Contents

 
• Introduction
  • What are algebraic calculations?
  • Why do algebraic calculations?
  • History
    • Chronology of computer algebra systems
• Algorithms for algebraic computation
  • Algebraic structures
    • Number domains
    • Algebraic expression domains
  • Representation of algebraic structures
    • Representation of integers
    • Representation of polynomials
      • Prefix representation
      • Dense representation
      • Sparse representation
      • Recursive representation
      • Recursive representation in Macsyma
      • Recursive representation in Reduce
    • Representation of Expressions
  • Arithmetic
    • Numeric vs symbolic arithmetic
    • Arithmetic of integers
    • Arithmetic of polynomials
      • Lists
  • Simplification
    • Canonical simplification on algebraic domains
    • Complexity of expressions
  • Greatest common divisor
    • GCD of integers
    • GCD of polynomials with rational coefficients
      • Example of GCD in Q[x]
    • GCD of polynomials with integer coefficients
      • Example of GCD in Z[x]
  • Resultant
  • Solving polynomial equations
  • Differentiation
  • Summation
    • Simple example
    • Gosper algorithm
    • Examples using the Gosper algorithm
  • Integration
    • Integration of rational functions
    • Integration of elementary transcendental functions
    • Integration examples
  • Ordinary differential equations
  • Polynomial factorization
  • Quantifier elimination
    • Example with QEPCAD program
• Integrated mathematical systems
  • Computer algebra systems
  • Peculiarities of programming in computer algebra systems
  • Expression Swell
    • An Example of Expression Swell
    • An Example of Intermediate Expression Swell
    • Expression Size
    • Another Example of Intermediate Expression Swell
    • Expression Swell Analysis
• Basic possibilities of integrated mathematical systems
  • Axiom
    • Number domains
      • Big integers
      • Rational numbers
      • Complex numbers
      • Algebraic numbers
      • Big floating point numbers
    • Polynomials
      • Basic operations
      • Factorization
      • Grobner bases
    • Rational functions
    • Solving equations
      • Linear systems
      • Nonlinear equations
      • Nonlinear systems
    • Analytical operations
      • Limits
      • Taylor series
      • Summation and Products
      • Integration
      • Ordinary differential equations
      • Substitutions - pattern matching
    • Matrices
    • Graphics
      • 2D Graphics
      • 3D Graphics
      • Parametric plots
  • Derive
    • Number domains
      • Big integers
      • Rational numbers
      • Complex numbers
      • Radicals
      • Big floating point numbers
    • Polynomials
      • Basic operations
      • Factorization
    • Rational functions
    • Solving equations
      • Linear systems
      • Nonlinear equations
    • Analytical operations
      • Limits
      • Taylor series
      • Summation and Products
      • Integration
    • Matrices
    • Graphics
      • 2D Graphics
      • 3D Graphics
      • Parametric plots
  • Macsyma
    • Number domains
      • Big integers
      • Rational numbers
      • Complex numbers
      • Algebraic numbers
      • Big floating point numbers
    • Polynomials
      • Basic operations
      • Factorization
      • Decomposition
      • Grobner bases
    • Rational functions
    • Solving equations
      • Linear systems
      • Nonlinear equations
    • Analytical operations
      • Limits
      • Taylor series
      • Summation and Products
      • Integration
      • Ordinary differential equations
      • Substitutions - pattern matching
    • Matrices
    • Code generation
      • Generated FORTRAN program
    • Graphics
      • 2D Graphics
      • 3D Graphics
      • Parametric plots
      • Contour maps
    • Graphical presentation of formulas
  • Maple
    • Number domains
      • Big integers
      • Rational numbers
      • Complex numbers
      • Big floating point numbers
    • Polynomials
      • Basic operations
      • Factorization
      • Decomposition
      • Grobner bases
    • Rational functions
    • Solving equations
      • Linear systems
      • Nonlinear equations
      • Nonlinear systems
    • Analytical operations
      • Limits
      • Taylor series
      • Summation and Products
      • Integration
      • Ordinary differential equations
    • Matrices
    • Graphics
      • 2D Graphics
      • 3D Graphics
      • Parametric plots
      • Contour maps
      • Polytopes
    • Graphical presentation of formulas
  • Mathematica
    • Number domains
      • Big integers
      • Rational numbers
      • Complex numbers
      • Big floating point numbers
    • Polynomials
      • Basic operations
      • Factorization
      • Decomposition
      • Grobner bases
    • Rational functions
    • Solving equations
      • Linear systems
      • Nonlinear equations
      • Nonlinear systems
    • Analytical operations
      • Limits
      • Taylor series
      • Summation and Products
      • Integration
      • Ordinary differential equations
    • Matrices
    • Graphics
      • 2D Graphics
      • 3D Graphics
      • Parametric plots
      • Contour maps
      • Polytopes
    • Graphical presentation of formulas
  • Reduce
    • Number domains
      • Big integers
      • Rational numbers
      • Complex numbers
      • Algebraic numbers
      • Big floating point numbers
    • Polynomials
      • Basic operations
      • Factorization
      • Decomposition
      • Simplification with side relations
      • Grobner bases
    • Rational functions
    • Solving equations
      • Linear systems
      • Nonlinear equations
      • Nonlinear systems
    • Analytical operations
      • Limits
      • Taylor series
      • Summation and Products
      • Integration
      • Ordinary differential equations
      • Substitutions - pattern matching
    • Matrices
    • Code generation
      • Generated FORTRAN program
    • Graphics
      • 2D Graphics
      • 3D Graphics
      • Parametric plots
      • Contour maps
    • Graphical presentation of formulas
• Applications of computer algebra
  • Classical application areas
  • Other application areas
  • Case study 1. - Perturbation methods
    • Celestial mechanics - nonlinear algebraic equations
    • Mechanics - nonlinear ordinary differential equations
    • Quantum mechanics - eigenvalue problem
  • Case study 2. - General theory of relativity
    • Basic notions
    • Examples
    • Other problems and references
  • Case study 3. - Collision integrals in plasma physics
    • Basic notions
    • Analytical calculation of collision integrals
  • Case study 4. - Numerical solving of partial differential equations
    • References
  • Survey articles on applications
• Another sources of study
  • Basic references
  • Other references
  • Journals
  • Electronic information sources
    • General electronic information sources
    • Electronic resourses related to particular systems
      • Axiom electronic information sources
      • Derive electronic information sources
      • Macsyma electronic information sources
      • Maple electronic information sources
      • Mathematica electronic information sources
      • REDUCE electronic information sources
  • References for CAS Comparisons
  • Computer Algebra Conferences
    • International
      • ISSAC
      • IMACS-ACA
    • Systems meetings
      • Axiom
      • Derive
      • Macsyma
      • Maple
      • Mathematica
  • Manuals
  • Distributors addresses
• About this document ...