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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:

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]-*may be reducible*-
     dim 1:  w1

o5 : NumericalVariety
 
i6 : V = oo

o6 = V

o6 : NumericalVariety
 
i7 : peek V

o7 = NumericalVariety{0 => {[dim=0,deg=1]-*may be reducible*-}}
                      1 => {w1}
 
i8 : peek w1

o8 = WitnessSet{cache => CacheTable{...1...}        }
                Equations => {-3} | x3+xy2+2x |
                             {-3} | x2y+y3+2y |
                Points => {{.999999*ii, .999999*ii}}
                          {{-ii, -ii}              }
                Slice => | 1 -1 0 |
 
i9 : peek O

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

Author

Version

This documentation describes version 1.21 of NAGtypes.

Citation

If you have used this package in your research, please cite it as follows:

@misc{NAGtypesSource,
  title = {{NAGtypes: types used in Numerical Algebraic Geometry. Version~1.21}},
  author = {Anton Leykin},
  howpublished = {A \emph{Macaulay2} package available at
    "http://people.math.gatech.edu/~aleykin3/NAG4M2"}
}

Exports

  • Types
    • AbstractPoint -- a type used to store a point in complex space
    • Point -- see AbstractPoint -- a type used to store a point in complex space
    • DualSpace -- a dual functional vector subspace
    • NumericalVariety -- a numerical variety
    • PolySpace -- a polynomial vector subspace
    • PolySystem -- a polynomial system
    • ProjectiveWitnessSet -- a projective witness set
    • System -- a system of functions
    • 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
    • coordinates(AbstractPoint) -- see AbstractPoint -- a type used to store a point in complex space
    • coordinates(Point) -- see AbstractPoint -- a type used to store a point in complex space
    • matrix(AbstractPoint) -- see AbstractPoint -- a type used to store a point in complex space
    • net(AbstractPoint) -- see AbstractPoint -- a type used to store a point in complex space
    • status(AbstractPoint) -- see AbstractPoint -- a type used to store a point in complex space
    • addition(PolySpace,PolySpace) -- see addition -- Union of polynomial spaces
    • AbstractPoint == AbstractPoint -- see areEqual -- determine if solutions are equal
    • areEqual(AbstractPoint,AbstractPoint) -- see areEqual -- determine if solutions are equal
    • areEqual(BasicList,BasicList) -- see areEqual -- determine if solutions are equal
    • areEqual(CC,CC) -- see areEqual -- determine if solutions are equal
    • areEqual(List,List) -- see areEqual -- determine if solutions are equal
    • areEqual(Matrix,Matrix) -- see areEqual -- determine if solutions are equal
    • areEqual(MutableMatrix,MutableMatrix) -- see areEqual -- determine if solutions are equal
    • areEqual(Number,Number) -- see areEqual -- determine if solutions are equal
    • areEqual(DualSpace,DualSpace) -- approximate equality of dual spaces
    • areEqual(PolySpace,PolySpace) -- approximate equality of subspaces spanned by polynomials
    • 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
    • declareIrreducible(WitnessSet) -- see declareIrreducible -- declares a component represented by the witness set irreducible
    • 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,AbstractPoint) -- see dualSpace -- construct a DualSpace
    • dualSpace(PolySpace,AbstractPoint) -- see dualSpace -- construct a DualSpace
    • evaluate(Matrix,AbstractPoint) -- see evaluate -- evaluate a polynomial system or matrix at a point
    • evaluate(Matrix,Matrix) -- 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
    • evaluateJacobian(PolySystem,AbstractPoint) -- see evaluate -- evaluate a polynomial system or matrix at a point
    • generalEquations(WitnessSet) -- see generalEquations -- random linear combinations of equations/generators
    • generalEquations(ZZ,Ideal) -- see generalEquations -- random linear combinations of equations/generators
    • generalEquations(ZZ,List) -- see generalEquations -- random linear combinations of equations/generators
    • homogenize(PolySystem,Ring,RingElement) -- homogenize a polynomial system
    • intersect(PolySpace,PolySpace) -- Intersection of polynomial spaces
    • isContained(DualSpace,DualSpace) -- see isContained -- Is one space contained in the other
    • isContained(PolySpace,PolySpace) -- see isContained -- Is one space contained in the other
    • isGEQ(AbstractPoint,AbstractPoint) -- see isGEQ -- compare two points
    • isGEQ(List,List) -- see isGEQ -- compare two points
    • isRealPoint(AbstractPoint) -- see isRealPoint -- determine whether a point is real
    • net(Point) (missing documentation)
    • norm(Thing,AbstractPoint) -- p-norm of the point
    • numericalAffineSpace(PolynomialRing) -- see numericalAffineSpace -- affine space as a numerical variety
    • 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
    • numericalVariety(List) -- construct a numerical variety
    • projectiveNumericalVariety(List) -- see numericalVariety(List) -- construct a numerical variety
    • point(AbstractPoint) -- see point -- construct a Point
    • point(List) -- see point -- construct a Point
    • point(List,CacheTable) -- see point -- construct a Point
    • point(Matrix) -- 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(Matrix) -- see polySpace -- construct a PolySpace
    • 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
    • numFunctions(PolySystem) -- see PolySystem -- a polynomial system
    • numParameters(PolySystem) -- see PolySystem -- a polynomial system
    • numVariables(PolySystem) -- see PolySystem -- a polynomial system
    • parameters(PolySystem) -- see PolySystem -- a polynomial system
    • ring(PolySystem) -- see PolySystem -- a polynomial system
    • polySystem(Ideal) -- see polySystem -- construct a polynomial system
    • polySystem(List) -- see polySystem -- construct a polynomial system
    • polySystem(Matrix) -- see polySystem -- construct a polynomial system
    • polySystem(PolySystem) -- see polySystem -- construct a polynomial system
    • project(AbstractPoint,ZZ) -- project a point
    • project(Point,ZZ) -- see project(AbstractPoint,ZZ) -- project a point
    • projectiveWitnessSet(Ideal,Matrix,Matrix,List) -- see projectiveWitnessSet -- construct a ProjectiveWitnessSet
    • random(DualSpace) -- see random(PolySpace) -- random element of a subspace
    • random(PolySpace) -- random element of a subspace
    • random(ZZ,DualSpace) -- see random(PolySpace) -- random element of a subspace
    • random(ZZ,PolySpace) -- see random(PolySpace) -- random element of a subspace
    • realPoints(List) -- see realPoints -- select real points
    • reduceSpace(DualSpace) -- see reduceSpace -- reduce the generators of a space
    • reduceSpace(PolySpace) -- see reduceSpace -- reduce the generators of a space
    • residual(List,Point) -- see residual -- residual of a polynomial function at a point
    • residual(Matrix,Matrix) -- see residual -- residual of a polynomial function at a point
    • residual(System,AbstractPoint) -- see residual -- residual of a polynomial function at a point
    • residual(System,Point) -- see residual -- residual of a polynomial function at a point
    • projectiveSliceEquations(Matrix,Ring) -- see sliceEquations(Matrix,Ring) -- slicing linear functions
    • sliceEquations(Matrix,Ring) -- slicing linear functions
    • solutionsWithMultiplicity(List) -- see solutionsWithMultiplicity -- replaces clusters of approximately equal points by single points with multiplicity
    • sortSolutions(List) -- sort the list of solutions
    • status(Point) (missing documentation)
    • substitute(PolySystem,Ring) -- substitute a ring in a polynomial system
    • evaluate(System,AbstractPoint) -- see System -- a system of functions
    • evaluate(System,AbstractPoint,AbstractPoint) -- see System -- a system of functions
    • evaluate(System,Matrix) -- see System -- a system of functions
    • evaluate(System,Matrix,Matrix) -- see System -- a system of functions
    • evaluateJacobian(System,AbstractPoint) -- see System -- a system of functions
    • evaluateJacobian(System,AbstractPoint,AbstractPoint) -- see System -- a system of functions
    • evaluateJacobian(System,Matrix) -- see System -- a system of functions
    • evaluateJacobian(System,Matrix,Matrix) -- see System -- a system of functions
    • numFunctions(System) -- see System -- a system of functions
    • numParameters(System) -- see System -- a system of functions
    • numVariables(System) -- see System -- a system of functions
    • toAffineChart(ZZ,List) -- coordinates of a point in the projective space in an affine chart
    • 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(Ideal,Ideal,List) -- see witnessSet -- construct a WitnessSet
    • witnessSet(Ideal,Matrix,List) -- see witnessSet -- construct a WitnessSet
    • 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
    • ConditionNumber -- see AbstractPoint -- a type used to store a point in complex space
    • Coordinates -- see AbstractPoint -- a type used to store a point in complex space
    • DecreasePrecision -- see AbstractPoint -- a type used to store a point in complex space
    • DeflationNumber -- see AbstractPoint -- a type used to store a point in complex space
    • ErrorBoundEstimate -- see AbstractPoint -- a type used to store a point in complex space
    • IncreasePrecision -- see AbstractPoint -- a type used to store a point in complex space
    • Infinity -- see AbstractPoint -- a type used to store a point in complex space
    • LastT -- see AbstractPoint -- a type used to store a point in complex space
    • MaxPrecision -- see AbstractPoint -- a type used to store a point in complex space
    • MinStepFailure -- see AbstractPoint -- a type used to store a point in complex space
    • Multiplicity -- see AbstractPoint -- a type used to store a point in complex space
    • NumberOfSteps -- see AbstractPoint -- a type used to store a point in complex space
    • NumericalRankFailure -- see AbstractPoint -- a type used to store a point in complex space
    • Origin -- see AbstractPoint -- a type used to store a point in complex space
    • RefinementFailure -- see AbstractPoint -- a type used to store a point in complex space
    • Regular -- see AbstractPoint -- a type used to store a point in complex space
    • Singular -- see AbstractPoint -- a type used to store a point in complex space
    • SolutionStatus -- see AbstractPoint -- a type used to store a point in complex space
    • WindingNumber -- see AbstractPoint -- a type used to store a point in complex space
    • Norm -- p in the p-norm
    • Reduced -- see polySpace -- construct a PolySpace
    • ContinuationParameter -- 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
    • 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

For the programmer

The object NAGtypes is a package, defined in NAGtypes.m2, with auxiliary files in NAGtypes/.

The source of this document is in NAGtypes/doc-NAGtypes.m2:36:0.