Topics in the Theory and Practice of Computable Analysis

This dissertation deals with a breadth of computational aspects of analysis, from obtaining computable information from classical proofs in analysis to im-plementing real number computations efficiently in practice. The main contri-butions of the paper can be summarized as follows: • A generalization of a set of conditions given by Matiyasevich under which a recursively defined sequence of real numbers can be shown to converge computably. The generalization replaces monotonicity with a weaker con-dition and allows errors in the computation of the sequence. • The first explicit computable bound on the asymptotic regularity of Kras-noselski-Mann iterations with error terms of asymptotically quasi nonex-pansive self-mappings of closed convex subsets of uniformly convex Ba-nach spaces. The bound permits the computation of approximate fixed points of the function and relaxes the conditions under which the iteration can be shown to be asymptotically regular