Contents

• NCAlgebra
  • ReleaseNotes NCAlgebra
  • The Most Basic NCAlgebra Commands -- A short tutorial
    • Pretty Output
  • Getting Started in NCAlgebra and NCGB
    • NC On the UNIX Platform
    • On the Windows platform
      • When trouble strikes
    • On Screen Help
  • Installing NCGB
    • Installing NCGB the Easy Way
    • Getting latex and a dvi previewer
    • Running SYStems, OldMmaGB
    • Warning: $NC$* variables
  • Examples
    • Simplifying a Complicated Expression
    • Vector Differentiation
  • Helpful Tricks
    • Expanding
    • Simplifying Expressions
      • Simplifying Rules
      • Orders
      • Automatic generation of rules
    • Edit - For those without Notebooks
    • Conventions
  • NC Commands
    • Manipulating an expression
      • ExpandNonCommutativeMultiply[expr]
      • NCCollect[expr, aListOfVariables]
      • NCStrongCollect[expr, aListOfVariables]
      • NCCollectSymmetric[expr]
      • NCTermsOfDegree[expr,aListOfVariables,indices]
      • NCSolve[expr1==expr2,var]
      • Substitute[expr,aListOfRules,(Optional On)]
      • SubstituteSymmetric[expr, aListOfRules, (optional On)]
      • SubstituteSingleReplace[expr, aListOfRules, (optional On)]
      • SubstituteAll[expr, aListOfRules, (optional On)]
      • Transform[expr,aListOfRules]
      • GrabIndeterminants[ aListOfPolynomialsOrRules]
      • GrabVariables[ aListOfPolynomialsOrRules ]
      • NCBackward[expr]
      • NCForward[expr]
      • NCMonomial[expr]
      • NCUnMonomial[expr]
    • Simplification
      • NCSimplifyRational[ expr ], NCSimplify1Rational[ expr ], and NCSimplify2Rational[ expr ]
      • NCSimplify1Rational[expr]
      • NCSimplify2Rational[expr]
    • Vector Differentiation
      • DirectionalD[expr, aVariable, h]
      • Grad[expr, aVariable]
      • CriticalPoint[expr, aVariable]
      • NCHessian[afunction, ]
    • Block Matrix Manipulation
      • MatMult[x, y, ]
      • ajMat[u]
      • coMat[u]
      • tpMat[u]
      • NCMToMatMult[expr]
      • TimesToNCM[expr]
      • Partitioning a Matrix into a Block Matrix
      • Special Operations with Block Matrices
      • NCLDUDecomposition[aMatrix, Options]
      • NCAllPermutationLDU[aMatrix]
      • NCInverse[aSquareMatrix]
      • NCPermutationMatrix[aListOfIntegers]
      • NCMatrixToPermutation[aMatrix]
      • NCCheckPermutation[SizeOfMatrix, aListOfPermutations]
      • Diag[aMatrix]
      • invMat2[mat]
      • GaussElimination[X]
      • Cascade[P, K]
      • Chain[P]
      • Redheffer[P]
      • DilationHalmos[x]
      • SchurComplementTop[M]
      • SchurComplementBtm[M]
    • Generating Random Matrices
      • RandomMatrix[m,n,min,max,options]
    • Complex Analysis
      • A tutorial
      • ComplexRules
      • ComplexCoordinates[expr]
      • ComplexD[expr, aVariable]
    • Setting symbols to commute or not commute
      • SetNonCommutative[A, B, C, ]
      • CommuteEverything[expr]
      • SetCommutative[a, b, c, ]
      • SetCommutingOperators[b,c]
      • LeftQ[expr]
      • CommutativeQ[X]
      • CommutativeAllQ[expr]
    • Operations on elements in an algebra
      • inv[x]
      • invL[x]
      • invR[x]
      • invQ[x]
      • ExpandQ[inv]
      • ExpandQ[tp]
      • OverrideInverse
      • aj[expr]
      • tp[expr]
      • co[expr]
    • Convexity of a NC function
      • NCConvexityRegion[afunction,alistOfVars,opts]
      • NCMatrixOfQuadratic[ , ]
      • NCIndependenceCheck[aListofLists,variable]
      • NCBorderVectorGather[alist,varlist]
    • NCGuts
      • NCStrongProduct1
      • NCStrongProduct2
      • NCSetNC
    • Setting Properties of an element in an algebra
      • SetInv[a, b, c, ]
      • SetSelfAdjoint[Symbols]
      • SelfAdjointQ[aSymbol]
      • SetIsometry[Symbols]
      • IsometryQ[aSymbol]
      • SetCoIsometry[Symbols]
      • CoIsometryQ[aSymbol]
      • SetUnitary[Symbols]
      • UnitaryQ[aSymbol]
      • SetProjection[Symbols]
      • ProjectionQ[S]
      • SetSignature[Symbols]
      • SignatureQ[Symbol]
    • Setting Properties of functions on an algebra
      • SetSesquilinear[Functions]
      • SesquilinearQ[aFunction]
      • SetBilinear[Functions]
      • BilinearQ[aFunction]
      • SetLinear[Functions]
      • LinearQ[aFunction]
      • SetConjugateLinear[Functions]
      • ConjugateLinearQ[aFunction]
      • SetIdempotent[Functions]
      • IdempotentQ[aFunction]
      • SetCommutingFunctions[ aFunction, anotherFunction]
      • SetNonCommutativeMultiplyAntihomomorphism[ Functions]
    • Manipulating an Expression -- less useful commands
      • NCTermArray[expr,aList,anArray]
      • NCReconstructFromTermArray[anArray]
      • NCCompose[aVerySpecialList]
      • NCDecompose[expr, listofsymbols]
    • Utilities
      • CEEP
      • SaveRules[expression, 'optional tag "message"']
      • SaveRulesQ[]
      • FunctionOnRules[Rules, Function1, Function2, (optional On)]
  • Pretty Output and Tex Commands
    • SetOutput[ optionlist,]
    • NC enhancement of Mathematica's TeX settings
    • Simple TeX Commands
    • Real time Tex
      • SeeTeX[] or SeeTeX[anInteger]
      • NoTeX[]
      • KillTeX[]
      • See[aListOfIntegers]
      • Keep[anInteger]
      • Kill[anInteger]
    • One shot commands
      • LookAtMatrix[aMatrix]
      • LookAtLongExpression[anExpression]
    • Informing TEX about notebook.sty, etc.
    • HTML output - not Supported
  • An NCAlgebra demo - The Bounded Real Lemma
    • Introduction
    • System Definition
    • Demo Initialization
    • Lemma
    • Proof (by computation)
  • Aliases
  
  
• NONCOMMUTATIVE CONTROL SYSTEM PROFESSIONAL
  • State Space Systems Constructions
    • System Interconnections
      • SeriesConnect[ System1, System2 ]
      • FeedbackConnect[ System1, System2 ]
      • ParallelConnect[ System1, System2 ]
    • Continuous vs. Discrete
      • ContinuousTimeQ[ System1]
      • DiscreteTimeQ[ System1]
    • Transfer Function
      • TransferFunction[ System1]
    • Systems from Systems
      • Dual[ System1]
      • InverseSystem[ System1]
  
  
• NONCOMMUTATIVE GRÖBNER BASES-NCGB
• NCGB: Easy Introduction
  • Introduction
  • Simple Demos of Basic Commands
    • To start a C++ GB session
      • NCGBSetIntegerOverflow[False]
    • Simplifying Expressions
    • Making a Groebner Basis
    • Reducing a polynomial by a GB
      • Simplification via GB's
  • NCGB Facilitates Natural Notation
    • A Simplification example
    • MakingGB's and Inv[], Tp[]
    • Simplification and GB's revisited
    • Saving lots of time when typing
      • Saving time when typing relations involving inverses:NCMakeRelations
      • Saving time working in algebras with involution: NCAddTranspose, NCAddAdjoint
      • Saving time when setting orders: NCAutomaticOrder
  • Demo on NCGB - Matrix Computation
  • To Run NCGB - Template.nb
  • NCProcess: What It Does
    • NCProcess: Input and Output
      • When to stop
    • Changing Variables
  • NCProcess: An Example
    • Background
    • The Problem
    • Solution via a Prestrategy
    • The end game
      • Concluding Remarks
    • Conclusions
  • NCProcess: The Commands
    • SetKnowns and SetUnknowns
    • NCProcess
      • NCProcess[aListOfPolynomials,iterations,fileName, Options ]
      • Examples
    • Commonly Used NCProcess Options and Commands
      • UserSelect aListOfPolynomials
      • DegreeCap and DegreeSumCap
      • MainUnknowns
      • NCShortFormula
      • Getting Categories
    • Typical use of the NCProcess command
    • Details of NCProcess
      • NCProcess1 command
      • NCProcess2 command
    • NCProcess1 and NCProcess2: The technical descriptions
  
  
• NCGB: FOR THE MORE ADVANCED USER
  • NCProcess: The Concepts
    • NCProcess: Input and Output
      • Almost an Algorithm
    • Elimination
    • What is a prestrategy?
      • Prestrategy
      • When to stop
      • Redundant Equations
      • Summary of a Prestrategy
    • A strategy
  • Another Example: Solving the Control Problem
    • Engineering motivation for the problem
      • System definitions
      • Energy balance equations
    • Problem statement
    • The key relations: executable form
    • Solving using NCProcess
      • Step 1
      • Step 2: The user attacks
      • Step 3
      • Step 4
    • End Game
  
  
• NCGB: LISTS OF COMMANDS AND OTHER DETAILS
  • Ordering on variables and monomials
    • Lex Order: The simplest elimination order
    • Graded lex ordering: A non-elimination order
    • Multigraded lex ordering : A variety of elimination orders
    • The list of commands
      • SetMonomialOrder[aListOfListsOfIndeterminates, ...]
      • SetUnknowns[aListOfIndeterminates]
      • SetUnKnowns[aListOfVariables]
      • ClearMonomialOrder[]
      • PrintMonomialOrder[]
      • NCAutomaticOrder[ aMonomialOrder, aListOfPolynomials ]
    • Fancier Order Setting Commands
      • SetMonomialOrder[aListOfIndeterminants, n]
      • ClearMonomialOrderN[n]
      • ClearMonomialOrderAll[]
      • WhatIsMultiplicityOfGrading[]
      • WhatIsSetOfIndeterminants[n]
  • More NCProcess Options
    • Creating small generating sets: RR, RRByCat, SB, SBByCat
    • NCCollectOnVars
      • NCCollectOnVars[aListOfExpressions, aListOfVariables]
    • Turning screen output off
    • Output Options
      • Turning screen output off: PrintScreenOutput
      • TeX
      • ASCII
      • NCGBFastRegularOutput
      • NCShortFormulas
    • NCProcess Summary Table
  • Commands for Making and Using GB's
    • Simplification
      • NCSimplifyAll[expressions, startRelations, iterations]
      • NCSimplifyRationalX1[expressions, startRelations, iterations]
      • NCSimplifyRationalX1[expressions,startRelations]
      • NCSimplifyRationalX1[expressions,iterations]
      • NCSimplifyRationalX1[expressions]
    • Making a Gröbner Basis and various options (with their defaults)
      • NCMakeGB[aListOfPolynomials, iterations]
      • UserSelect(Distinguishing important relations)
      • ClearUserSelect[]
      • Deselect (DISABLED)
      • FinishedComputingBasisQ[] - Untested in 1999
      • WhatIsPartialGB[]
      • NCGBSetIntegerOverflow[False]
      • PartialBasis[aNumber] - Untested in 1999
      • IterationNumber[aList] or IterationNumber[ aNumber ] - UNTESTED 1999
      • CleanUp
      • SetCleanUpBasis[n] - Untested in 1999
      • CleanUpBasisQ[] - Untested in 1999
      • History Off
      • Correspondence to sections `Simplification' and `Reduction'
      • Setting Monomial Orders- See Chapter 21
      • ReinstateOrder[]
    • Reduction
      • Reduction[aListOfPolynomials, aListOfRules]
      • PolyToRule[aPolynomial]
      • RuleToPoly[aRule]
  • Commands for Making Small Bases for Ideals: Small Basis, Shrink Basis
    • Brute Force: Shrinking
      • SmallBasis[aListOfPolynomials, anotherListOfPolynomials, iter]
      • SmallBasisByCategory[aListOfPolynomials, iter]
      • ShrinkOutput[aListOfPolynomials,fileName]
    • Brute Force: Many shrinks
      • ShrinkBasis[aListOfPolynomials,iterations]
    • First Example
    • Second Example
    • Smaller Bases and the Spreadsheet command
    • How Small Basis commands relate to the similar NCProcess Options
  • Help in Typing Relations .
    • NCMakeRelations[aSpecialList, aSpecialList, ...]
    • Output notation for pseudoinverse and perp's
      • NCAddTranspose[aListOfExpressions]
      • NCAddAdjoint[aListOfExpressions]
      • Pulling important equations into your session from an NCProcess output - See GetCategories in §26.0.5.
      • Help in typing Monomial Orders - See NCAutomaticOrder Section 21.4.6
  • Retrieving Categories and Regular Output
    • GetCategory[aListOfVariables, NCPAns]
    • GetCategory[aCharString,NCPAns]
    • Clear[NCPAns]
    • Example
    • Creating Categories
      • CreateCategories[aListOfPolynomials, aName]
    • RegularOutput[aListOfPolynomials,``fileName"]
      • RegularOutput[aListOfPolynomials,``fileName"]
    • How to Really Change Regular Output
  • The Dimension of the Solution Set of a Set of Polynomial Equations
    • The Commuting Case
    • Noncommutative Case: Gelfand-Kirillov dimension
    • References
    • Commands
      • NCHilbertCoefficient[integer1, aListOfExpressions, integer2, anOption]
      • NCX1VectorDimension[alist]
  • Commands which are not supported
    • A Mathematica Groebner Basis Package Without C++
    • NCXWholeProcess[ polys, orderList, fileName, grobIters]
  • Getting NCALGEBRA and NCGB
    • Getting NCAlgebra and NCGB off the web
    • Getting NCAlgebra and NCGB through anonymous ftp
      • The ``.Z" file
      • The ``.gz" file
      • The ``.zip" file
      • Look at the document
    • The last step
    • The NC directory structure
    • Directory structure of NCAlgebra alone
  • Testing your version of NCGB
    • Beginners
    • Turning On Screen Output
    • More Testing for Developers - DOES NOT WORK 2001
      • Setting the Testing Environment
  
  
• Index
  • References
  
  
• DEMOS - ONLY ON THE WEB
  • Singular Perturbation Demo
  
  
• DETAILS AND OLD OR NEW COMMANDS -ONLY ON THE WEB
  • History of the Production of a GB and Playing By Numbers
    • Play By Numbers
      • WhatAreGBNumbers[]
      • WhatAreNumbers[]
      • WhatIsPartialGB[aListOfIntegers]
      • NumbersFromHistory[aPolynomial,history]
    • History of the production of a GB
      • WhatIsHistory[aListOfIntegers]
      • WhatIsKludgeHistory[aListOfIntegers]
      • More on the History of how NCMakeGB produced its answer
      • The DAG associated with a History
  • Commands for Making Small Bases for Ideals: Remove Redundant
    • Removing excess relations
      • Introduction
      • RemoveRedundant[]
      • RemoveRedundant[aListOfPolynomials,history]
      • RemoveRedundentByCategory[]
      • RemoveRedundentByCategory[ aListOfPolynomials, history]
    • Discussion of RemoveRedundent command
    • Examples
    • First Example
    • Second Example
    • Smaller Bases and the Spreadsheet command
  • NCXFindChangeOfVariables: The Long Description
    • Details of the Algorithm
      • Preparation
      • Collect and extract
      • Eliminate candidates which are too small
      • Eliminate purely numerical terms from candidates - Default is Off
      • Sort list of candidates by number of terms
      • Multiply through by monomials - Default is off
      • Run the Grobner basis algorithm
      • Options
    • Future Improvements
    • Finding Coefficients of Variables in a Polynomial
      • NCCoefficientList[Expression, aListOfIndeterminants]
    • Main Change Of Variables Command
      • NCXFindChangeOfVariables[ aListofPolynomials, anInteger, aString, Options]
      • NCXPossibleChangeOfVariables[ aListofPolynomials, Options]
    • Less Valuable Change of Variables Commands
      • NCXMultiplyByMonomials[ aVerySpecialList]
      • NCXAllPossibleChangeOfVariables[ aListOfPolynomials]
  • Representing Noncommutative Expressions with Commutative Ones.
    • NCXRepresent[aListOfExpressions, aListOfVariables, aListOfDims, aListOfFunctions, aListOfExtraRules]
  
  
• DETAILS ON INSTALLATION AND MAINTENANCE - ONLY ON THE WEB
  • NCAlgebra and NCGB Installation
    • Running NCAlgebra
    • Running NCAlgebra
    • Running NCGB
    • Running SYStems
    • Running OldMmaGB (which we do not support)
    • Environment Settings
      • NCAlgebra $NC$ Variables
      • NCGB $NC$ Variables
    • How to set up if someone else compiled the code -- UNIX
      • When your system administrator installed the code
      • When your friend installed the code
    • Informing TEX about notebook.sty
    • Testing NCGB
  • Installing NCGB the Hard Way
    • GNU C++ Version 2.6.3
      • Compiling the C++ part of NCGB
    • Running NCGB
      • Loading NCGB more quickly
  • For the Home Team
    • How to make the PC version of NC
  
  
• TRULY OLD MATERIAL - ONLY ON THE WEB
  • An Old Example of Get Categories- Lots of info
  • Example of Hypothesis Selection in Discovering
  • Possibly Obsolete Command Descriptions
    • NCMakeGB Options -maybe obsolete
      • ReturnRelationsToMmaTrue - DEAD??
      • SupressCOutput False (less output to the screen )
      • SupressAllCOutput False (very little outp ut to the screen)
      • NCContinueMakeGB[iterationNumber]
    • Special GB related commands- may not work
    • Starting Relations
      • SortRelations[aListOfRules]
      • SortMonomials[aListOfVariables]
    • Changing the default options for NCMakeGB
      • ChangeNCMakeGBOptions[option,value] -- need to update description
  • Generating Universal Gröbner Basis- MAY NOT WORK - untested in 1999
    • AllOrders[aListofPolynomials, aListofIndeterminants]
    • EquivalenceClasses[aListOfPolynomials] or EquivalenceClasses[aListOfPolynomials, Simpler]
    • UniversalBasis[aListOfPolynomials, NumberOfIterations]
    • Very Technical Commands
      • GroebnerCutOffFlag[n_Integer]
      • GroebnerCutOffMin[n_Integer]
      • GroebnerCutOffSum[n_Integer]
  • Commands for Producing HTML Output
    • .
    • ToHTMLString[expression]
    • MakeGif[file,expression]
    • HTML
    • Using an input file
  
  
• THE PACKAGE SYStems - ONLY ON THE WEB
  • Preface
  • How To Run The Systems Package
  • How To Contribute
  • What SYStems Does
  • Sample Applications
    • Bounded Real Lemma
    • Measurement Feedback Control
      • Derivation of CritW and sHWo
      • The MIN/MAX in U
      • Derivations of IAX, Critc, and Hopt
      • Derivation of IAYI
      • Derivation of critical q, k, and bterm
    • Specializing to less general systems
      • Specializing to linear systems
      • Specializing Using The Doyle Glover Khargonekar Francis Simplifying Assumptions
      • Demo: Linear Doyle Glover Kargonekar Francis Equations
  • References
  • Glossary For System Hamiltonian Calculations
  
  
• Index
  • References
  
  
• DEMOS - ONLY ON THE WEB
  • Singular Perturbation Demo
  
  
• DETAILS AND OLD OR NEW COMMANDS -ONLY ON THE WEB
  • History of the Production of a GB and Playing By Numbers
    • Play By Numbers
      • WhatAreGBNumbers[]
      • WhatAreNumbers[]
      • WhatIsPartialGB[aListOfIntegers]
      • NumbersFromHistory[aPolynomial,history]
    • History of the production of a GB
      • WhatIsHistory[aListOfIntegers]
      • WhatIsKludgeHistory[aListOfIntegers]
      • More on the History of how NCMakeGB produced its answer
      • The DAG associated with a History
  • Commands for Making Small Bases for Ideals: Remove Redundant
    • Removing excess relations
      • Introduction
      • RemoveRedundant[]
      • RemoveRedundant[aListOfPolynomials,history]
      • RemoveRedundentByCategory[]
      • RemoveRedundentByCategory[ aListOfPolynomials, history]
    • Discussion of RemoveRedundent command
    • Examples
    • First Example
    • Second Example
    • Smaller Bases and the Spreadsheet command
  • NCXFindChangeOfVariables: The Long Description
    • Details of the Algorithm
      • Preparation
      • Collect and extract
      • Eliminate candidates which are too small
      • Eliminate purely numerical terms from candidates - Default is Off
      • Sort list of candidates by number of terms
      • Multiply through by monomials - Default is off
      • Run the Grobner basis algorithm
      • Options
    • Future Improvements
    • Finding Coefficients of Variables in a Polynomial
      • NCCoefficientList[Expression, aListOfIndeterminants]
    • Main Change Of Variables Command
      • NCXFindChangeOfVariables[ aListofPolynomials, anInteger, aString, Options]
      • NCXPossibleChangeOfVariables[ aListofPolynomials, Options]
    • Less Valuable Change of Variables Commands
      • NCXMultiplyByMonomials[ aVerySpecialList]
      • NCXAllPossibleChangeOfVariables[ aListOfPolynomials]
  • Representing Noncommutative Expressions with Commutative Ones.
    • NCXRepresent[aListOfExpressions, aListOfVariables, aListOfDims, aListOfFunctions, aListOfExtraRules]
  
  
• DETAILS ON INSTALLATION AND MAINTENANCE - ONLY ON THE WEB
  • NCAlgebra and NCGB Installation
    • Running NCAlgebra
    • Running NCAlgebra
    • Running NCGB
    • Running SYStems
    • Running OldMmaGB (which we do not support)
    • Environment Settings
      • NCAlgebra $NC$ Variables
      • NCGB $NC$ Variables
    • How to set up if someone else compiled the code -- UNIX
      • When your system administrator installed the code
      • When your friend installed the code
    • Informing TEX about notebook.sty
    • Testing NCGB
  • Installing NCGB the Hard Way
    • GNU C++ Version 2.6.3
      • Compiling the C++ part of NCGB
    • Running NCGB
      • Loading NCGB more quickly
  • For the Home Team
    • How to make the PC version of NC
  
  
• TRULY OLD MATERIAL - ONLY ON THE WEB
  • An Old Example of Get Categories- Lots of info
  • Example of Hypothesis Selection in Discovering
  • Possibly Obsolete Command Descriptions
    • NCMakeGB Options -maybe obsolete
      • ReturnRelationsToMmaTrue - DEAD??
      • SupressCOutput False (less output to the screen )
      • SupressAllCOutput False (very little outp ut to the screen)
      • NCContinueMakeGB[iterationNumber]
    • Special GB related commands- may not work
    • Starting Relations
      • SortRelations[aListOfRules]
      • SortMonomials[aListOfVariables]
    • Changing the default options for NCMakeGB
      • ChangeNCMakeGBOptions[option,value] -- need to update description
  • Generating Universal Gröbner Basis- MAY NOT WORK - untested in 1999
    • AllOrders[aListofPolynomials, aListofIndeterminants]
    • EquivalenceClasses[aListOfPolynomials] or EquivalenceClasses[aListOfPolynomials, Simpler]
    • UniversalBasis[aListOfPolynomials, NumberOfIterations]
    • Very Technical Commands
      • GroebnerCutOffFlag[n_Integer]
      • GroebnerCutOffMin[n_Integer]
      • GroebnerCutOffSum[n_Integer]
  • Commands for Producing HTML Output
    • .
    • ToHTMLString[expression]
    • MakeGif[file,expression]
    • HTML
    • Using an input file
  
  
• THE PACKAGE SYStems - ONLY ON THE WEB
  • Preface
  • How To Run The Systems Package
  • How To Contribute
  • What SYStems Does
  • Sample Applications
    • Bounded Real Lemma
    • Measurement Feedback Control
      • Derivation of CritW and sHWo
      • The MIN/MAX in U
      • Derivations of IAX, Critc, and Hopt
      • Derivation of IAYI
      • Derivation of critical q, k, and bterm
    • Specializing to less general systems
      • Specializing to linear systems
      • Specializing Using The Doyle Glover Khargonekar Francis Simplifying Assumptions
      • Demo: Linear Doyle Glover Kargonekar Francis Equations
  • References
  • Glossary For System Hamiltonian Calculations
  • THE MOST BASIC COMMANDS -- TEAR OFF SECTION
    • Pretty Output
    • Setting properties of elements and functions
  
  
• About this document ...