About 20 years ago I went through much of Henrici's work on "circular arithmetic", and while it was quite elegant, I didn't find in my experiments that it gained in either speed or stability. I may have missed something, because I wasn't able to fully grok all of his analysis. BTW, "circular arithmetic" is a version of interval arithmetic in which you use a center (in the complex plane) & radius instead of an upper & lower bound. The cool thing is that 1/z converts circles to circles; unfortunately, it also turns them inside out! Since circular arithmetic is a version of interval arithmetic, it shares many of the same problems: i.e., x^2 is (or should be) a different function from x*x, etc. At 10:29 AM 9/16/2010, Fred lunnon wrote:

On 9/16/10, Joerg Arndt <arndt@jjj.de> wrote:

...

... This may not be what you have in mind but there are methods to find _all_ roots simultaneously, see e.g. http://en.wikipedia.org/wiki/Durand-Kerner_method and the references at the bottom of the page

A very elegant method which computes all roots simultaneously, yielding circles in which they are guaranteed to lie, is buried (I don't remember exactly where --- anybody else know?) in the strangely neglected

Henrici P., Applied and Computational Complex Analysis (Wiley). [Three volumes: 1974, 1977, 1986.]

It's interesting that these methods actually gain in speed and stability by simultaneity, as opposed to attempting to exclude the other roots.

WFL