S. R. Czapor

Solving Systems of Equations

In this chapter we consider the classical problem of solving (exactly) a system of algebraic equations over a field F. This problem, along with the related problem of solving single univariate equations, was the fundamental concern of algebra until the beginning of the “modern” era (roughly, in the nineteenth century); it remains today an important, widespread concern in mathematics, science and engineering. Although considerable effort has been devoted to developing methods for numerical solution of equations, the develop- ment of exact methods is also well motivated. Obviously, exact methods avoid the issues of conditioning and stability. Moreover, in the case of nonlinear systems, numerical methods cannot guarantee that all solutions will be found (or prove that none exist). Finally, many systems which arise in practice contain “free” parameters and hence must be solved over non-numerical domains.

Integration of Rational Functions

The problem of indefinite integration is one of the easiest problems of mathematics to describe: given a function f(x), find a function g(x) such that g´(x) =f(x)

The Risch Integration Algorithm

When solving for an indefinite integral, it is not enough simply to ask to find an antiderivative of a given function f(x). After all, the fundamental theorem of integral calculus gives the area function A(x)=∭ x a f(t) dt as an antiderivative of f (x). One really wishes to have some sort of closed expression for the antiderivative in terms of well-known functions (e.g. sin(x), e x, log(x)) allowing for common function operations (e.g. addition, multiplication, composition). This is known as the problem of integration in closed form or integration in finite terms. Thus, one is given an elementary function f(x), and asks to find if there exists an elementary function g(x) which is the antiderivative of f(x) and, if so, to determine g(x)

On the derivation and reduction of C 1 trigonometric basis functions using Maple

Abstract A heuristic methodology for the reduction of very complicated trigonometric expressions is presented in the context of deriving C 1 basis functions, for the finite element method, from a trial function which has trigonometric terms. These results are then compared to those obtained through canonical simplification by the method of Grobner bases.

Gröbner Bases for Polynomial Ideals

We have already seen that, among the various algebraic objects we have encountered, polynomials play a central role in symbolic computation. Indeed, many of the (higher-level) algorithms discussed in Chapter 9 (and later in Chapters 11 and 12) depend heavily on com putation with multivariate polynomials. Hence, considerable effort has been devoted to improving the efficiency of algorithms for arithmetic, GCDs and factorization of polynomials. It also happens, though, that a fairly wide variety of problems involving polynomials (among them, simplification and the solution of equations) may be formulated in terms of polynomial ideals. This should come as no surprise, since we have already used particular types of ideal bases (i.e. those derived as kernels of homomorphisms) to obtain algorithms based on interpolation and Hensels lemma. Still, satisfactory algorithmic solutions for many such problems did not exist until the fairly recent development of a special type of ideal basis, namely the Grobner basis.

Normal Forms and Algebraic Representations

This chapter is concerned with the computer representation of the algebraic objects discussed in Chapter 2. The zero equivalence problem is introduced and the important concepts of normal form and canonical form are defined. Various normal forms are presented for polynomials, rational functions, and power series. Finally data structures are considered for the representation of multiprecision integers, rational numbers, polynomials, rational functions, and power series.

Introduction to Computer Algebra

The desire to use a computer to perform a mathematical computation symbolically arises naturally whenever a long and tedious sequence of manipulations is required. We have all had the experience of working out a result which required page after page of algebraic manipulation and hours (perhaps days) of our time. This computation might have been to solve a linear system of equations exactly where an approximate numerical solution would not have been appropriate. Or it might have been to work out the indefinite integral of a fairly complicated function for which it was hoped that some transformation would put the integral into one of the forms appearing in a table of integrals. In the latter case, we might have stumbled upon an appropriate transformation or we might have eventually given up without knowing whether or not the integral could be expressed in terms of elementary functions. Or it might have been any one of numerous other problems requiring symbolic manipulation.

Algebra of Polynomials, Rational Functions, and Power Series

In this chapter we present some basic concepts from algebra which are of central importance in the development of algorithms and systems for symbolic mathematical computation. The main issues distinguishing various computer algebra systems arise out of the choice of algebraic structures to be manipulated and the choice of representations for the given algebraic structures.

Newton’s Iteration and the Hensel Construction

In this chapter we continue our discussion of techniques for inverting modular and evaluation homomorphisms defined on the domain Z[x 1, . . ., x v ]. The particular methods developed in this chapter are based on Newtons iteration for solving a polynomial equation. Unlike the integer and polynomial Chinese remainder algorithms of the preceding chapter, algorithms based on Newtons iteration generally require only one image of the solution in a domain of the form Z p [x 1] from which to reconstruct the desired solution in the larger domain Z[x 1, . . . , x v]. A particularly important case of Newtons iteration to be discussed here is the Hensel construction. It will be seen that multivariate polynomial computations (such as GCD computation and factorization) can be performed much more efficiently (in most cases) by methods based on the Hensel construction than by methods based on the Chinese remainder algorithms of the preceding chapter

Homomorphisms and Chinese Remainder Algorithms

In the previous three chapters we have introduced the general mathematical framework for computer algebra systems. In Chapter 2 we discussed the algebraic domains which we will be working with. In Chapter 3 we concerned ourselves with the representations of these algebraic domains in a computer environment. In Chapter 4 we discussed algorithms for performing the basic arithmetic operations in these algebraic domains.