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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.

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
    • 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
    • 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
    • intersection(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
    • 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
    • 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.