On Mon, Oct 11, 2004 at 09:53:16PM -0400, Scott Turner wrote:
Evenutally I realized that calculating with lazy lists is not as smooth as you might expect. For example, the square root of 2 has a simple representation as a lazy continued fraction, but if you multiply the square root of 2 by itself, your result lazy list will never get anywhere. The calculation will keep trying to determine whether or not the result is less than 2, this being necessary to find the first number in the representation. But every finite prefix of the square root of 2 leaves uncertainty both below and above, so the determination will never be made.
Right, one way to think about this problem is that the representations by continued fractions are unique, so there's no way to compute the prefix of a representation for something right on the boundary. Representing numbers by lazy strings of, say, decimal digits has the same problem. There are known solutions, but they lack the elegance of continued fraction representations. You fundamentally have to have non-unique representations, and that causes some other problems. One popular version is to use base 2 with digits -1, 0, and +1. Simon Peyton-Jones already posted the references. These methods appear to lose out in practice to using a large fixed precision and interval arithmetic, increasing the precision and recomputing as necessary. Peace, Dylan