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