  We establish primitive recursive versions of some known facts about
computable ordered fields of reals and computable reals, and then apply them to
proving primitive recursiveness of some natural problems in linear algebra and
analysis. In particular, we find a partial primitive recursive analogue of
Ershov-Madison's theorem about real closures of computable ordered fields,
relate the corresponding fields to the primitive recursive reals, give
sufficient conditions for primitive recursive root-finding, computing normal
forms of matrices, and computing solution operators of some linear systems of
PDE.
