A Computer Oriented Approach to Get Sharp Reliable Error Bounds

We present interval methods to get reliable a priori error bounds for the machine evaluation of algorithms implementing some mathematical expression. The term expression not only means simple arithmetical expressions but also more complex program parts including loops or recursive structures (e.g. a complete elementary function routine).We sketch a method that can be used to get an upper bound for the approximation error of a polynomial or a rational approximation. We also discuss a method to compute worst case a priori error estimates for arbitrary IEEE double floating-point computations. Our theoretical results lead to reliable and easy to use public domain software tools. The application of these tools to an accurate table method shows that error bounds of high quality can be derived.