next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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,
i1 : R = CC[x,y,z];
i2 : F = {x^2+y^2+z^2-1, y-x^2, z-x^3};
i3 : s = solveSystem F 

o3 = {{.540536+1.03152*ii, -.771845+1.11514*ii, -1.5675-.193395*ii},
     ------------------------------------------------------------------------
     {-.540536+1.03152*ii, -.771845-1.11514*ii, 1.5675-.193395*ii},
     ------------------------------------------------------------------------
     {.540536-1.03152*ii, -.771845-1.11514*ii, -1.5675+.193395*ii},
     ------------------------------------------------------------------------
     {-.737353, .543689, -.400891}, {-.540536-1.03152*ii,
     ------------------------------------------------------------------------
     -.771845+1.11514*ii, 1.5675+.193395*ii}, {.737353, .543689, .400891}}

o3 : List
i4 : realPoints s

o4 = {{-.737353, .543689, -.400891}, {.737353, .543689, .400891}}

o4 : List
and describe positive-dimensional complex algebraic varieties,
i5 : R = CC[x,y,z];
i6 : sph = x^2+y^2+z^2-1; 
i7 : I = ideal {x*sph*(y-x^2), sph*(z-x^3)};

o7 : Ideal of R
i8 : numericalIrreducibleDecomposition I 

o8 = a numerical variety with components in
     dim 1:  (dim=1,deg=1) (dim=1,deg=3)
     dim 2:  (dim=2,deg=2)

o8 : NumericalVariety

Basic types (such as Point and WitnessSet) are defined in the package NAGtypes.

Basic functions:

Optionally, the user may outsource some basic routines to Bertini and PHCpack (look for Software option).

Service functions:

Functions related to scheme analysis:

  • isPointEmbedded -- determine if the point is an embedded component of the scheme
  • isPointEmbeddedInCurve -- determine if the point is an embedded component of a 1-dimensional scheme
  • colon -- colon of a (truncated) dual space

Functions related to Certified tracking:

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", Experimental Mathematics 21(1): 69-83 (2012)
  • R. Krone and A. Leykin, "Numerical algorithms for detecting embedded components.", arXiv:1405.7871

Authors

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, svn://svn.macaulay2.com/Macaulay2/trunk/M2/Macaulay2/packages/NumericalAlgebraicGeometry.m2, release number 13254.

Version

This documentation describes version 1.11 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