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NumericalAlgebraicGeometry :: NumericalAlgebraicGeometry

NumericalAlgebraicGeometry -- Numerical Algebraic Geometry

Description

The package NumericalAlgebraicGeometry, also known as NAG4M2 (Numerical Algebraic Geometry for Macaulay2), implements methods of polynomial homotopy continuation to solve systems of polynomial equations and describe positive-dimensional complex algebraic varieties. The current version focuses on solving square systems with a finite number of solutions.

Basic types Point and WitnessSet are defined in the package NAGtypes.

Basic functions:

  • track -- track a user homotopy
  • solveSystem -- solve a square system of polynomial equations
  • refine -- refine numerical solutions to a system of polynomial equations
  • totalDegreeStartSystem -- construct a start system for the total degree homotopy

Service functions:

  • setDefault -- set/get the default parameters for continuation algorithms
  • getDefault -- set/get the default parameters for continuation algorithms
  • areEqual -- determine if solutions are equal
  • sortSolutions -- sort the list of solutions
  • toAffineChart -- coordinates of a point in the projective space in an affine chart
  • NAGtrace -- set the trace level in NumericalAlgebraicGeometry package

Functions related to Certified tracking:

Other functions:

References:

  • A.J. Sommese, J. Verschelde, and C.W. Wampler, "Introduction to numerical algebraic geometry", in "Solving polynomial equations" (2005), 301--338
  • A.J. Sommese and C.W. Wampler, "The numerical solution of systems of polynomials", World Scientific Publishing (2005)
  • C. Beltran and A. Leykin, "Certified numerical homotopy tracking", arXiv:0911.1783

Author

Certification a gold star

Version 1.4 of this package was accepted for publication in volume 3 of the journal The Journal of Software for Algebra and Geometry: Macaulay2 on 2011-05-20, in the article Numerical Algebraic Geometry. That version can be obtained from the journal or from the Macaulay2 source code repository, after installing subversion, with the following shell command:

   svn export -r 13254 svn://svn.macaulay2.com/Macaulay2/trunk/M2/Macaulay2/packages/NumericalAlgebraicGeometry.m2

The following command will display the log messages accompanying any changes to the file in the repository since publication.

   svn log -r 13255:HEAD svn://svn.macaulay2.com/Macaulay2/trunk/M2/Macaulay2/packages/NumericalAlgebraicGeometry.m2

The following command will summarize the changes to the file in the repository since publication, in the format the program diff uses: lines starting with + have been added, and lines starting with - have been removed. (Changes to white space or end of line style will not be reported.)

   svn diff -x "-b --ignore-eol-style" -r 13254:HEAD svn://svn.macaulay2.com/Macaulay2/trunk/M2/Macaulay2/packages/NumericalAlgebraicGeometry.m2

The differences between two releases in the repository mentioned in the log can be displayed by replacing 13254:HEAD by the pair of release numbers separated by a colon.

Version

This documentation describes version 1.5.0.1 of NumericalAlgebraicGeometry.

Source code

The source code from which this documentation is derived is in the file NumericalAlgebraicGeometry.m2. The auxiliary files accompanying it are in the directory NumericalAlgebraicGeometry/.

Exports