From: trebla@io.org (Albert Y.C. Lai) Newsgroups: sci.math Subject: Re: General Cubic Solution Date: Sat, 22 Jun 1996 15:05:03 -0400 In article <585540551wnr@ynot1.demon.co.uk>, Anthony Hugh Back wrote: >The above working should be enough to discourage you from >analytical solutions of cubics. Why discourage that? There are times when it is useful. Instead we should discourage people from expecting answers (and life) to be easy. Encourage them to be aware of different approaches and different points of view to the same problem, so that they can choose an approach well-suited to the actual situation. On this line, many people think that Newton's method gives an easy answer to cubic equations. As I have suggested many times, these people should try to solve the equation x^3 - 1.265 x + 1 = 0 with Newton's method. -- Albert Y.C. Lai trebla@io.org http://www.io.org/~trebla/ ============================================================================== From: bruck@pacificnet.net (Ronald Bruck) Newsgroups: sci.math Subject: Re: General Cubic Solution Date: Sat, 22 Jun 1996 15:20:38 -0700 In article , bruck@pacificnet.net (Ronald Bruck) wrote: :In article , trebla@io.org (Albert Y.C. Lai) wrote: : ::On this line, many people think that Newton's method gives an easy ::answer to cubic equations. As I have suggested many times, these ::people should try to solve the equation :: :: x^3 - 1.265 x + 1 = 0 :: ::with Newton's method. : :??? I don't quite understand the point of this example? There are two :intervals, each a little shorter than one unit in length, where the Newton :iteration function x - f(x)/f'(x) isn't a contraction -- near the points :where f'(x) = 0, of course -- but from any point OTHER than those two :obvious singularities, Newton's Method converges to the real root. Quite :rapidly, usually. Ahh, I see; you mean the CYCLING. --Ron Bruck Now 100% ISDN from this account