Symbolic vs Numeric: relating computations in discrete and continuous structures

Symbolic algorithms provide precise solutions (presented in a suitable form) and are widely used in computer algebra systems. Numeric algorithms provide approximate solutions and are widely used in numerical packages.
This talk is a short informal survey of some results on computable fields of real numbers and their applications to some computation problems in linear algebra and analysis (symmetric hyperbolic systems of PDEs with constant coefficients). Along with general computability (without any bound on computational resources) we discuss primitive recursive and polynomial-time computability. We trace how more efficient algorithms on fields of reals (which are symbolic algorithms on countable discrete structures) yield more efficient numeric algorithms on continuous structures (like the reals and other relevant metric spaces). From available competing models of computation on continuous structures we choose the model going back to Turing that arguably better corresponds to numeric algorithms (namely, to algorithms with arbitrary guaranteed precision). Our results suggest that adequate mathematical foundations for symbolic and numeric computations are provided by, respectively, computable model theory and computable analysis. All results mentioned in the talk are joint with either Svetlana Selivanova or Pavel Alaev.