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NAGtypes :: NAGtypes

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

Version

This documentation describes version 1.6.0.1 of NAGtypes.

Source code

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

Exports

  • Types
  • 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) (missing documentation)
    • 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
    • segmentHomotopy(PolySystem,PolySystem), see polynomial homotopy -- basic methods for manipulating polynomial homotopies
    • specializeContinuationParameter(PolySystem,Number), see polynomial homotopy -- basic methods for manipulating polynomial homotopies
    • substituteContinuationParameter(PolySystem,RingElement), see polynomial homotopy -- basic methods for manipulating polynomial homotopies
    • 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
    • BasePoint (missing documentation)
    • Gens (missing documentation)
    • 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
    • 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
    • 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
    • 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
    • Reduced (missing documentation)
    • Space (missing documentation)
    • 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