You made some good points.

At the moment, I would like to focus on the quadratic equation.

The algorithms that deal with higher roots, and for which source code is available, seem to compute roots, deflate the polynomial until it is a quadratic equation, and then solve that quadratic equation. So now I am wondering if there is one best solver for quadratic equations. In other words, among the various algorithms for solving quadratic equations, is there one that accepts the widest range of floating point data for the coefficients, properly deals with multiple zeros, etc. If, say, the RPOLY algorithm fails to find the proper roots 2% of the time, but the IMSL routine fails 4% of the time, for any and all floating point data that can be input as coefficients, that is something I'd be interested in knowing; I would like to go with the odds when selecting an algorithm for use.

(The program is for use on a standard PC)

I am back with an additional question.

Since my previous post I have done some further research on the subject of numerical computations, specifically for the problem of finding the roots of polynomials. It seems that no techniques exist that take advantage of the fact closed-form solutions exist for the cubic and quartic equations when computing their roots. In fact, I am also reconsidering the quality of a program I had written for computing the roots of a quadratic equation.

A question: is there a piece of software that you consider to incorporate the best algorithm for computing the roots of a quadratic equation (Maple; MATLAB; the TOMS routine RPOLY; the IMSL routine, etc.)? In other words, an algorithm that achieves the highest accuracy possible on a particular machine, and successfully computes roots for the widest range of inputs?

Looking forward to hearing your recommendations.
 

Thanks for the information, Roman. Wow, that stuff goes over my head! And I thought I was pretty well-informed when it comes to numerical routines! I don't recall having come across mention of those methods before. Even my two favorite sources for background material, MathWorld and Wikipedia, are sparse on those topics (perhaps it is time somebody wrote an article for Wikipedia on those topics). I definitely have some research ahead of me, to learn more about these topics.