# [PURRS-devel] Extended arithmetic with CoStLy

Alessandro Zaccagnini zaccagni at prmat.math.unipr.it
Tue Nov 19 12:04:08 CET 2002

```Dear Markus,

thank you very much for your reply. We apologize for our delay, but we
have just started looking into the problem of complex interval
arithmetic. Indeed, our main problem is to approximate as closely as
possible to the (possibly complex) roots of polynomials. Our polynomials
have usually integer coefficients, and we compute an exact formula for
the roots whenever possible, but we do not insist on the fact that this
is the best (more accurate, faster...) way of dealing with this kind of approximations.

More precisely, we always want to find exact formulas if it is at all
possible, but we also want to compute approximations. It may be the case
that, for our problem, it is better to approximate to the roots of the
polynomial ignoring the exact (but sometimes cumbersome) formula given
by our system. Indeed, some expressions that we obtain are several MB in size!

We probably have no problems with iterated logarithms, or with nested
square roots (beside the ones arising from the solution of polynomial
equations as explained above), since the majority of expressions that we
want to approximate are (as of now; it may change in the future) of the
general shape

a * x^n * n^k

where a is a numeric coefficient, x is a root of a polynomial equation,
k is a positive integer and n is an integer variable. Of course, we also
need to approximate sums of expressions like that. The reason why we
mentioned sines and cosines is that we may use the trigonometric
representation of complex numbers, if it happens to be useful.

There is at least one more thing that we would like to have: it would be
more convenient for us if CoStLy answered with the whole complex plane
instead of throwing exceptions (in the cases where the interval contains
0 or some real negative number, for instance). Would it be possible to
have something like this (even as a temporary solution)?

Thank you very much again.

Tatiana Zolo & Alessandro Zaccagnini

```