sortSolutions(List) -- sort the list of solutions

Synopsis

Function: sortSolutions
Usage:

t = sortSolutions s
Inputs:
- s, a list, contains solutions (represented either by lists of coordinates or points)
Optional inputs:
- Tolerance => ..., default value .000001, the tolerance of a numerical computation
- Weights => ..., default value null
Outputs:
- t, a list, contains solutions sorted as described below

Description

The sorting is done lexicographically regarding each complex n-vector as real 2n-vector. The output format of track and solveSystem is respected.
For the corresponding coordinates a and b (of two real 2n-vectors) a < b if b-a is larger than Tolerance.

i1 : needsPackage "NumericalAlgebraicGeometry"

o1 = NumericalAlgebraicGeometry

o1 : Package

i2 : R = CC[x,y];

i3 : s = solveSystem {x^2+y^2-1, x*y}

o3 = {{3.02251e-20+2.05294e-20*ii, -1}, {-3.02251e-20-2.05294e-20*ii, 1}, {1,
     ------------------------------------------------------------------------
     8.30314e-21+4.44918e-20*ii}, {-1, -8.30314e-21-4.44918e-20*ii}}

o3 : List

i4 : sortSolutions s

o4 = {{-1, -8.30314e-21-4.44918e-20*ii}, {3.02251e-20+2.05294e-20*ii, -1},
     ------------------------------------------------------------------------
     {-3.02251e-20-2.05294e-20*ii, 1}, {1, 8.30314e-21+4.44918e-20*ii}}

o4 : List

Caveat

The sorting described above does not possess good properties, since there may be near ties in specific coordinate values between several points. A better way is to specify a random weight (of length 2n where n=#points) as an optional parameter sortSolutions(...,Weights=>...), which provides a linear functional that evaluates to distinct (and sufficiently) real numbers on the given points.

Ways to use this method: