Introduction to GB Packages

Differences

The three types of GB Packages (NCGB, NCGB and NGBMma) can be used to find new matrix identities from a given list of matrix identities. The basic difference between these program are:

The development of NCGB started in 1993. NCGB is implemented in C++ and is used via Mathematica running NCAlgebra.
The development of NGB started in 2000 and is intended to replace NCGB. The code should be faster. NGB is implemented in C++ and can be used via Mathematica running NCAlgebra or in a stand-alone mode (without Mathematica).
The development of NGBMma started in 2004 and is intended to coexist with NGB. NGBMma is implemented only in Mathematica, but will be slower than NGB.

General Description

The following is a general (but suggestive) description of what these packages do:

Computes NonCommutative Groebner Bases and has extensive sorting and display features as well as algorithms for automatically discarding "redundant" polynomials, as well as "kludgy" methods for suggesting changes of variables (which work better than one would expect). NCGB runs in conjunction with NCAlgebra. A very brief TEMPLATE/DEMO is given here. The whole story appears in the rather long NCGBDOCUMENT obtainable as Dvi or Postscript.

NonCommutative Inequatities

Since there is a notion of semi-positive definite matrix, matrix inequalities can be considered. In fill in, the authors define what it means for a noncommutative expression to be positive and what it means for a noncommutive expression convex. See here for more.

Groups

Problems which do not seem to involve an algebra can be solved using NCGB. One such example is that of groups.

Demos of NCGB

This page shows both the input and output of NCGB for a short and simple problem. The output has been modified from its original form (a dvi file) to an html file for easy viewing.
This page shows the input of NCProcess in the form of a notebook for the inverse matrix completion problem mentioned above. If you have installed NCGB, download this notebook,start Mathematica, load NCGB and then load this notebook, then you can execute this example.
This page demonstrates the use of the NCMakeGB command. It too can be executed. (See the longer directions above.)