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AbstractPoint -- a type used to store a point in complex space

Description

The type Point inherited from AbstractPoint is used to store a solution to a polynomial system obtained by such functions as solveSystem, track. The following methods can be used to access a AbstractPoint: Possible return values of status reflect the status with respect to a homotopy continuation procedure that obtained this point: Only coordinates are displayed (by net); other attributes of a Point p are stored in p.cache. Different algorithms attach different information describing the point. For example, solveSystem produces the following.

i1 : loadPackage "NumericalAlgebraicGeometry";
i2 : R = CC[x,y];
i3 : sols = solveSystem{x^2+y^2-3, x^3-y^3-7}

o3 = {{-1.7957-1.31322*ii, 1.7957-1.31322*ii}, {-.101284+.779159*ii,
     ------------------------------------------------------------------------
     -1.89699-.041601*ii}, {-1.7957+1.31322*ii, 1.7957+1.31322*ii},
     ------------------------------------------------------------------------
     {-.101284-.779159*ii, -1.89699+.041601*ii}, {1.89699+.041601*ii,
     ------------------------------------------------------------------------
     .101284-.779159*ii}, {1.89699-.041601*ii, .101284+.779159*ii}}

o3 : List
i4 : pt = first sols

o4 = pt

o4 : Point
i5 : peek pt

o5 = Point{cache => CacheTable{...5...}                          }
           Coordinates => {-1.7957-1.31322*ii, 1.7957-1.31322*ii}
i6 : coordinates pt

o6 = {-1.7957-1.31322*ii, 1.7957-1.31322*ii}

o6 : List
i7 : status pt

o7 = Regular

o7 : Symbol

The other keys that may be attached include

Basic service functions:

Types of point:

  • Point

Methods that use a point:

  • coordinates(AbstractPoint)
  • matrix(AbstractPoint)
  • net(AbstractPoint)
  • status(AbstractPoint)
  • AbstractPoint == AbstractPoint -- see areEqual -- determine if solutions are equal
  • areEqual(AbstractPoint,AbstractPoint) -- see areEqual -- determine if solutions are equal
  • 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
  • evaluateJacobian(PolySystem,AbstractPoint) -- see evaluate -- evaluate a polynomial system or matrix at a point
  • isGEQ(AbstractPoint,AbstractPoint) -- see isGEQ -- compare two points
  • isRealPoint(AbstractPoint) -- see isRealPoint -- determine whether a point is real
  • norm(Thing,AbstractPoint) -- p-norm of the point
  • point(AbstractPoint) -- see point -- construct a Point
  • project(AbstractPoint,ZZ) -- project a point
  • residual(System,AbstractPoint) -- see residual -- residual of a polynomial function at a point
  • evaluate(System,AbstractPoint) -- see System -- a system of functions
  • evaluate(System,AbstractPoint,AbstractPoint) -- 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

For the programmer

The object AbstractPoint is a type, with ancestor classes HashTable < Thing.


The source of this document is in NAGtypes/doc-NAGtypes.m2:103:0.