Indeed, naive evaluation of even the quadratic formula can lead to problems.
Remember -b (+/-) sqrt(b^2-4ac) ?
One of these could be the difference of two very nearly equal quantities,
resulting
in roundoff error dominating the result. Fortunately, the other choice of
sign
would then be ok, and if you know one root, the other is computed by noting
that
the product of the roots is c/a.
RJF
> -----Original Message-----
> From: maxima-bounces at math.utexas.edu
> [mailto:maxima-bounces at math.utexas.edu] On Behalf Of Albert Reiner
> Sent: Thursday, August 24, 2006 7:59 AM
> To: maxima at math.utexas.edu
> Subject: Re: [Maxima] cubics and quartics
>
> [sen1 at math.msu.edu, Wed, 23 Aug 2006 12:24:20 -0400 (EDT)]:
> > For me, (aside from teaching tools), the main reason to
> find symbolic
> > solutions of algebraic equations would be to be able to
> compute those
> > solutions to arbitrary precision at some point.
>
> At the risk of pointing out the obvious: This opens yet
> another, rather unpleasant can of worms, viz., numerical
> stability of a naive evaluation of the expression. The
> symbolic expressions that come out of some CAS computation
> may not be very well suited for numerical processing,
> especially if you also want to know the precision of the
> result may be.
>
> Of course, for a given problem you may already know that
> things are well-behaved, or that they ought to be.
>
> Not even EUR 0.02,
>
> Albert.
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