Numerical Integration with Exact Real Arithmetic

We show that the classical techniques in numerical integration (namely the Darboux sums method, the compound trapezoidal and Simpson's rules and the Gauss-Legendre formulae) can be implemented in an exact real arithmetic framework in which the numerical value of an integral of an elementary function is obtained up to any desired accuracy without any round-off errors. Any exact framework which provides a library of algorithms for computing elementary functions with an arbitrary accuracy is suitable for such an implementation; we have used an exact real arithmetic framework based on linear fractional transformations and have thereby implemented these numerical integration techniques. We also show that Euler's and Runge-Kutta methods for solving the initial value problem of an ordinary differential equation can be implemented using an exact framework which will guarantee the convergence of the approximation to the actual solution of the differential equation as the step size in the partition of the interval in question tends to zero.