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Bertini :: bertiniTrackHomotopy

bertiniTrackHomotopy -- track a user-defined homotopy

Synopsis

Description

This method calls Bertini to track a user-defined homotopy. The user needs to specify the homotopy H, the path variable t, and a list of start solutions S1. In the following example, we solve x2-2 by moving from x2-1 with a linear homotopy. Bertini tracks homotopies starting at t=1 and ending at t=0. Final solutions are of type Point.

i1 : R = CC[x,t]; -- include the path variable in the ring
i2 : H = { (x^2-1)*t + (x^2-2)*(1-t)};
i3 : sol1 = point {{1}};
i4 : sol2 = point {{-1}};
i5 : S1= { sol1, sol2  };--solutions to H when t=1
i6 : S0 = bertiniTrackHomotopy (t, H, S1) --solutions to H when t=0

o6 = {{-1.41421}, {1.41421}}

o6 : List
i7 : peek S0_0

o7 = Point{ConditionNumber => 1           }
           Coordinates => {-1.41421}
           CycleNumber => 1
           FunctionResidual => 4.44089e-16
           LastT => .0015625
           MaximumPrecision => 52
           Multiplicity => 1
           NewtonResidual => 0
           SolutionNumber => -1
           SolutionStatus => Regular
i8 : R=CC[x,y,t]; -- include the path variable in the ring
i9 : f1=(x^2-y^2);
i10 : f2=(2*x^2-3*x*y+5*y^2);
i11 : H = { f1*t + f2*(1-t)}; --H is a list of polynomials in x,y,t
i12 : sol1=    point{{1,1}}--{{x,y}} coordinates

o12 = sol1

o12 : Point
i13 : sol2=    point{{ -1,1}}

o13 = sol2

o13 : Point
i14 : S1={sol1,sol2}--solutions to H when t=1

o14 = {sol1, sol2}

o14 : List
i15 : S0=bertiniTrackHomotopy(t, H, S1, IsProjective=>1) --solutions to H when t=0

o15 = {{-1.45358+1.47955*ii, .387705+1.25319*ii}, {-1.14653-.745214*ii,
      -----------------------------------------------------------------------
      .070959-.861925*ii}}

o15 : List

Caveat

Variables must begin with a letter (lowercase or capital) and can only contain letters, numbers, underscores, and square brackets.

Ways to use bertiniTrackHomotopy :