trackPaths -- tracks paths defined by a typical homotopy between start and end systems

Synopsis

• Usage:

  solsT = trackPaths(T,S,solsS)

• Inputs:
  • T, a list, of polynomials, called the target system
  • S, a list, of polynomials, called the start system
  • solsS, a list, solutions of S
• Optional inputs:
  • gamma => ..., default value 0, Option to specify the value of the constant in the gamma trick
  • interactive => ..., default value false, Option to run phc -p in interactive mode.
  • intermediateSolutions => ..., default value false, option to get all intermediate solutions on a path
  • loadSettingsPath => ..., default value "", option to load the settings of the path trackers for a reproducible rerun
  • numThreads => ..., default value 0, Option to define the number of threads in the path tracker.
  • saveSettingsPath => ..., default value "", option to save the settings of the path trackers for a reproducible rerun
  • seeProgress => ..., default value false, Option to follow the progress of the multithreaded path tracker.
  • tDegree => ..., default value 2, Option to specify the degree of the continuation parameter
  • Verbose => ..., default value false, option to specify whether additional output is wanted
• Outputs:
  • solsT, a list, containing the solutions of T that are at the ends of the paths starting at solsS
• Consequences:

  • Writes the start system and its solutions to temporary files
  • Invokes the command phc -p
  • After running phc -z, stores output of phc in temporary file
  • Parses and outputs the solutions.

Description

i1 : R = CC[x,y];

i2 : f = { x^3*y^5 + y^2 + x^2*y, x*y + x^2 - 1};

i3 : (m,q,qsols) = mixedVolume(f,StartSystem=>true);

i4 : fsols = trackPaths(f,q,qsols)

o4 = {{-.894935-.624334*ii, .143333+1.14868*ii}, {-1.59272, .964857},
     ------------------------------------------------------------------------
     {.742585-.425943*ii, .270685+1.00715*ii}, {1.33076+.335184*ii,
     ------------------------------------------------------------------------
     -.62414-.513163*ii}, {1.33076-.335184*ii, -.62414+.513163*ii},
     ------------------------------------------------------------------------
     {-.764107, -.544612}, {.742585+.425943*ii, .270685-1.00715*ii},
     ------------------------------------------------------------------------
     {-.894935+.624334*ii, .143333-1.14868*ii}}

o4 : List

See also

• gamma -- the constant in the gamma trick, optional input for trackPaths
• interactive -- flag to run phc -p or phc -m in interactive mode
• intermediateSolutions -- option of trackPaths to get all intermediate solutions on a path
• loadSettingsPath -- option of trackPaths to load the settings for a reproducible rerun
• saveSettingsPath -- option of trackPaths to save the settings for a reproducible rerun
• numThreads -- the number of threads in the path tracker
• seeProgress -- flag to monitor the progress of the multithreaded path tracker
• tDegree -- the degree of the continuation parameter