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# areEqual -- determine if solutions are equal

- Usage:
    `b = areEqual(x,y)`
- Inputs:
  - x, a solution or a list of solutions
  - y, a solution or a list of solutions
- Optional inputs:
  - Projective => ..., default value false, if true then solutions are considered as representatives of points in the projective space
  - Tolerance => ..., default value .000001, the tolerance of a numerical computation
- Outputs:
  - b, tells if x and y are approximately equal

## Description

The inputs can be complex numbers, points, or lists of points (presented as points or lists of coordinates). The function returns false if the distance between x and y exceeds Tolerance and true, otherwise.

If Projective=>true then 1-\cos\alpha is compared with the Tolerance, where \alpha is the angle between x and y.

    i1 : areEqual({{-1,1e-7},{1e-7*ii,-1}}, {{-1, 0}, {0, -1}})

    o1 = true

    i2 : areEqual({3*ii,2*ii,1+ii}, {-6,-4,-2+2*ii}, Projective=>true)

    o2 = false

For two points A and Bcalling A == Bis equivalent to areEqual(A,B), however, there is no way to specify the optional parameter.

    i3 : A = point {{-1,1e-7}, {1e-7*ii,-1}}

    o3 = A

    o3 : Point

    i4 : B = point {{-1,0}, {0, -1}}

    o4 = B

    o4 : Point

    i5 : A == B

    o5 = true

## See also

- solveSystem -- solve a system of polynomial equations
- track -- track a linear segment homotopy given start and target system
- sortSolutions -- sort the list of solutions

## Ways to use areEqual:

- `areEqual(AbstractPoint,AbstractPoint)`
- `areEqual(BasicList,BasicList)`
- `areEqual(CC,CC)`
- `areEqual(List,List)`
- `areEqual(Matrix,Matrix)`
- `areEqual(MutableMatrix,MutableMatrix)`
- `areEqual(Number,Number)`
- areEqual(DualSpace,DualSpace) -- approximate equality of dual spaces
- areEqual(PolySpace,PolySpace) -- approximate equality of subspaces spanned by polynomials

## For the programmer

The object areEqual is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NAGtypes/doc-NAGtypes.m2:227:0.