SpecializedParameterHomotopy -- a homotopy obtained from a parameter homotopy by specializing parameters

Description

An object GateParameterHomotopy with parameters can be specified to this type by specialize with given parameter values. This type can be passed to trackHomotopy.

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

o4 = GateParameterHomotopy{...2...}

o4 : GateParameterHomotopy

i5 : (a0,b0) = (1,2); startSolution = point{{1,1}};

i7 : (a1,b1) = (2,1);

i8 : H01 = specialize(PH, matrix{{a0,b0,a1,b1}});

i9 : targetSolution = first trackHomotopy(H01,{startSolution})

o9 = targetSolution

o9 : Point

i10 : assert(norm evaluate(F,matrix{{a1,b1}},matrix targetSolution) < 0.0001)

Methods that use an object of class SpecializedParameterHomotopy:

evaluateH(SpecializedParameterHomotopy,Matrix,Number) -- see evaluateH -- evaluate a homotopy and its derivatives
evaluateHt(SpecializedParameterHomotopy,Matrix,Number) -- see evaluateH -- evaluate a homotopy and its derivatives
evaluateHx(SpecializedParameterHomotopy,Matrix,Number) -- see evaluateH -- evaluate a homotopy and its derivatives
numVariables(SpecializedParameterHomotopy) -- the number of variables in the parameter homotopy

For the programmer

The object SpecializedParameterHomotopy is a type, with ancestor classes Homotopy < MutableHashTable < HashTable < Thing.

The source of this document is in NumericalAlgebraicGeometry/doc.m2:1433:0.