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In article <1992Aug25.161234.13275@magnus.acs.ohio-state.edu>
kiyer@magnus.acs.ohio-state.edu (Karthik A Iyer) writes:
>	            I am looking for a routine(s) or a place where I can find 
>them (FTP sites, etc) to solve quartic and cubic eqations. I realize that 
>analytic solutions exist, but my problem is that the coefficients of the 
>powers vary by as much as 10-15 orders of magnitude. So, a small error is
>solving the for the cubic's root leads to large errors when that root is
>used to solve for the four roots of the quartic. Any help or leads  would be
>appreciated. I would prefer to get my hands on a FORTRAN double precision 
>routine. Thanks in advance,

Unfortunately, the "classical" closed-form analytical solutions
tend to be quite ill-conditioned, *especially* when the coefficients
are widely varying in magnitude.

I suspect you would get much more accurate results if you simply
used a good numerical zero-finding code to numerically locate the
(a) zero of your quartic/cubic.  (Of course, you still have to be
able to *evaluate* the quartic/cubic, which is itself rather tricky
when the coefficients are so disparate in magnitude.  But that's
still a much easier problem than trying to find the roots directly.)

An excellent zero-finding code is ZEROIN, available from netlib
via

	echo 'send zeroin from fmm' | Mail netlib@ornl.gov

This will get you a Fortran subroutine with all the documentation
you need to use it.  For more details, see either the (excellent,
superb, ++good, highly recommended) book

        G. E. Forsythe, M. A. Malcolm, and C. B. Moler
        "Computer Methods for Mathematical Computations"
        Prentice-Hall, Englewood Cliffs, 1977

or section 9.3 of Numerical Recipes.  Be warned, though, that the
code for ZEROIN given in Numerical Recipes (where it's called ZBRENT)
has been "improved", and is of somewhat lower quality than the
original one you'll get from netlib.

- Jonathan Thornburg
  <jonathan@einstein.ph.utexas.edu> or <jonathan@hermes.chpc.utexas.edu>
  University of Texas at Austin / Physics Dept / Center for Relativity
  and (for a few more months) U of British Columbia / {Astronomy,Physics}