I'm interested in exact real arithmetic (and by extension constructive analysis). A nice representation of real numbers is via Cauchy Sequences. The basic idea being that you have a function which, given a rational error bound, returns a rational number within the the error bound (I realize this isn't quite a sequence but it is isomorphic to one).
Currently I have figured out how to compute addition, subtraction, multiplication, division, taylor series, and non infinite limits from the left and right (and thus derivatives by the limit definition). I have not figured out how to compute definite integrals however (which is sort of a gapping whole).
I've looked up a bunch of numerical integration techniques but they all seem have error bounds that are expressed in terms of the second derivative of the function being integrated. Moreover there is a term
Is there a numerical integration technique which gives a rational error bound which can be made arbitrarily small by selecting some other parameter (like step size)? In particular, is there a there a technique that only relays off of the operations that I have already mentioned (+, -, *, /, taylor series, finite limits, derivatives)? It doesn't have to be a good technique, it just has to have rational upper bounds on error.
So I found "foundations of constructive analysis" by Bishop at my local university after it was suggested that I find a book on the matter.
Bishop defines a partition $P$ of an interval $[a, b]$ to be a finite sequence of real numbers $$a < a_1 < ... < a_{n-1} < b$$.
Integration of a function $f$ over a partition $P={a_0,...,a_n}$ as defined as follows: $$S(f, P) - \sum^{n-1}_{i=0}f(x_i)(a_{i+1} - a_i)$$ where $x_i \in [a_{i+1},a]$
He the goes on to state the following
$$|S(f, P) - \int_a^b f(x)dx| < \epsilon(b-a)$$ as long as the maximum distance between points in your partition is less than $\omega({\epsilon})$ where $\omega$ is the modulus of continuity. You need to keep track of the modulus of continuity for many proofs/algorithms so its existence is assumed.
So for a given error bound , compute $\epsilon$ by dividing by $b-a$. Compute $\omega(\epsilon)$ to get the maximum distance between points in your partition and then make your partition the uniform partition with that delta (finessing the last value in the partition).
Interestingly this is all defined in terms of real numbers and is thus independent of the underlying representation of real numbers. Additionally it lets you specify a real valued error (which in turn lets you easily specify a rational valued error). So this is actually a fantastically general answer!