Reducing root precision so Union can work

• To: MATHGROUP at yoda.physics.unc.edu
• Subject: Reducing root precision so Union can work
• From: "Mathematical Institute, (0865) 2-73525" <AKYILDIZ at vax.ox.ac.uk>
• Date: Tue, 6 Oct 92 15:27 BST

```>Date: Thu, 1 Oct 92 10:57:21 -0400
>To: mathgroup at yoda.physics.unc.edu
>From: wiscombe at climate.gsfc.nasa.gov
>Subject: Reducing root precision so Union can work

>Dear MathGroupers,

>   I was trying to simulate a function that I think should be built into

> Mma, but is not:  one that makes a best effort to find ALL the roots of

> nonlinear equations on a given real interval,......
> To do this, I used FindRoot in a loop over starting values (fairly densely
> nested on the interval of interest).  Naturally this produces multiple
> hits on the same root, but I presumed Union could take care of that.
> Unfortunately, it does not, and it led to some serious questions about
> just how to reduce the precision of approximate solutions so that Union
> can do its job.  N didn't work .....

One way to get around all the problems mentioned above is to use
Interval Arithmetic.  In a recent paper, submitted to Mma Journal, we used
the builtin Mma function RealInterval to implement an interval version of
the bisection method to find ALL the (real) roots of an equation within a
given range, (complex roots under preparation). Copies of the paper can be
obtained from akyildiz at vax.oxford.ac.uk

Y. Akyildiz  &  M.C. Bortholomew-Biggs

```

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