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PHCpack :: cascade

cascade -- runs a cascade of homotopies to get witness sets for the variety

Synopsis

Description

Given a list of generators of an ideal I, this function creates an embedding and then runs a cascade of homotopies. The output is a NumericalVariety that contains a WitnessSet for each pure dimensional variety contained in V(I).

i1 : R = CC[x,y,z];
i2 : L = { z*(x+y), z*(x-y) };
i3 : WitSets = cascade(L)

o3 = WitSets

o3 : NumericalVariety
i4 : W=first WitSets#2

o4 = W

o4 : WitnessSet

The function cascade extends the ring of the inputted system with slack variables beginning with zz. Each witness set in contains the equations, points, and slices of the embedded system.

i5 : W#Equations

o5 = ideal ((- .380139 + .924929*ii)zz1 + (.547606 + .836736*ii)zz2 - x*z - y*z, (.45571 - .890128*ii)zz1 + (- .663772 - .747935*ii)zz2 - x*z + y*z, - zz1 + (.537862 + .843033*ii)zz2)

o5 : Ideal of CC  {x, y, z, zz1, zz2}
                53
i6 : W#Points

o6 = {{.148619-.161802*ii, .45886-1.03272*ii, 3.21921e-32+7.65273e-33*ii, -2.28184e-32-4.87389e-33*ii, -1.6382e-32+1.66152e-32*ii}}

o6 : VerticalList
i7 : W#Slice

o7 = | .205377+.978683ii  .673112+.739541ii .943327-.331864ii  -.911067-.412259ii .497812+.867285ii  -.883724+.468009ii |
     | -.759495+.650513ii -.89022-.45553ii  -.972487+.232959ii -.595572-.803302ii -.236638-.971598ii .871298-.490754ii  |

                2          6
o7 : Matrix CC    <--- CC
              53         53

Caveat

Coefficient ring of the polynomial system must be of type ComplexField.

See also

Ways to use cascade :