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}}```

• Anton Leykin

Version

This documentation describes version 1.11 of NAGtypes.

Source code

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

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
• ProjectiveWitnessSet -- a projective witness set
• WitnessSet -- a witness set
• Ambient, see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• SlicingVariety, see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• Functions and commands
• 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
• ambient(SlicingVariety), see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• ambient(WSet), see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• codim(SlicingVariety), see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• codim(WSet), see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• degree(WSet), see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• dim(Ambient), see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• dim(SlicingVariety), see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• dim(WSet), see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• map(SlicingVariety), see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• net(Ambient), see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• net(SlicingVariety), see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• net(WSet), see WSet -- (under construction!) new types and methods needed to generalize WitnessSet
• points(WSet), see WSet -- (under construction!) new types and methods needed to generalize 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
• Reduced, see polySpace -- construct a PolySpace
• 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
• ProjectiveNumericalVariety -- a projective numerical variety
• 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