I know the Blum-Shub-Smale model. It is claimed to provide a theoretical framework for algorithms in real and complex algebra and analysis.
A very general question: Most algorithms compromise of finite input - finite computation steps - finite output
However, in mathematics most schemes which one could call algorithmic actually compromise infinitely many iterations. Just to name a few:
- Iteration to construct fix-points of contractive maps
- Iterative solvers in numerical analysis
- Newtons method
I know the last one is treated in the classical book "Complexity and Real Computation" by Blum, Cucker, Shub & Smale, but there does not seem to exist a generalization which puts "iterative" algorithms onto solid foundations.
Edit: Yes, the question:
How can you deal with these iterative algorithms (which as such are not necessarily supposed to finish)?
Addendum: I always wondered whether there might be something like a BSS-like model, which is tailored for numerical analysis - in that it does formalize errors, inexactness, approximate solutions, and does not allow for "unreasonable" computing power. And is easy to access.
The book mentioned above is concerned about this gap, too: "There is not even a formal definition of algorithm in the subject [of numerical analysis]" (p. 23). However, it does not seem their approach suits this need. On the other hand, the monograph Andrej mentions does include a book which heads towards this direction.