Keith Briggs: Reliable real arithmetic and one-dimensional dynamical systems
Abstract:
Significant advances continue to be made in the field of software for accurately emulating real-number arithmetic. These have potential applications for experimental studies of dynamical systems, and for computer-assisted theorem proving. The main issue here is how to get reliability without an intolerable performance penalty.
My talk has two parts: first a quick survey of the state-of-the-art, with a description of my own "lazy stochastic" approach to computing with reals (where the aim is strict preservation of the order relation while minimizing the amount of computation); and second an application to the study of the statistical distribution of blocks of partial quotients in continued fractions. The latter concerns a simple dynamical system on the real line, for which there are still simply-stated questions to which the answer is unknown (such as whether cubic irrationals are normal). I suggest that exact computations (i.e., where every partial quotient is guaranteed correct) may provide some illumination on such problems.