bertiniUserHomotopy -- a main method to track a user-defined homotopy

Synopsis

• Usage:

  S0=bertiniUserHomotopy(t, P, H, S1)

• Inputs:
  • t, a ring element, a path variable
  • P, a list, a list of options that set the parameters
  • H, a list, a list of polynomials that define the homotopy
  • S1, a list, a list of solutions to the start system
• Optional inputs:
  • AffVariableGroup => ..., default value {}, an option to group variables and use multihomogeneous homotopies
  • B'Constants (missing documentation) => ..., default value {},
  • B'Functions (missing documentation) => ..., default value {},
  • BertiniInputConfiguration (missing documentation) => ..., default value {},
  • HomVariableGroup (missing documentation) => ..., default value {},
  • M2Precision (missing documentation) => ..., default value 53,
  • OutputStyle (missing documentation) => ..., default value OutPoints,
  • RandomComplex (missing documentation) => ..., default value {},
  • RandomReal (missing documentation) => ..., default value {},
  • TopDirectory => ..., default value /tmp/M2-107405-0/0, Option to change directory for file storage.
  • Verbose => ..., default value false, Option to silence additional output
• Outputs:
  • S0, a list, a list of solutions to the target system

Description

i1 : R = CC[x,a,t]; -- include the path variable in the ring

i2 : H = { (x^2-1)*a + (x^2-2)*(1-a)};

i3 : sol1 = point {{1}};

i4 : sol2 = point {{-1}};

i5 : S1= { sol1, sol2  };--solutions to H when t=1

i6 : S0 = bertiniUserHomotopy (t,{a=>t}, H, S1) --solutions to H when t=0

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

o6 : List

i7 : peek S0_0

o7 = Point{AccuracyEst => 1.21205e-14        }
           AccuracyEstInternal => 1.43214e-14
           ConditionNumber => 69.2692
           Coordinates => {1.41421}
           CycleNumber => 1
           FinalTValue => .000390625
           FunctionResidual => 3.97205e-15
           MaxPrecisionUtilized => 52
           Multiplicity => 1
           NewtonResidual => 1.04182e-15
           PathNumber => 0
           PathsWithSameEndpoint => {}
           PrecisionIncreased => 0
           SolutionNumber => 1

i8 : R=CC[x,y,t,a]; -- 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*a + f2*(1-a)}; --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=bertiniUserHomotopy(t,{a=>t}, H, S1, HomVariableGroup=>{x,y}) --solutions to H when t=0

o15 = {{1, .3+.556776*ii}, {1, .3-.556776*ii}}

o15 : List