On factoring polynomials  (Page 2/2)

One can look at the zeros of a polynomial as being a second parameterization of the polynomial with the coefficient form of [link] being the first. A combination would be expressing an even degree polynomial as the product of two $N/2$ degree polynomials. Factoring and “unfactoring" can create a variety of parameterizations forboth discrete-time signals and systems. It should be noted that the coefficients of the polynomial are nonlinear functions of all of the zeros, and the zeros of the polynomial are nonlinear functions of all of the coefficients. In some cases, a very small change in one zero can cause a huge change in the coefficients and/or a very small change in a coefficient can cause a huge change in the zeros. In these cases, the polynomials are called "poorly conditioned" and are very difficult to factor by any means.

The four problems for polynomials

1. Evaluating the polynomial and its derivatives. “polyval" in Matlab or Horner's method
2. Factoring the polynomial. “roots" in Matlab
3. Composing the factors. “poly" in Matlab or convolution perhaps using the FFT.
4. Deflating the polynomial by a root. Horner's method or FFT division.

If the polynomial is represented in factored form [link] , then the coefficient sum form [link] may be easily found by computing the product of the $M$ linear polynomial factors. This is done by multiplying the linear factors one at a time and collecting all coefficients of likepowers in the product. As will be shown later, this is equivalent to a cascaded discrete convolution of all of the polynomial coefficients where the $r^{th}$ linear polynomial is represented as a doublet set of two coefficients $\{-z_r,1\}$ where the leading coefficient $-z_r$ is usually complex. This computation allows the reconstruction of thecoefficients 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 [link] , find the factored form [link] .

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 $r^{th}$ complex root $z_r=x_r+i{y}_r$ , its complex conjugate is given by $z_r^{*}=x_r-i{y}_r$ .

Thus for $y_r\ne 0$ , a complex root $z_r$ will always be associated with its conjugate $z_r^{*}$ to form a conjugate root pair. This pair of roots represents the product of two linear factors forming a realquadratic or second degree factor:

From [link] a polynomial may be defined as the product of all of its factors. If the factors have only real coefficients, then the product ofall the factors will have only real coefficients $a_n$ . Thus only half of the complex roots need be found, say in the upper complex half-planewith positive imaginary parts, i.e. $y_r\ge 0$ . The associated conjugate roots in the lower half-plane may be trivially derived by simplenegation 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 withempirically derived rules and algorithms that perform well on well-conditioned polynomials and, in many cases, specifically applied to signal processingapplications with random coefficients.

Factoring polynomials

Below are three approaches to factoring polynomials:

• Find and Deflate: The usual algorithms for factoring polynomials start by guessing or estimating the value of a zero, thenusing some descent method on |f(z)|, find a single zero. This zero is then removed from f(z) by "deflation" (f(z) is divided by the factor represented by the zero) and the process repeated on thereduced degree quotient polynomial. The descent method is often Newton's method which is implemented with Horner's method . There may be problems with error accumulation which, after several deflations, results infailure. Also, if the zeros are not found and removed in a carefully chosen order, the quotient polynomial becomes poorly conditioned. This set of procedures isoften simply called Newton's method. Methods presented in these notes include "Random Argument", "Chosen Argument", and "Pre-Whitening", all ofwhich try to maintain or improve the conditioning of the quotient polynomial produced by deflation. The deflation itself must be done in a"stable" way to prevent error accumulation. In some cases, this involves using Horner's method and in others, the DFT.
• Eigenvalue Method: If the companion matrix for the polynomial is created, its eigenvalues are the zeros of the polynomial. Because verysophisticated algorithms have been developed for finding eigenvalues, this is a powerful and robust approach. However, it requires considerablecomputer memory for the matrices and is computationally inefficient and, therefore, slow. Matlab uses this approach.
• Grid Search: If one has knowledge from the structure of the polynomial what regions in the complex plane contain the zeros, a directsearch can be used. Because a large class of polynomials, including those with random coefficients, have their zeros clusters around the unitcircle, a very efficient polar coordinate grid search can be conducted to find good estimates for the zeros which are then found accurately by aNewton's or Laguerre's algorithm [18,19]. The Lindsey-Fox algorithm uses thisapproach. A description of the Lindsey-Fox program

Bibliography

1. J. Derbyshire, Unknown Quantity: A Real and Imaginary History of Algebra. Joseph Henry Press, an imprint of the National Academies Press,Washington, 2006.
2. V. Y. Pan, "Solving a polynomial equation: some history and recent progress". SIAM Review. No. 2,Vol. 39, June1997. pp. 187–220.
3. V. Y. Pan, "Solving Polynomials with Computers". American Scientist. No. 1,Vol. 86, January-February1998. pp. 62–69.
4. G. Sitton, C. S. Burrus, J. W. Fox, and S. Treitel, "Factoring Very High Degree Polynomials in Signal Processing". Signal Processing Magazine . No. 6,Vol. 20, November2003. pp. 27–42.
5. J. F. Traub, "Another Algorithm". American Scientist . No. 2,Vol. 86, March-April1998. pp. 108–109.
6. J. M. McNamee "A Bibliography on Roots of Polynomials", J. Comput. Appl. Math. , 47 (1993) pp. 391-394.78 (1997). 110 (1999) pp. 305-306.