gateHomotopy(GateMatrix,GateMatrix,InputGate) -- homotopy implemented via straight line programs

• Function: gateHomotopy

• Usage:

  HS = gateHomotopy(H,X,T)

• Inputs:
  • H, an instance of the type GateMatrix, a family of systems (given by a column vector)
  • X, an instance of the type GateMatrix, (a row vector of) variables
  • T, an instance of the type InputGate, homotopy (continuation) parameter
• Optional inputs:
  • Parameters => ..., default value null, specifies parameter names
  • Software => ..., default value null, specifies where evaluation should be done (M2=top level, M2engine=core)
  • Strategy => ..., default value compress, strategy is either to "compress" or not (any other value)
• Outputs:
  • HS
  • an instance of the type GateHomotopy or an instance of the type GateParameterHomotopy, a homotopy that can be used with some routines of NumericalAlgebraicGeometry

i1 : X = inputGate symbol X

o1 = X

o1 : InputGate

i2 : Y = inputGate symbol Y

o2 = Y

o2 : InputGate

i3 : T = inputGate symbol T

o3 = T

o3 : InputGate

i4 : F = {X*X-1, Y*Y*Y-1}

o4 = {((X * X) + -1), (((Y * Y) * Y) + -1)}

o4 : List

i5 : G = {X*X+Y*Y-1, X*X*X+Y*Y*Y-1}

o5 = {(((X * X) + (Y * Y)) + -1), ((((X * X) * X) + ((Y * Y) * Y)) + -1)}

o5 : List

i6 : H = (1 - T) * F + T * G

o6 = {(((1 + (-1 * T)) * ((X * X) + -1)) + (T * (((X * X) + (Y * Y)) + -1))),
     ------------------------------------------------------------------------
     (((1 + (-1 * T)) * (((Y * Y) * Y) + -1)) + (T * ((((X * X) * X) + ((Y *
     ------------------------------------------------------------------------
     Y) * Y)) + -1)))}

o6 : List

i7 : HS = gateHomotopy(transpose matrix {H},matrix{{X,Y}},T)

o7 = GateHomotopy{...11...}

o7 : GateHomotopy