### D.8.7 recover_lib

Library:
  recover.lib
Purpose:
  Hybrid numerical/symbolical algorithms for algebraic geometry
Author:
  Adrian Koch (kocha at rhrk.uni-kl.de)

Overview:
  In this library you'll find implementations of some of the algorithms presented in the paper listed below: Bertini is used to compute a witness set of a given ideal I. Then a lattice basis reduction algorithm is used to recover exact results from the inexact numerical data. More precisely, we obtain elements of prime components of I, the radical of I, or an elimination ideal of I.

  NOTE that Bertini may create quite a lot of files in the current directory (or overwrite files which have the same names as the files it wants to create). It also prints information to the screen.

  The usefulness of the results of the exactness recovery algorithms heavily depends on the quality of the witness set and the quality of the lattice basis reduction algorithm.
  The procedures requiring a witness set as part of their input use a simple, unsofisticated version of the LLL algorithm.

References:
  Daniel Bates, Jonathan Hauenstein, Timothy McCoy, Chris Peterson, and Andrew Sommese; Recovering exact results from inexact numerical data in algebraic geometry; Published in Experimental Mathematics 22(1) on pages 38-50 in 2013

Procedures:

> D.8.7.1 substAll |  | poly: ring variables in v substituted by elements of p
> D.8.7.2 veronese |  | ideal: image of p under the degree d Veronese embedding
> D.8.7.3 getRelations |  | list of ideals: homogeneous polynomial relations between components of p
> D.8.7.4 getRelationsRadical |  | modified version of getRelations
> D.8.7.5 gaussRowWithoutPerm |  | matrix: a row-reduced form of M
> D.8.7.6 gaussColWithoutPerm |  | matrix: a column-reduced form of M
> D.8.7.7 getWitnessSet |  | extracts the witness set from the file "main_data" produced by Bertini
> D.8.7.8 writeBertiniInput |  | writes the input-file for bertini with the polynomials in J as functions
> D.8.7.9 num_prime_decom |  | is supposed to compute a prime decomposition of the radical of I
> D.8.7.10 num_prime_decom1 |  | is supposed to compute a prime decomposition for the ideal represented by the witness point set P
> D.8.7.11 num_radical_via_decom |  | compute elements of the radical of I by using num_prime_decom
> D.8.7.12 num_radical_via_randlincom |  | computes elements of the radical of I by using a different method
> D.8.7.13 num_radical1 |  | computes elements of the radical via num_prime_decom1
> D.8.7.14 num_radical2 |  | computes elements of the radical using a different method
> D.8.7.15 num_elim |  | computes elements of the elimination ideal of I w.r.t. the variables specified by f
> D.8.7.16 num_elim1 |  | computes elements of the elimination ideal of the ideal represented by the witness point set P (w.r.t. the variables specified in v)
> D.8.7.17 realLLL |  | simple version of the LLL-algorithm;works only over real numbers

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