parametricSegmentHomotopy -- creates an ansatz for a segment homotopy

Synopsis

Usage:

PH = parametricSegmentHomotopy F
Inputs:
- F, an instance of the type System, either a GateSystem or a PolySystem
Outputs:
- PH, an instance of the type GateParameterHomotopy,

Description

This method returns a homotopy that after specialization of parameters is akin to the output of segmentHomotopy. There are 2 m parameters inPH where m is the number of parameters in F. The first m parameters correspond to the starting point A in the parameter space. The last m parameters correspond to the end point B in the parameter space.

i1 : variables = declareVariable \ {x,y}

o1 = {x, y}

o1 : List

i2 : params = declareVariable \ {a,b}

o2 = {a, b}

o2 : List

i3 : F = gateSystem(matrix{params}, matrix{variables}, matrix{{a*x*y-1},{x^3+y^2-b}})

o3 = gate system: 2 input(s) ---> 2 output(s) (with 2 parameters)

o3 : GateSystem

i4 : PH = parametricSegmentHomotopy F;

i5 : parameters PH

o5 = {{A , A , B , B }}
        a   b   a   b

o5 : GateMatrix

i6 : (a0,b0) = (1,2); startSolution = point{{1,1}};

i8 : (a1,b1) = (2,1);

i9 : H01 = specialize(PH, matrix{{a0,b0,a1,b1}});

i10 : targetSolution = first trackHomotopy(H01,{startSolution})

o10 = targetSolution

o10 : Point

i11 : assert(norm evaluate(F,matrix{{a1,b1}},matrix targetSolution) < 0.0001)

Ways to use parametricSegmentHomotopy :

parametricSegmentHomotopy(GateSystem)
parametricSegmentHomotopy(PolySystem)

For the programmer

The object parametricSegmentHomotopy is a method function.