From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Newton-Raphson question. Date: 22 Mar 1999 13:37:37 -0500 Newsgroups: sci.math Keywords: Examples of failure of Newton's method In article , Fredrik Conradson wrote: >I'm implementing a solver for non-linear systems using the Newton-Raphson >method and I have a question: Does this method always converge to a root >regardless of the inital-values? > >thanks for any help! >Fredrik >conne@mindless.com It's a potential disaster area, even in single variable. It helps if you sketch the graphs and the effect of initial gusses. Example 1. f(x) = x^3 - 5*x Starting with x_0 = 1, you will get into a 2-cycle {1, -1 , -1, ...} This cycle is unstable, and starting nearby may lead to hopping back and forth before settling near one of the three roots. And you may not know (because of round-off) where you land. Example 2. f(x) = x^3 - 2*x + 2 Starting in the interval [-0.1, 0.1] , you will get into a stable asymptotic 2-cycle, and very quickly so (used as an example by S. Smale). Of course, there is a negative root to which Newton's method will converge if started, say, to the left from the root. Example 3 . f(z) = z^3 - 1 (z complex, equation f(z)=0 can be re-written as a real system). The regions of attraction have boundaries that puzzled Cayley because he did not have the graphical or topological machinery. These boundaries, placed symmetrically around the straight lines theta=2*k*pi/3, k in {-1, 0, 1} were later identified as fractals. If you start at some points of that boundary set, the iterations will wander around without converging anywhere. Hope it helps, ZVK(Slavek).