On 14-Jul-03 Ted Harding wrote: > Spot on, Baz! > > An intriguing example to set people is the following: > > Write a program to generate a sequence of numbers x(0) ( = anything in > the range 0 <= x <= 1, chosen by user), x(1), x(2), ... , x(n), ... > where > x(n+1) = f(x(n)), ... > and f() is defined by > f(x) = 2*x if 0 <= x <= 1/2 > f(x) = 2*(1-x) if 1/2 <= x <= 1 > [...] > Mathematically, all other numbers are not equilibria and the sequence > is "chaotic", yet this cannot be observed either. In fact the sequence > generated will arrive at 0 (and stay there) in at most (N+1) steps > where N is the number of bits in the abscissa of the floating-point > representation.
Sorry! This is inaccurate (I was thinking in terms of a zero exponent 2^0 in the FP representation of x(0) -- and by the way of course it's "mantissa" and not "abscissa"!). Anyway, the above is true, even with a very small x(0) = 2^(-E)*0.1... , once the multplications by 2 have taken it up to the crossover from < 1/2 to > 1/2. Ted. -------------------------------------------------------------------- E-Mail: (Ted Harding) <[EMAIL PROTECTED]> Fax-to-email: +44 (0)870 167 1972 Date: 14-Jul-03 Time: 18:22:01 ------------------------------ XFMail ------------------------------ ______________________________________________ [EMAIL PROTECTED] mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help
