polynomial homotopy

polynomial homotopy -- basic methods for manipulating polynomial homotopies

Construct a polynomial system representing a segment homotopy H = (1-t) F + t G for t ∈[0,1].

i1 : R = CC[x,y]; 
i2 : S = polySystem {x^2-1, y^2-1};
i3 : T = polySystem {x^2+y^2-6, 2*x^2-y};
i4 : H := segmentHomotopy(S,T)

o4 = {-2} | y2t+x2-5t-1        |
     {-2} | 2x2t-y2t+y2-yt+t-1 |

o4 : PolySystem

Specialize the continuation parameter:

i5 : specializeContinuationParameter(H,0)

o5 = {-2} | x2-1 |
     {-2} | y2-1 |

o5 : PolySystem
i6 : specializeContinuationParameter(H,1)

o6 = {-2} | x2+y2-6 |
     {-2} | 2x2-y   |

o6 : PolySystem
i7 : specializeContinuationParameter(H,2+3*ii)

o7 = {-2} | x2+(2+3i)y2-11-15i             |
     {-2} | (4+6i)x2-(1+3i)y2-(2+3i)y+1+3i |

o7 : PolySystem

Substitute 1-t for the continuation parameter to swap the ends of the homotopy.

i8 : t := H.ContinuationParameter

o8 = t

o8 : CC  [x, y, t]
       53
i9 : H' := substituteContinuationParameter(H,1-t)

o9 = {-2} | -y2t+x2+y2+5t-6      |
     {-2} | -2x2t+y2t+2x2+yt-y-t |

o9 : PolySystem

See also