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PHCpack :: cascade

cascade -- runs a cascade of homotopies to get witness sets for each component of the variety (invokes "phc -c")

Synopsis

Description

Given an ideal, I, this function creates an embedding and runs the cascades of homotopies. The output is a list of witness points for each pure dimensional variety contained in V(I). For each pure dimensional subvariety of dimension d, a file is created with the ending targetsw"d". This file contains the complete witness set: the witness points and the d general hyperplanes. Each entry in the list is a hashtable, which lists the dimension, filename of the witness set, and witness points for the stated dimension.
i1 : R=QQ[x11,x22,x21,x12,x23,x13,x14,x24]

o1 = R

o1 : PolynomialRing
i2 : I=ideal(x11*x22-x21*x12,x12*x23-x22*x13,x13*x24-x23*x14)

o2 = ideal (x11*x22 - x21*x12, x12*x23 - x22*x13, - x23*x14 + x13*x24)

o2 : Ideal of R
i3 : dim I

o3 = 5
i4 : degree I

o4 = 8
i5 : cascade I
using temporary files /tmp/M2-5155-2PHCtarget, /tmp/M2-5155-1PHCmonodromy, /tmp/M2-5155-4PHCsolfile

o5 = {5 => /tmp/M2-5155-2PHCtarget_sw5 => {{.678386-1.05795*ii,
     ------------------------------------------------------------------------
     .0856455-.457005*ii, -.790111-.111277*ii, .597942+.422849*ii,
     ------------------------------------------------------------------------
     .0023372-.108661*ii, .154014+.0747344*ii, -1.48854-1.01442*ii,
     ------------------------------------------------------------------------
     -.191149+1.12756*ii}, {.0962048-.471522*ii, -1.0595-.306157*ii,
     ------------------------------------------------------------------------
     -.95181-.39778*ii, .044556-.512545*ii, .559397+1.27516*ii,
     ------------------------------------------------------------------------
     -.533178+.371061*ii, -.465753+.316977*ii, .473599+1.11094*ii},
     ------------------------------------------------------------------------
     {1.35402-.7637*ii, -1.58741e-46, .876649+1.34286*ii,
     ------------------------------------------------------------------------
     -1.75162e-46+2.62743e-46*ii, .867162-.170381*ii, 1.62712-1.33578*ii,
     ------------------------------------------------------------------------
     -.118882-1.06341*ii, .167466-.416806*ii}, {1.92209-1.0919*ii,
     ------------------------------------------------------------------------
     -.798517+.437563*ii, 1.91204+.954772*ii, -.084433+.938033*ii,
     ------------------------------------------------------------------------
     1.93353-.393398*ii, .776563-1.88742*ii, .05922-.621966*ii,
     ------------------------------------------------------------------------
     .531202-.287529*ii}, {.416855-1.04416*ii, .053077-.0441538*ii,
     ------------------------------------------------------------------------
     .0421552-.0857575*ii, .582647-.566013*ii, .0407108-.0426077*ii,
     ------------------------------------------------------------------------
     .438983-.536678*ii, -.232636-.724812*ii, -.0149729-.0629437*ii},
     ------------------------------------------------------------------------
     {1.833-1.39733*ii, -.324332-2.03502*ii, -1.29435+.801983*ii,
     ------------------------------------------------------------------------
     .785856+3.01868*ii, -1.05097e-45+5.66541e-46*ii,
     ------------------------------------------------------------------------
     1.22614e-45+1.4013e-45*ii, -5.09086-.987602*ii, -.265666+4.00356*ii},
     ------------------------------------------------------------------------
     {.49752-1.00084*ii, .057045-.329093*ii, -.680502-.352465*ii,
     ------------------------------------------------------------------------
     .481269+.0752278*ii, 1.2737e-32+2.9574e-34*ii,
     ------------------------------------------------------------------------
     -9.50036e-33+1.88418e-32*ii, -1.06987-.985715*ii, -.260132+.838919*ii},
     ------------------------------------------------------------------------
     {.388581-.731399*ii, 1.55046e-32-4.18513e-32*ii, -.971063-.848175*ii,
     ------------------------------------------------------------------------
     2.84448e-32+3.58005e-33*ii, .340641+.104212*ii, -.44289+.35038*ii,
     ------------------------------------------------------------------------
     -.521069-1.27628*ii, -.475511+.728047*ii}}}

o5 : List

Caveat

This function should be returning witness sets using the type WitnessSet from NAGtypes. This will be completed in version 1.1.

Ways to use cascade :