From pdxgate!qiclab!psgrain!ogicse!cs.uoregon.edu!news.uoregon.edu!cie.uoregon.edu!scavo Wed Mar  3 09:23:01 PST 1993

In article <1993Mar1.170648.19294@math.ucla.edu> szhong@joshua.math.ucla.edu (Sifen Zhong) writes:
>
>Is Newton's method the best method to compute a cubic root?

Depends on what you mean by "best".  Newton's method is a
quadratically converging algorithm, but there are higher-
order techniques like, say, Halley's method.  The iteration

         2x^3 + 4r
   x' = ----------- x
         4x^3 + 2r

is cubically converging to the cube root of  r , for example
(it's generalization to nth roots is due to Lambert in 1770).
And there's no end to higher-order iterative methods such as
this (cf. I. Kiss.  A generalization of Newton's approximation
procedure [German].  _Z. angew. Math. Mech._ 34, 68--69, 1954.)
but of course the price you pay is more computation per iteration.

-- 
Tom Scavo
scavo@cie.uoregon.edu