Primitive recursive ordered fields and some applications

We establish primitive recursive (PR) versions of some known facts about computable ordered fields of reals and computable reals, and apply them to prove primitive recursiveness of several important problems in linear algebra and analysis. One of the central results of this paper is a partial PR analogue of Ershov-Madison's theorem about real closures of computable ordered fields. It allows us, in particular, to obtain PR root-finding algorithms in the PR real and algebraic closures of PR fields with a certain property (PR splitting). We also relate the corresponding fields to the PR reals, as well as introduce and study the notion of a PR metric space. It enables us to derive sufficient conditions for PR computing of normal forms of matrices and solution operators of symmetric hyperbolic systems of PDEs. The methods represent a mix of symbolic and approximate algorithms.