New slope methods for sharper interval functions and a note on Fischer's acceleration method

This paper presents algorithms evaluating sharper bounds for interval functionsF(X) :IR n →IR. We revisit two methods that use partial derivatives of the function, and develop four other inclusion methods using the set of slopesS f (x, z) off atx eX with respect to somez eIR n . All methods can be implemented using tools that automatically evaluate gradient and slope vectors by using a forward strategy, so the complex management of reverse accumulation methods is avoided. The sharpest methods compute each component of gradients and slopes separately, by substituting each interval variable at a time. Backward methods bring no great advantage in the sharpest algorithms, since object-oriented forward implementations are easy and immediate.