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