Abstract
In many applications, observations are prone to imprecise measurements. When constructing a model based on such data, an approximation rather than an interpolation approach is needed. Very often a least squares approximation is used. Here we follow a different approach. A natural way for dealing with uncertainty in the data is by means of an uncertainty interval. We assume that the uncertainty in the independent variables is negligible and that for each observation an uncertainty interval can be given which contains the (unknown) exact value. To approximate such data we look for functions which intersect all uncertainty intervals. In the past this problem has been studied for polynomials, or more generally for functions which are linear in the unknown coefficients. Here we study the problem for a particular class of functions which are nonlinear in the unknown coefficients, namely rational functions. We show how to reduce the problem to a quadratic programming problem with a strictly convex objective function, yielding a unique rational function which intersects all uncertainty intervals and satisfies some additional properties. Compared to rational least squares approximation which reduces to a nonlinear optimization problem where the objective function may have many local minima, this makes the new approach attractive.
Similar content being viewed by others
References
Allouche, H., Cuyt, A.: Unattainable points in multivariate rational interpolation. J. Comput. Appl. Math. 72, 159–173 (1993)
Gass, S.I.: Comments on the possibility of cycling with the simplex method. Oper. Res. 27(4), 848–852 (1979)
Goldman, A.J., Tucker, A.W.: Polyhedral convex cones. In: Kuhn, H.W., Tucker, A.W. (eds.) Linear Inequalities and Related Systems. Annals of Mathematics Studies, vol. 38. Princeton University Press, Princeton NJ (1956)
Grünbaum, B.: Convex polytopes. Pure and Applied Mathematics, vol. XVI. Wiley, New York (1967)
Markov, S.: Polynomial interpolation of vertical segments in the plane. In: Kaucher, E., Markov, S., Mayer, G. (eds.) Computer Arithmetic, Scientific Computation and Mathematical Modelling, IMACS, pp. 251–262. J.C. Baltzer AG, Scientific, Basel, Switzerland (1991)
Markov, S., Popova, E., Schneider, U., Schulze, J.: On linear interpolation under interval data. Math. Comput. Simul. 42, 35–45 (1996)
Milanese, M., Belforte, G.: Estimations theory and uncertainty intervals evaluation in the presence of unknown but bounded errors:linear families of models and estimators. IEEE Trans. Automat. Contr. 27(2), 408–414 (1982)
Motulsky, H., Christopoulos, A.: Fitting Models to Biological Data Using Linear and Nonlinear Regression: A Practical Guide to Curve Fitting. Oxford University Press, Oxford, UK (2004)
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)
Press, W., Teukolsky, S., Vetterling, W., Flannery, B.: Numerical Recipes in C, 2nd edn. Cambridge University Press, Cambridge, UK (1992)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Walter Gautschi for his 50 years of valuable work in rational approximation theory.
Rights and permissions
About this article
Cite this article
Salazar Celis, O., Cuyt, A. & Verdonk, B. Rational approximation of vertical segments. Numer Algor 45, 375–388 (2007). https://doi.org/10.1007/s11075-007-9077-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-007-9077-3