From: Peter Percival Subject: Re: Square Root Error Date: Wed, 24 May 2000 22:15:32 +0100 Newsgroups: sci.math Summary: [missing] "Zdislav V. Kovarik" wrote: > > In article <8ges9v$32e$1@nnrp1.deja.com>, wrote: > :Using the formula x[new] = (1/2)(x + a/x), one can approximate the > :square root of a. But how quickly does it converge, and what is the > :maximum possible error? > > This is a classic - and no chaos is involved. > > Change the variable by > > (x - sqrt(a)) / (x + sqrt(a)) = y > > (x[new] - sqrt(a)) / (x[new] + sqrt(a)) = y[new] > > simplify and see what happens. (I mean, establish an equation between > y[new] and y.) Do the algebra. > > As for "maximum error", the the question needs restatement. Guessing what > it might be, I'd answer that the maximum error between the first two > iterations does not exist (individual errors can be arbitrarily large). > Any extra conditions on the starting guess? > > Fractal boundary, in complex case, occurs for cube root calculations > using Newton's method. I am told that it was Cayley who found out that he > did not have the machinery to describe the regions of attraction, and it A one-page paper "The Newton-Fourier Imaginary Problem" by Sir Arthur Cayley in the American Journal of Mathematics, 2, 1879 describes the "Newton formula" x_next = x - f(x)/f'(x) [his notation is different] and proposes considering equations with complex coefficients thus generating a sequence of points P, P_1, ... in the plane. His paper ends: "... The problem is to determine the regions of the plane, such that P being taken at pleasure anywhere within one region we arrive ultimately at the point A; anywhere within another region at point B; and so for the several points representing the roots of the equation. The solution is easy and elegant in the case of a quadratic equation, but the next succeeding case of the cubic equation appears to present considerable difficulty." Quoted without permission. > was Julia anf Fatou who started the study of iterations along these lines. > > Cheers, ZVK(Slavek)