Repeated Knots in Least Squares Multiquadric Functions

A previous paper by the authors [FH93] noted that there was a strong tendency to obtain near-repeated knots in their algorithm for least squares approximation of scattered data by multiquadric functions. In this paper we observe that this leads naturally to the inclusion of derivatives of the multiquadric basis function in the approximation, and give an algorithm for accomplishing this. A comparison of the results obtained with this algorithm and the previous one is made. While the multiple knot algorithm usually has the advantage in terms of accuracy and computational stability, there are datasets for which this is reversed. * Department of Mathematics, Naval Postgraduate School, Monterey, CA 93943 t FB Informatik, Universitit Kaiserslautern, 67653 Kaiserslautern, Germany t Department of Computer Science, Arizona State University, Tempe, AZ 85287 This work was performed in part while the first and last authors were visiting Universitit Kaiserslautern and were supported by a DFG-grant Ka 477/13-1.