This article is rated Start-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
Why are we only using examples of period 1? Septentrionalis PMAnderson 03:54, 13 December 2006 (UTC)
Your general theorem cannot be true as stated; x2 - 2 has a perfectly good continued fraction for the positive root (and for that matter, the negative root) despite having b = 0; I think I see what is meant, but it's not what is said. Septentrionalis PMAnderson 04:08, 13 December 2006 (UTC)
The section about complex coefficients has been added. There's a red link to convergence problem, which should get fixed in the next couple of days. DavidCBryant 20:26, 14 December 2006 (UTC)
Fine, good article, but... I hope, someday, might be in a non distant future, people will finally realize that traditional continued fractions are just the same second-order Lineal Homogeneous Recurrence Relations than those arising from the well-known Daniel Bernoulli's root-solving method, and consequently such tradional concept constitutes just a particular case of the Generalized Continued Fractions as can be seen at: http://mipagina.cantv.net/arithmetic/gencontfrac.htm Moreover, it might be people someday will also realize that all the Means, Continued Fractions, Daniel Bernoulli's method, Newton's method, Householder's method, and the whole Science of Quantity is ruled by an extremely simple arithmetical operation called: The Rational Mean. Domingo Gomez Morin Civil Engineer. Structural Engineer.
A continued fraction algorithm is also nice for approximating real roots of polynomials of higher degree. The terms, however, are a chaotic succession of positive integers, and it is an open question (at least in 1973, probably still is) whether these are a bounded set. A few iterations throws all the real conjugates of a real root into the open interval (-1,1); that greatly simplifies things. Scott Tillinghast, Houston TX 21:27, 8 April 2007 (UTC)