last modified Jan 2010

For the impatient

GET 2010 Alpha Version of NCAlgebra and NCGB Releases for Windows, Mac OSX, Linux, Solaris

For the others

Our Non Commutative Algebra Packages run under Mathematica© and give it the capability of manipulating noncommuting algebraic expressions. NCGB Computes Non Commutative Groebner Bases and has extensive sorting and display features as well as algorithms for automatically discarding "redundant" polynomials...

Introduction
Some Accomplishments Of NonCommutative Computer Algebra
Non-Commuters (Links to others)
Dell Kronewitter's Ph.D. thesis in postscript or PDF
Links to other NCAlgebra sites

Introduction

Our NonCommutative Algebra Packages run under Mathematica© and give it the capability of manipulating noncommuting expressions.

To get them click NCAlgebra and NCGB download.

If you have any trouble getting our software, then send us email at ncalg@ucsd.edu.

NCAlgebra

For an introduction to NCAlgebra see the short tutorial of some of the most basic commands in HTML

or a Mma notebook;

or a brief Demo of NCAlgebra Mathematica Notebook (you need Mma to read it and you must load NCAlgebra to run it). Also the palette NCPalette for NCAlgebra contains the main NCAlgebra and NCGB commands, so is informative.

The rather extensive NC DOCUMENT is available in Pdf

NCGB

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 NC DOCUMENT obtainable as PDF

SIMPLE DEMOS

(You NEED Mma too view all but (1a)):

(1a) The output of NCProcess
(1b) The input of NCProcess load NCGB.m to run
(2) Demo of NCMakeGB load NCGB.m to run or Postscript format (Aug 1998).

NONCOMMUTATIVE INEQUATITIES

Is a given noncommutative function "convex"? You type in a function of noncommutative variables; the command NCConvexityRegion[Func, ListOfVariables] tells you where the (symbolic) Function is convex in the Variables. This corresponds to papers of Camino Helton and Skelton.

GROUPS

You can compute a complete list of rewrite rules for Groups using NCGB. See demos below.

System Requirements

AS OF JAN 2010 TO RUN NCGB YOU NEED

Mathematica running under the Solaris operating system and GNU C++ (free)

OR Windows on a PC

Linux

An Apple Also you need one of the above to run (all but the first of) these DEMOS, downloaded. We use Mathematica 4.0 and higher. The codes have been tested on Mma 6 and 7
WE REPEAT TO RUN NCAlgebra you only need Mathematica.

SOME ACCOMPLISHMENTS OF NON COMMUTATIVE COMPUTER ALGEBRA

This part of the site contains examples of problems which have been investigated with the aid of the features in NCAlgebra for DISCOVERING FORMULAS. Exactly what can be done for engineering systems theory and operator theory with NonCommuting GB's (Mora's algorithm) and techniques we are developing is thoroughly unexplored. Our goal is to test these methods on a variety of problems, most of which are classic theorems in some field. Classifying existing mathematics according to what is required to discover it is an extremely valuable gauge of these symbolic techniques. However, some of the results described here are new, and a few contain open questions.

SYSTEMS ENGINEERING

NCGB Does Singular Perturbation Well

The most basic control problem (Demo on ) April 1999
Singular Perturbation for the infostate equation in postscript or PDF (An article) April 1999

Helton and Kronewitter

Factoring systems

Minimal factorization (Theorem of Bart, Gohberg, Kaashoek, Van Dooren) Jun 96

Solution by Helton, Schneider, Stankus and Yoshinobu

Model reduction problems

Balanced truncation Jun 96

Contributed by Keith Glover

Solutions by Helton, Schneider and Stankus

H-infinity control

Derivation of the X and Y Ricatti equations Jun 96

Solution by Helton, Schneider, Stankus and Yoshinobu

Orbit problems

Orbits of U(n,n) acting on Mn Jun 96

Contributed by Nicholas Young

Solution by Helton, Schneider and Stankus

Other

OPERATOR THEORY

Matrix completion problems

2 x 2 Invertible Matrix Completion Problem Jun 96

Contributed by Hugo Woerdeman

Solution by Helton, Schneider, Stankus and Yoshinobu

3 x 3 Invertible Matrix Completion Problem; 31,000 New Theorems April 1999

Solutions by Kronewitter

Parrot matrix completion problem - the equality case Jun 96

Solutions by Helton, Schneider and Stankus

Pseudoinverses : Gröebner Bases for basic situations

Some notes(.dvi) (.tex) on the runs below which give finite and, in one case, an infinite Gröebner Basis for situations involving pseudoinverses.

PseudoInverse of A has a finite GB. (.dvi) (.tex)
PseudoInverse of A,B, and A + B has a finite GB. (.dvi) (.tex)
PseudoInverse of A,B, and AB does NOT have a finite GB. (Four recursion laws.)
dvi: (3 iters) (4 iters) (5 iters)
tex: (3 iters) (4 iters) (5 iters)
PseudoInverse of A,B, and A**Pinv[A]+B**Pinv[B] has a finite GB. (.dvi) (.tex)
PseudoInverse of A,B, and A**Pinv[A] + Pinv[B]**B has a finite GB. (.dvi) (.tex)
PseudoInverse of A,B, and (1-Pinv[B]**B)**A**Pinv[A] has a finite GB. (.dvi) (.tex)
PseudoInverse of A,B, and (1-Pinv[B]**B)**A**Pinv[A] AND A**Pinv[A]+Pinv[B]**B has a finite GB. (.dvi) (.tex)
Here is the mathematica file with all of the previous runs.

Pseudoinverse of a column matrix.

Here is the mathematica file for the pseudoinverse of a column matrix.
Here is the (.dvi) (.tex) file for its output.
Note that we are solving for the pseudoinverse of a 2x1 matrix with entries a and b where the pseudoinverse is a 1x2 matrix with entries c and d. It turns out that c = Pinv[tp[a]**a+tp[b]**b]**tp[a] and d = Pinv[tp[a]**a+tp[b]**b]**tp[b].

Other

Sum of Idempotents Jun 96

Contributed by H. Bart and J. Stampfli

Solutions by Stankus and Sloboda

H. Bart and J. Kaashoek problem (open)

A solution to Yongge Tian's EP Matrices problem proposed in Image April 2001 volume 26.

PROBLEMS ORDERED ACCORDING TO METHODOLOGY REQUIRED TO SOLVE

Gröbner basis only

NCGB Does Singular Perturbation Well

The most basic control problem (Demo on ) April 1999
Singular Perturbation for the infostate equation (An article) April 1999

Helton and Kronewitter

Sum of Idempotents Jun 96

1-strategies

2 x 2 Invertible Matrix Completion Problem Jun 96
3 x 3 Invertible Matrix Completion Problem; 31,000 New Theorems April 1999
Parrot matrix completion problem - the equality case Jun 96
Minimal factorization Jun 96
Balanced truncation (parts a and b) Jun 96
Orbits of U(n,n) acting on Mn Jun 96

2-strategies

Derivation of the X and Y Ricatti equations (requires decompose) Jun 96

Open

Balanced truncation (part c) Jun 96
H. Bart and J. Kaashoek problem

USE OF NCGB ON GROUPS-- to generate a complete list of rewrite rules

Gröbner basis only-demos by E. Rowell

Some of these examples along with more about ``strategies'' is in the paper Computer assistance for ``discovering'' formulas in system engineering and operator theory by J. William Helton and Mark Stankus, Journal of Functional Analysis 1999. It is available via anonymous ftp in either Dvi or PostScript formats. It is also on the World Wide Web in HTML

Click here for complete documentation for NCAlgebra and NCGB in HTML.