track(List,List,List) -- track a user homotopy

Synopsis

• Usage:

  solsT = track(S,T,solsS)

• Function: track
• Inputs:

  • S, a list, List of polynomials in the start system
  • T, a list, List of polynomials in the target system
  • solsS, a list, List of start solutions
• Outputs:

  • a list, solsT, solutions of T=0 obtained by continuing solsS
• Optional inputs:

  • gamma => default value 1, a parameter in the homotopy: H(t)=(1-t)^tDegree S + gamma t^tDegree T
  • tDegree => default value 1, a parameter in the homotopy: H(t)=(1-t)^tDegree S + gamma t^tDegree T
  • tStep => default value .05, initial step size
  • tStepMin => default value .000001, minimal step size
  • stepIncreaseFactor => default value 2, determine how step size is adjusted
  • numberSuccessesBeforeIncrease => default value 4, determine how step size is adjusted
  • Predictor => default value RungeKutta4, a method to predict the next point on the homotopy path: choose between RungeKutta4, Tangent, Euler, Secant, ProjectiveNewton. The latter provides certified tracking.
  • maxCorrSteps => default value 3, max number of steps corrector takes before a failure is declared
  • CorrectorTolerance => default value .000001, corrector succeeds if the relative error does not esceed this tolerance
  • EndZoneFactor => default value .05, size of `end zone'
  • InfinityThreshold => default value 100000, paths are truncated if the norm of the approximation exceeds the threshold
  • Projectivize => default value false, if true then the system is homogenized and projective tracker is executed
  • Normalize => default value false, normalize the start and target systems w.r.t. Bombieri-Weyl norm
  • NoOutput => default value false, if true, no output is produced (useful in combination with getSolution)
  • AffinePatches => ..., -- track a user homotopy
  • MultistepDegree => ..., -- track a user homotopy
  • SLP => ..., -- track a user homotopy
  • SLPcorrector => ..., -- track a user homotopy
  • SLPpredictor => ..., -- track a user homotopy
  • Software => ..., -- specify software for the solver

Description

i1 : R = CC[x,y];

i2 : S = {x^2-1,y^2-1};

i3 : T = {x^2+y^2-1, x*y};

i4 : solsS = {(1,-1),(1,1),(-1,1),(-1,-1)};

i5 : track(S,T,solsS) / first 

o5 = {{-3.8337e-16, -1}, {1, 1.17962e-15}, {3.8337e-16, 1}, {-1,
     ------------------------------------------------------------------------
     -1.17962e-15}}

o5 : List

Caveat