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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]:
==> (2.729346128702428481085144733769373625469+i*2.38174314427933903805069763\
   3725154582206)
==>    [2]:
==> (-1.787414716397637513930796572666616399481+i*2.9551592082255551721303565\
   43245714058138)
==>    [3]:
==> (2.729346128702428481085144733769373625469+i*2.38174314427933903805069763\
   3725154582206)
==> [2]:
==>    [1]:
==> (-0.1503274250788516642393781363924204729043+i*0.786080542010574751862885\
   0393744034708897)
==>    [2]:
==> (0.003765094002973004558627554736661096411504+i*0.03485663903781037414109\
   117259611529131364)
==>    [3]:
==> (0.003765094002973004558627554736661096411504+i*0.03485663903781037414109\
   117259611529131364)
==> [3]:
==>    [1]:
==> (0.282883198471590716381016707500789766424+i*0.49843463973822043800436178\
   28609138018253)
==>    [2]:
==> (0.282883198471590716381016707500789766424+i*0.49843463973822043800436178\
   28609138018253)
==>    [3]:
==> (0.5310872780686717415655250254822846407835-i*0.1765334114995975497833853\
   991956888702414)


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