Computer Algebra and Symbolic Computation: Mathematical Methods
The book is a readable introduction to computer algebra. It presents a theoretical basis of recently published Computer Algebra and Symbolic Computation: Elementary Algorithms, of the same author. The book explores applications of algorithms to automatic simplification, greatest common divisor calculation, resultant computation, polynomial decomposition, and factorisation. After a review of basic background concepts, algorithms for manipulation and evaluation of numerical objects are given. Automatic algebraic and trigonometric simplification is worked out thoroughly and it gives the reader rather detailed information about one of the essential algorithms implemented in computer algebra software. Another of the important numerical objects - single variable polynomials - is discussed in two chapters. Attention is paid in particular to the algorithm for polynomial division, which is the basis to the algorithm for polynomial expansion, Euclidean algorithm, algorithm for partial fraction expansion, and algorithm for performing arithmetic operations for expressions in simple algebraic number fields. This is followed by a generalization of these concepts and algorithms to multivariate polynomials. Two important concepts of computer algebra, the resultant and Gröbner basis, are introduced. The book culminates in the last chapter, devoted to polynomial factorisation and a modern algorithm for it. The book will be useful for undergraduate students of mathematics and computer science. The text is also accessible to a more general audience interested in computer algebra and its applications.