  We introduce a new type of generalized Turing machines (GTMs), which are
intended as a tool for the mathematician who studies computability in Analysis.
In a single tape cell a GTM can store a symbol, a real number, a continuous
real function or a probability measure, for example. The model is based on TTE,
the representation approach for computable analysis. As a main result we prove
that the functions that are computable via given representations are closed
under GTM programming. This generalizes the well known fact that these
functions are closed under composition. The theorem allows to speak about
objects themselves instead of names in algorithms and proofs. By using GTMs for
specifying algorithms, many proofs become more rigorous and also simpler and
more transparent since the GTM model is very simple and allows to apply
well-known techniques from Turing machine theory. We also show how finite or
infinite sequences as names can be replaced by sets (generalized
representations) on which computability is already defined via representations.
This allows further simplification of proofs. All of this is done for
multi-functions, which are essential in Computable Analysis, and
multi-representations, which often allow more elegant formulations. As a
byproduct we show that the computable functions on finite and infinite
sequences of symbols are closed under programming with GTMs. We conclude with
examples of application.
