Jim Fox

Read cnx.org/content/m43398/latest/. I give Matlab code for 2 alternative ways to do polynomial division. They do not seem to have the problem. By the way, one of them does polynomial division by solving the matrix equation Ax=b. b is the product polynomial and A is constructed from the denominator. The other way is to do division in the Fourier domain. Indeed, I have written several programs using Michael Rings multi-precision math package. However, this problem could take an awful lot of significant digits. I just repeated the experiment of creating a 99 degree random coefficient polynomial. Then I multiplied it by z-1000. Then I divided it by z-1000. The answer should be the original array of random values between 0 and 1. Instead the final value was 1E280. So I believe you would need 296 digits of accuracy to do this correctly. The errors in the deflation increase by a factor of 1000 at each step of the division. If the root was 10,000 or if it was a 200 degree polynomial you would need vastly more significant digits. Multi-precision math is not the solution. You asked, " do you have something to discuss about this topic? Or a new algorithm to avoid the errors?" Yes, the first article I mentioned, cnx.org/content/m43398/latest/, gives Matlab code for two alternative methods of doing polynomial division and they do not have this problem. This article in wikepedia discusses "vanity publishers" who charge authors a fee. Rice does not charge a fee.

Yes indeed. If all your coefficients are integers, you are safe. You will only have problems if the coefficients are 16 digit numbers that can experience round off error when they are manipulated on the computer. Indeed, peer reviewed journals are the gold standard of publishing. However, if a university puts something out on the web, I do not know what you can call it except publishing. Some of the things they published are extremely un-interesting to me. However, some are quite good. I think my 2 articles are new and interesting, especially the first which is the subject of my post.

I suspect that everyone does polynomial division the same way, the way we learned in high school. However, this can lead to large exponentially increasing errors. Problems will occur if the denominator has a large magnitude root and the quotient is long. For example, if 1000 is a root of a long polynomial and it is deflated, expect large errors. If the polynomial has high enough degree, expect infinity at some point in the deflated polynomial. For an explanation of why this occurs and what you can do about it see: cnx.org/content/m43398/latest/, "Exponential errors in polynomial evaluation, deflation, & division" P.S. Understanding this article is helped immensely if you know the program Matlab. While you are there, you might check out my other article: cnx.org/content/m15595/latest/, "Information in the Spectrum of the Polynomial Coefficients" Both articles are posted on Connexions which is a project of Rice University. Connexions allows anyone to publish, on the web, anything about any subject. There are no referees to shoot down your paper. Your paper does not need to be new information. If you feel that you are knowledgeable about some well researched subject, then write a summary paper. Connections will publish it for you on the web. Connexions makes it easy for anyone to become a published author. Do you have a paper in you?