Introduction to Symbolic Computation

Contents

• Notations and summary definitions
• Preliminary considerations
• Mathematical induction
• Algebraic preliminaries
• Groups, rings, fields and ideals
• Domains
• Factorization in integral domains
• Greatest Common Divisor
• Unique Factorization Domains
• Ideals and principal ideal domains
• Computing in Z
• Ordinary Euclidean algorithm
• Relatively prime numbers
• Extended Euclidean algorithm
• Efficiency of the Euclidean algorithm
• Properties of the Fibonacci numbers
• Generating function for Fibonacci numbers
• An explicit formula for Fibonacci numbers
• Chinese Remainder Theorem
• Exercises
• Elementary linear algebra
• Vectors and matrices
• Square matrices and inverse matrices
• Systems of linear equations
• Determinants
• Exercises
• Numerical solving
• Newton's method
• Convergence of Newton's iteration
• Gerlach's iteration
• Exercises
• Polynomials
• Polynomial division in the univariate case
• Euclidean algorithm for polynomials
• Memebership problem for ideals of univariate polynomials
• Fundamental Theorem of Algebra
• Affine varieties
• Properties of affine varieties
• Parametrizing and plotting of affine varieties
• Ideal generated by multivariate polynomials
• Review for Exam #2
• Multivariate polynomials
• Monomial orderings
• Multivariate division with remainder
• Groebner basis
• Applications of Groebner basis
• Resultants
• Properties of resultants
• Applications of resultants
• Integration
• Bernoulli Algorithm
• Hermite reduction

Copyright © 1998 - 2000  Mika Seppälä, all rights reserved.