Letter to the EditorThe mystery of the shape parameter III

This is a continuation of our earlier study of the shape parameter c contained in the multiquadrics (−1)⌈β2⌉(c2+‖x‖2)β2, β>0, and the inverse multiquadrics (c2+‖x‖2)β, β<0. In the previous two papers we presented criteria for the optimal choice of c, based on the exponential-type error bound. In this paper a new set of criteria is developed, based on the improved exponential-type error bound. This results in much sharper error estimates when c is chosen appropriately, with the same size of fill distance as in the previous two papers. What is important is that the optimal value of c can be successfully predicted without any search when fill distance is of reasonable size, making it practically useful. The drawback is that the distribution of the data points is not purely scattered. However, for practical purposes, it does not seem to be a big problem.