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D.8.7.7 getWitnessSet

Procedure from library recover.lib (see recover_lib).

Usage:
getWitnessSet();

Assume:
There is a text-document "main_data" in the current directory which was produced by Bertini.
The basefield is the field of real numbers or the field of complex numbers.

Return:
list; a list P of lists p_i of numbers: P a set of witness points

Note:
Reads the file "main_data", searches the strings containing the witness points, and converts them into floating point numbers.

Example:
 
LIB "recover.lib";
//First, we write the input file for bertini, then run bertini
ring r=0,(x,y,z),dp;
ideal I=(x-y)*(y-z)*(x-z);
writeBertiniInput(I,40);
system("sh","bertini input");
==> 
==>    Bertini(TM) v1.6
==>     (May 22, 2018)
==> 
==>  D.J. Bates, J.D. Hauenstein,
==>  A.J. Sommese, C.W. Wampler
==> 
==> (using GMP v6.0.0, MPFR v3.1.2)
==> 
==> 
==> 
==> NOTE: You have requested to use adaptive path tracking.  Please make sure\
    that you have
==> setup the following tolerances appropriately:
==> CoeffBound: 6.000000000000e+00, DegreeBound: 3.000000000000e+00
==> AMPSafetyDigits1: 1, AMPSafetyDigits2: 1, AMPMaxPrec: 160
==> 
==> 
==> Tracking regeneration codim 1 of 1: 3 paths to track.
==> Tracking path 0 of 3
==> Tracking path 1 of 3
==> Tracking path 2 of 3
==> 
==> Sorting codimension 1 of 1: 3 paths to sort.
==> Sorting 0 of 3
==> Sorting 1 of 3
==> Sorting 2 of 3
==> 
==> 
==> ************ Regenerative Cascade Summary ************
==> 
==> NOTE: nonsingular vs singular is based on rank deficiency and identical e\
   ndpoints
==> 
==> |codim|   paths   |witness superset| nonsingular | singular |nonsolutions\
   | inf endpoints | other bad endpoints
==> ----------------------------------------------------------------------------------------------------------------
==> | 1   |   3       |   3            |  3          |  0       |  0         \
   |   0           |  0
==> ----------------------------------------------------------------------------------------------------------------
==> |total|   3
==> 
==> ****************************************************
==> 
==> 
==> 
==> *************** Witness Set Summary ****************
==> 
==> NOTE: nonsingular vs singular is based on rank deficiency and identical e\
   ndpoints
==> 
==> |codim| witness points | nonsingular | singular 
==> -------------------------------------------------
==> | 1   |   3            |  3          |  0       
==> -------------------------------------------------
==> 
==> ****************************************************
==> 
==> 
==> Calculating traces for codimension 1.
==> Calculating 0 of 3
==> Calculating 1 of 3
==> Calculating 2 of 3
==> 
==> Using combinatorial trace test to decompose codimension 1.
==> 
==> 
==> ************* Witness Set Decomposition *************
==> 
==> | dimension | components | classified | unclassified
==> -----------------------------------------------------
==> |   1       |   0        |   0        |  3
==> -----------------------------------------------------
==> 
==> ************** Decomposition by Degree **************
==> 
==> *****************************************************
==> 
==> 0
//Then we change the ring and extract the witness set from main_data
ring R=(complex,40,i),(x,y,z),dp;
list P=getWitnessSet();
P;
==> [1]:
==>    [1]:
==> (0.9204961223691798830909895684013744828808+i*0.5342831358250098594560295\
   718953423685735)
==>    [2]:
==> (0.9204961223691798830909895684013744828808+i*0.5342831358250098594560295\
   718953423685735)
==>    [3]:
==> (0.4688458586727205794699338951380045538809+i*0.7488043688504113527687355\
   139867524795476)
==> [2]:
==>    [1]:
==> (0.732973020961725612108864307379187202837+i*0.92901824613026946367504929\
   03466603600946)
==>    [2]:
==> (1.317930159917983164747765256534314768904+i*0.82808460302767851281435870\
   06577929909908)
==>    [3]:
==> (0.732973020961725612108864307379187202837+i*0.92901824613026946367504929\
   03466603600946)
==> [3]:
==>    [1]:
==> (0.1113800254169985225652794186495944141468-i*0.4154379939142527483119000\
   494150292979989)
==>    [2]:
==> (-0.3092207074954687880536511214115958511511+i*1.226243700175612253893611\
   263708492500593)
==>    [3]:
==> (-0.3092207074954687880536511214115958511511+i*1.226243700175612253893611\
   263708492500593)


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