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NumericalAlgebraicGeometry -- Numerical Algebraic Geometry


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

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 Certified tracking:



Certification a gold star

Version 1.4 of this package was accepted for publication in volume 3 of 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.


This documentation describes version 1.23 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/.


For the programmer

The object NumericalAlgebraicGeometry is a package.