specialize -- specialize parameters

Usage:

F = specialize(G,p)

Hp = specialize(H,p)
Inputs:
- G, a gate system with parameters
- H, a parameter homotopy
- p, values of the parameters
Outputs:
- F, a gate system with parameters specialized
- Hp, a specialized homotopy

Description

This method specializes parameters by assigning them the values in p.

If G is a GateSystem that is defined with parameters, then the output is a GateSystem in which the parameters of G have been replaced by the values in p.

i1 : variables = declareVariable \ {x,y}

o1 = {x, y}

o1 : List

i2 : params = declareVariable \ {a,b}

o2 = {a, b}

o2 : List

i3 : Fab = 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 : F = specialize(Fab, point{{1,2}})

o4 = gate system: 2 input(s) ---> 2 output(s)

o4 : GateSystem

i5 : p0 = point{{0.1,0.2+ii}}

o5 = p0

o5 : Point

i6 : evaluate(F,p0)

o6 = | -1 -.959+.4ii |

                1         2
o6 : Matrix CC    <-- CC
              53        53

i7 : evaluateJacobian(F,p0)

o7 = | 0   0      |
     | .03 .4+2ii |

                2         2
o7 : Matrix CC    <-- CC
              53        53

If H is a ParameterHomotopy, then the output is a SpecializedParameterHomotopy, which can be passed to trackHomotopy.

i8 : variables = declareVariable \ {x,y}

o8 = {x, y}

o8 : List

i9 : params = declareVariable \ {a,b}

o9 = {a, b}

o9 : List

i10 : F = gateSystem(matrix{params}, matrix{variables}, matrix{{a*x*y-1},{x^3+y^2-b}})

o10 = gate system: 2 input(s) ---> 2 output(s) (with 2 parameters)

o10 : GateSystem

i11 : PH = parametricSegmentHomotopy F;

i12 : parameters PH

o12 = {{A , A , B , B }}
         a   b   a   b

o12 : GateMatrix

i13 : (a0,b0) = (1,2); startSolution = point{{1,1}};

i15 : (a1,b1) = (2,1);

i16 : H01 = specialize(PH, matrix{{a0,b0,a1,b1}});

i17 : targetSolution = first trackHomotopy(H01,{startSolution})

o17 = targetSolution

o17 : Point

i18 : assert(norm evaluate(F,matrix{{a1,b1}},matrix targetSolution) < 0.0001)

Ways to use specialize:

specialize(GateSystem,AbstractPoint)
specialize(ParameterHomotopy,Matrix)

For the programmer

The object specialize is a method function.

The source of this document is in NumericalAlgebraicGeometry/doc.m2:1187:0.