From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: Root finding
Date: 2 Jun 1999 10:13:50 GMT
Newsgroups: sci.math.num-analysis
Keywords: Hirano's method

In article <19990601231049.19947.00007545@ng-ft1.aol.com>,
 jemfinch02@aol.com (Jemfinch02) writes:
|> Again, I'm still working on numerical methods on my calculator  :-).  This
|> time, however, I'm trying to write a program that, given one guess, will find
|> the nearest root.  Newton's method is obvious, but doesn't work if there is a
|> critical point between the guess and the root, or if the root is a relative
|> max/min.  I need something that doesn't have these restrictions, but still
|> converges on a root quickly and still only takes one guess.

if you speak about polynomials, then either Laguerres or Hiranos method
(the latter in SIAM J. Num. Anal. 19, (1982), 793--799) are the solution.
Hiranos method is a damped Newtons method which also takes into account 
complex and multpile zeroes and needs a complete Horners scheme in every step.
If you speak about a "general" f, then the situation is much more involved. you
must be able to estimate f'' (say) in absolute value on an given interval in order
to provide save steps towards a zero. such methods are discussed in Brent:
minimization of functions without using derivatives (you are searching 
for the _global_ minimizer of (f)^2 ).
hope this helps
peter