THE NCAlgebra SUITE

NCAlgebra 4.0

Mathematica’s Control System Professional
Noncommutative support 1.0

NCGB NonCommutative Groebner Bases 4.0

SYStems - ON THE WEB

http://math.ucsd.edu/~ncalg
Download the software and
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January 6, 2010

  Table of Contents
Contents
  Preface
Preface
I  NCAlgebra
 0.1 ReleaseNotes NCAlgebra
1 A Short Tutorial on NCAlgebra
 1.1 Installing NCAlgebra
 1.2 Running NCAlgebra
 1.3 A Short Tutorial
 1.4 Where to Find More Examples
 1.5 Pretty Output
2 A More Sophisticated Demo
3 Helpful Tricks
 3.1 Expanding
 3.2 Simplifying Expressions
 3.3 Edit - For those without Notebooks
 3.4 Conventions
4 NC Commands
 4.1 Manipulating an expression
 4.2 Simplification
 4.3 Vector Differentiation
 4.4 Block Matrix Manipulation
 4.5 Complex Analysis
 4.6 Setting symbols to commute or not commute
 4.7 Operations on elements in an algebra
 4.8 Convexity of a NC function
 4.9 NCGuts
 4.10 Setting Properties of an element in an algebra
 4.11 Setting Properties of functions on an algebra
 4.12 Manipulating an Expression — less useful commands
 4.13 Utilities
 4.14 Deprecated Commands
5 Pretty Output and Tex Commands
 5.1 Pretty Output
 5.2 TEX Typesetting with NCTEXForm
 5.3 Simple TEX Commands with NCTEX
 5.4 Deprecated Commands
6 Aliases
II   NONCOMMUTATIVE CONTROL SYSTEM PROFESSIONAL
7 State Space Systems Constructions
 7.1 System Interconnections
 7.2 Continuous vs. Discrete
 7.3 Transfer Function
 7.4 Systems from Systems
III  NONCOMMUTATIVE GRÖBNER BASES–NCGB
  ReleaseNotesNCGB
IV  NCGB: Easy Introduction
8 Introduction
  How to read this document
9 Simple Demos of Basic Commands
 9.1 To start a C++ GB session
 9.2 Simplifying Expressions
 9.3 Making a Groebner Basis
 9.4 Reducing a polynomial by a GB
10 NCGB Facilitates Natural Notation
 10.1 A Simplification example
 10.2 MakingGB’s and Inv[], Tp[]
 10.3 Simplification and GB’s revisited
 10.4 Saving lots of time when typing
11 Demo on NCGB - Matrix Computation
 11.1 The Partially Prescribed Inverse Problem
12 To Run NCGB - Template.nb
 12.1 Making a Groebner basis and NCProcess template
13 NCProcess: What It Does
 13.1 NCProcess: Input and Output
 13.2 Changing Variables
14 NCProcess: An Example
 14.1 Background
 14.2 The Problem
 14.3 Solution via a Prestrategy
 14.4 The end game
15 NCProcess: The Commands
 15.1 SetKnowns and SetUnknowns
 15.2 NCProcess
 15.3 Commonly Used NCProcess Options and Commands
 15.4 Typical use of the NCProcess command
 15.5 Details of NCProcess
 15.6 NCProcess1 and NCProcess2: The technical descriptions
V  NCGB: FOR THE MORE ADVANCED USER
16 NCProcess: The Concepts
 16.1 NCProcess: Input and Output
 16.2 Elimination
 16.3 What is a prestrategy?
 16.4 A strategy
17 Another Example: Solving the H Control Problem
 17.1 Problem statement
 17.2 The key relations: executable form
 17.3 Solving (HGRAIL) using NCProcess
 17.4 End Game
VI  NCGB: LISTS OF COMMANDS AND OTHER DETAILS
18 Ordering on variables and monomials
 18.1 Lex Order: The simplest elimination order
 18.2 Graded lex ordering: A non-elimination order
 18.3 Multigraded lex ordering : A variety of elimination orders
 18.4 The list of commands
 18.5 Fancier Order Setting Commands
19 More NCProcess Options
 19.1 Creating small generating sets: RRTrue, RRByCatTrue, SBFalse, SBByCatTrue
 19.2 NCCollectOnVars
 19.3 Turning screen output off
 19.4 Output Options
 19.5 NCProcess Summary Table
20 Commands for Making and Using GB’s
 20.1 Simplification
 20.2 Making a Gröbner Basis and various options (with their defaults)
 20.3 Reduction
21 Commands for Making Small Bases for Ideals: Small Basis, Shrink Basis
 21.1 Brute Force: Shrinking
 21.2 Brute Force: Many shrinks
 21.3 First Example
 21.4 Second Example
 21.5 Smaller Bases and the Spreadsheet command
 21.6 How Small Basis commands relate to the similar NCProcess Options
22 Help in Typing Relations .
 22.1 Output notation for pseudoinverse and perp’s
23 Retrieving Categories and Regular Output
 23.1 Example
 23.2 Creating Categories
 23.3 RegularOutput[aListOfPolynomials,“fileName”]
 23.4 How to Really Change Regular Output
24 The Dimension of the Solution Set of a Set of Polynomial Equations
 24.1 The Commuting Case
 24.2 Noncommutative Case: Gelfand-Kirillov dimension
 24.3 References
 24.4 Commands
25 Commands which are not supported
 25.1 A Mathematica Groebner Basis Package Without C++
 25.2 NCXWholeProcess[ polys, orderList, fileName, grobIters]
26 Getting NCALGEBRA and NCGB
 26.1 Getting NCAlgebra and NCGB off the web
 26.2 Getting NCAlgebra and NCGB through anonymous ftp
 26.3 The last step
 26.4 The NC directory structure
 26.5 Directory structure of NCAlgebra alone
27 Testing your version of NCGB
 27.1 Beginners
 27.2 Turning On Screen Output
 27.3 More Testing for Developers - DOES NOT WORK 2001
28 References
VII  DETAILS AND OLD OR NEW COMMANDS -ONLY ON THE WEB
29 History of the Production of a GB and Playing By Numbers
 29.1 Play By Numbers
 29.2 History of the production of a GB
30 Commands for Making Small Bases for Ideals: Remove Redundant
 30.1 Removing excess relations
 30.2 Discussion of RemoveRedundent command
 30.3 Examples
 30.4 First Example
 30.5 Second Example
 30.6 Smaller Bases and the Spreadsheet command
31 NCXFindChangeOfVariables: The Long Description
 31.1 Details of the Algorithm
 31.2 Finding Coefficients of Variables in a Polynomial
 31.3 Main Change Of Variables Command
 31.4 Less Valuable Change of Variables Commands
32 Representing Noncommutative Expressions with Commutative Ones.
VIII   DETAILS ON INSTALLATION AND MAINTENANCE - ONLY ON THE WEB
33 NCAlgebra and NCGB Installation
 33.1 Running NCAlgebra
 33.2 Running NCGB
 33.3 Running SYStems
 33.4 Running OldMmaGB (which we do not support)
 33.5 Environment Settings
 33.6 How to set up if someone else compiled the code — UNIX
 33.7 Informing TEX about notebook.sty
34 Installing NCGB the Hard Way
 34.1 GNU C++ Version 2.6.3
 34.2 Running NCGB
35 For the Home Team
 35.1 How to make the PC version of NC
IX   TRULY OLD MATERIAL - ONLY ON THE WEB
36 An Old Example of Get Categories- Lots of info
37 Example of Hypothesis Selection in Discovering
38 Possibly Obsolete Command Descriptions
 38.1 NCMakeGB Options -maybe obsolete
 38.2 Special GB related commands- may not work
 38.3 Starting Relations
 38.4 Changing the default options for NCMakeGB
39 Generating Universal Gröbner Basis- MAY NOT WORK - untested in 1999
 39.1 Very Technical Commands
40 Commands for Producing HTML Output
 40.1 Using an input file
X  THE PACKAGE SYStems - ONLY ON THE WEB
41 Preface
42 How To Run The Systems Package
43 How To Contribute
44 What SYStems Does
45 Sample Applications
 45.1 Bounded Real Lemma
 45.2 Measurement Feedback H Control
 45.3 Specializing to less general systems
46 References
47 Glossary For System Hamiltonian Calculations
  Complete Index
Index