NAGtypes -- Common types used in Numerical Algebraic Geometry

Description

The package defines types used by the package NumericalAlgebraicGeometry as well as other numerical algebraic geometry packages: e.g., interface packages PHCpack and Bertini::Bertini.

Datatypes:

Point -- a numerical approximation of a point in a complex space
PolySystem -- a polynomial system (usually with complex coefficients)
WitnessSet -- a witness set representing (possibly positive-dimensional) solution components
NumericalVariety -- a numerical description of a variety
PolySpace -- a polynomial vector subspace
DualSpace -- a dual functional vector subspace

See the corresponding documentation nodes for description of provided service functions.

We display the objects of all new types showing only partial data. Moreover, if an object is assigned to a global variable, only the name of the variable is shown. Use peek for more information.

i1 : R = CC[x,y]     

o1 = R

o1 : PolynomialRing

i2 : I = ideal((x^2+y^2+2)*x,(x^2+y^2+2)*y);

o2 : Ideal of R

i3 : w1 = witnessSet(I , ideal(x-y), {point {{0.999999*ii,0.999999*ii}}, point {{-1.000001*ii,-1.000001*ii}}} )

o3 = w1

o3 : WitnessSet

i4 : O = point {{0.,0.}}

o4 = O

o4 : Point

i5 : numericalVariety {witnessSet(I, ideal R, {O}),w1}

o5 = a numerical variety with components in
     dim 0:  [dim=0,deg=1]
     dim 1:  w1

o5 : NumericalVariety

i6 : V = oo

o6 = V

o6 : NumericalVariety

i7 : peek V

o7 = NumericalVariety{0 => {[dim=0,deg=1]}}
                      1 => {w1}

i8 : peek w1

o8 = WitnessSet{IsIrreducible => null                             }
                Points => {{.999999*ii, .999999*ii}}
                          {{-ii, -ii}              }
                Slice => | 1 -1 0 |
                                     3      2        2     3
                Equations => ideal (x  + x*y  + 2x, x y + y  + 2y)

i9 : peek O

o9 = Point{Coordinates => {0, 0}}

Author

Anton Leykin <leykin@math.gatech.edu>

Version

This documentation describes version 1.9 of NAGtypes.

Source code

The source code from which this documentation is derived is in the file NAGtypes.m2.

Exports

Types
- DualSpace -- a dual functional vector subspace
- NumericalVariety -- a numerical variety
- Point -- a type used to store a point in complex space
- PolySpace -- a polynomial vector subspace
- PolySystem -- a polynomial system
- ProjectiveNumericalVariety -- a projective numerical variety
- ProjectiveWitnessSet -- a projective witness set
- WitnessSet -- a witness set
Functions and commands
- areEqual -- determine if solutions are equal
- dualSpace -- construct a DualSpace
- evaluate -- evaluate a polynomial system or matrix at a point
- generalEquations -- random linear combinations of equations/generators
- isGEQ -- compare two points
- isRealPoint -- determine whether a point is real
- numericalAffineSpace -- affine space as a numerical variety
- numericalVariety, see numericalVariety(List) -- construct a numerical variety
- projectiveNumericalVariety, see numericalVariety(List) -- construct a numerical variety
- coordinates, see Point -- a type used to store a point in complex space
- point -- construct a Point
- polySpace -- construct a PolySpace
- polySystem -- construct a polynomial system
- project, see project(Point,ZZ) -- project a point
- projectiveWitnessSet -- construct a ProjectiveWitnessSet
- realPoints -- select real points
- residual -- residual of a polynomial function at a point
- projectiveSliceEquations, see sliceEquations(Matrix,Ring) -- slicing linear functions
- sliceEquations, see sliceEquations(Matrix,Ring) -- slicing linear functions
- solutionsWithMultiplicity -- replaces clusters of approximately equal points by single points with multiplicity
- sortSolutions, see sortSolutions(List) -- sort the list of solutions
- toAffineChart, see toAffineChart(ZZ,List) -- coordinates of a point in the projective space in an affine chart
- equations, see WitnessSet -- a witness set
- points, see WitnessSet -- a witness set
- slice, see WitnessSet -- a witness set
- witnessSet -- construct a WitnessSet
Methods
- areEqual(BasicList,Point), see areEqual -- determine if solutions are equal
- areEqual(Point,BasicList), see areEqual -- determine if solutions are equal
- areEqual(Point,Point), see areEqual -- determine if solutions are equal
- Point == Point, see areEqual -- determine if solutions are equal
- components(NumericalVariety) -- list components of a numerical variety
- components(NumericalVariety,ZZ), see components(NumericalVariety) -- list components of a numerical variety
- components(NumericalVariety,ZZ,InfiniteNumber), see components(NumericalVariety) -- list components of a numerical variety
- components(NumericalVariety,ZZ,ZZ), see components(NumericalVariety) -- list components of a numerical variety
- dim(DualSpace), see DualSpace -- a dual functional vector subspace
- generators(DualSpace), see DualSpace -- a dual functional vector subspace
- net(DualSpace), see DualSpace -- a dual functional vector subspace
- point(DualSpace), see DualSpace -- a dual functional vector subspace
- ring(DualSpace), see DualSpace -- a dual functional vector subspace
- dualSpace(DualSpace), see dualSpace -- construct a DualSpace
- dualSpace(Matrix,Point), see dualSpace -- construct a DualSpace
- dualSpace(PolySpace,Point), see dualSpace -- construct a DualSpace
- evaluate(Matrix,Point), see evaluate -- evaluate a polynomial system or matrix at a point
- evaluate(PolySystem,Matrix), see evaluate -- evaluate a polynomial system or matrix at a point
- evaluate(PolySystem,Point), see evaluate -- evaluate a polynomial system or matrix at a point
- generalEquations(WitnessSet), see generalEquations -- random linear combinations of equations/generators
- homogenize(PolySystem,Ring,RingElement) -- homogenize a polynomial system
- isGEQ(Point,Point), see isGEQ -- compare two points
- isRealPoint(Point), see isRealPoint -- determine whether a point is real
- norm(Thing,Point) -- p-norm of the point
- check(NumericalVariety), see NumericalVariety -- a numerical variety
- degree(NumericalVariety), see NumericalVariety -- a numerical variety
- dim(NumericalVariety), see NumericalVariety -- a numerical variety
- net(NumericalVariety), see NumericalVariety -- a numerical variety
- coordinates(Point), see Point -- a type used to store a point in complex space
- matrix(Point), see Point -- a type used to store a point in complex space
- net(Point), see Point -- a type used to store a point in complex space
- status(Point), see Point -- a type used to store a point in complex space
- point(Point), see point -- construct a Point
- dim(PolySpace), see PolySpace -- a polynomial vector subspace
- generators(PolySpace), see PolySpace -- a polynomial vector subspace
- net(PolySpace), see PolySpace -- a polynomial vector subspace
- ring(PolySpace), see PolySpace -- a polynomial vector subspace
- polySpace(PolySpace), see polySpace -- construct a PolySpace
- equations(PolySystem), see PolySystem -- a polynomial system
- ideal(PolySystem), see PolySystem -- a polynomial system
- isHomogeneous(PolySystem), see PolySystem -- a polynomial system
- jacobian(PolySystem), see PolySystem -- a polynomial system
- net(PolySystem), see PolySystem -- a polynomial system
- ring(PolySystem), see PolySystem -- a polynomial system
- polySystem(PolySystem), see polySystem -- construct a polynomial system
- project(Point,ZZ) -- project a point
- residual(List,Point), see residual -- residual of a polynomial function at a point
- residual(PolySystem,Point), see residual -- residual of a polynomial function at a point
- substitute(PolySystem,Ring) -- substitute a ring in a polynomial system
- codim(WitnessSet), see WitnessSet -- a witness set
- degree(WitnessSet), see WitnessSet -- a witness set
- dim(WitnessSet), see WitnessSet -- a witness set
- equations(WitnessSet), see WitnessSet -- a witness set
- ideal(WitnessSet), see WitnessSet -- a witness set
- net(WitnessSet), see WitnessSet -- a witness set
- points(WitnessSet), see WitnessSet -- a witness set
- ring(WitnessSet), see WitnessSet -- a witness set
- slice(WitnessSet), see WitnessSet -- a witness set
- witnessSet(PolySystem,Matrix,List), see witnessSet -- construct a WitnessSet
- witnessSet(PolySystem,PolySystem,List), see witnessSet -- construct a WitnessSet
Symbols
- Norm -- p in the p-norm
- ConditionNumber, see Point -- a type used to store a point in complex space
- Coordinates, see Point -- a type used to store a point in complex space
- DecreasePrecision, see Point -- a type used to store a point in complex space
- DeflationNumber, see Point -- a type used to store a point in complex space
- ErrorBoundEstimate, see Point -- a type used to store a point in complex space
- IncreasePrecision, see Point -- a type used to store a point in complex space
- Infinity, see Point -- a type used to store a point in complex space
- LastT, see Point -- a type used to store a point in complex space
- MaxPrecision, see Point -- a type used to store a point in complex space
- MinStepFailure, see Point -- a type used to store a point in complex space
- Multiplicity, see Point -- a type used to store a point in complex space
- NumberOfSteps, see Point -- a type used to store a point in complex space
- NumericalRankFailure, see Point -- a type used to store a point in complex space
- Origin, see Point -- a type used to store a point in complex space
- RefinementFailure, see Point -- a type used to store a point in complex space
- Regular, see Point -- a type used to store a point in complex space
- Singular, see Point -- a type used to store a point in complex space
- SolutionStatus, see Point -- a type used to store a point in complex space
- WindingNumber, see Point -- a type used to store a point in complex space
- ContinuationParameter, see PolySystem -- a polynomial system
- Jacobian, see PolySystem -- a polynomial system
- NumberOfPolys, see PolySystem -- a polynomial system
- NumberOfVariables, see PolySystem -- a polynomial system
- PolyMap, see PolySystem -- a polynomial system
- SpecializationRing, see PolySystem -- a polynomial system
- AffineChart, see ProjectiveWitnessSet -- a projective witness set
- Tolerance -- the tolerance of a numerical computation
- Equations, see WitnessSet -- a witness set
- IsIrreducible, see WitnessSet -- a witness set
- Points, see WitnessSet -- a witness set
- ProjectionDimension, see WitnessSet -- a witness set
- Slice, see WitnessSet -- a witness set