Introduction

Polynomials form one of the oldest classes of mathematical functions in mathematics with an important history in science, engineering, and other quantitative fields [Pan, 1997], [Pan, 1998], [Traub, 1998], [Derbyshire, 2006], [Sitton, 2003]. Part of the polynomial's appeal comes from the fact that it may be numerically evaluated using a finite number of multiplications and additions. Another advantage it presents is its use in modeling physical processes or complicated mathematical functions. Polynomials are used to build expansion systems or basis sets in various vector spaces. Segments of polynomials are used to create splines. It is worth studying the basic ideas and the variety of applications.

Polynomials are introduced early in the teaching of algebra as a means of demonstrating basic principles and methods, e.g. substitution, simplification, and factoring. Most people are aware that there exist formulas for factoring quadratic (2nd degree), cubic (3rd degree), and quartic (4th degree) polynomials. In 1824 the mathematician Niels Abel proved that there is no possible "closed form" solution in terms of basic operations for the general pentic or quintic (5th degree) polynomial or those of higher degrees. This fact requires the development of effective numerical methods for iteratively factoring polynomials above degree 4.

The results reported in these modules are from a group at Rice University called the "Polynomial Club" (Jim Fox, Sidney Burrus, Gary Sitton, and Sven Treitel). The program was designed and written by Jim Fox.

Basic Definitions

The definition of an NNth degree polynomial is

where both f(z)f(z) and zz are complex valued and the coefficients akak can be complex but are often real valued. NN, which is the highest power of zz in the polynomial, is called the degree (or sometimes the order) of the polynomial. The number of coefficients is N+1N+1.

From the various forms of the Fundamental Theorem of Algebra, one can show that all polynomials can also be expressed in a “factored" form by

where the possibly complex valued zrzr are called the “zeros" of the polynomial since f(zk)=0f(zk)=0. Because the zeros are not necessarily distinct, a unique form of (Equation 2) is given by

where MkMk is the multiplicity of the kthkth zero, LL is the number of distinct zeros, and ∑kMk=N∑kMk=N. The first degree polynomials (z-zk)(z-zk) in (Equation 2) and (Equation 3) are called the factors of f(z)f(z). Note the analogy between polynomials and integers. Indeed, multiplying two polynomials is the same operation as multiplying two integers (except for carrying) or convolving two number sequences.

Creating (Equation 1) from (Equation 2) or (Equation 3) is fairly straight forward and requires only a finite number of arithmetic operations, but finding (Equation 2) or (Equation 3) from (Equation 1) is difficult. That process is called “factoring" the polynomial and is the topic of these notes. Actually, the effects of finite precision arithmetic sometimes make the "unfactoring" process of calculating (Equation 1) from (Equation 2) or (Equation 3) poorly defined because it depends on the sequence order of combining the zeros.

One can look at the zeros of a polynomial as being a second parameterization of the polynomial with the coefficient form of (Equation 1) being the first. A combination would be expressing an even degree polynomial as the product of two N/2N/2 degree polynomials. Factoring and “unfactoring" can create a variety of parameterizations for both discrete-time signals and systems.

The Four Problems for Polynomials

If the polynomial is represented in factored form (Equation 2), then the coefficient sum form (Equation 1) may be easily found by computing the product of the MM linear polynomial factors. This is done by multiplying the linear factors one at a time and collecting all coefficients of like powers in the product. As will be shown later, this is equivalent to a cascaded discrete convolution of all of the polynomial coefficients where the rthrth linear polynomial is represented as a doublet set of two coefficients {-zr,1}{-zr,1} where the leading coefficient -zr-zr is usually complex. This computation allows the reconstruction of the coefficients of the polynomial corresponding to this root set. We call this process unfactoring but it can lead to large numerical inaccuracies for a surprisingly small number of terms. The main concern here is with the inverse or factoring problem: given the coefficient form (Equation 1), find the factored form (Equation 2).

The common case of purely real polynomial coefficients leads to a simplification: All of the complex roots of a real polynomial occur in complex conjugate pairs. Given the rthrth complex root zr=xr+iyrzr=xr+iyr, its complex conjugate is given by zr*=xr-iyrzr*=xr-iyr.

Thus for yr≠0yr≠0, a complex root zrzr will always be associated with its conjugate zr*zr* to form a conjugate root pair. This pair of roots represents the product of two linear factors forming a real quadratic or second degree factor:

From (Equation 2) a polynomial may be defined as the product of all of its factors. If the factors have only real coefficients, then the product of all the factors will have only real coefficients anan. Thus only half of the complex roots need be found, say in the upper complex half-plane with positive imaginary parts, i.e. yr≥0yr≥0. The associated conjugate roots in the lower half-plane may be trivially derived by simple negation of their imaginary parts.

Most of the results in these modules are based on extensive numerical experimentation. We have built on existing theory and techniques with empirically derived rules and algorithms that perform well on well-conditioned polynomials and, in many cases, specifically applied to signal processing applications with random coefficients.

Factoring Polynomials

Below are three approaches to factoring polynomials: