A Newton-type method for constrained least-squares data-fitting with easy-to-control rational curves

• Published in: Journal of computational and applied mathematics Vol. 223; no. 2; pp. 672 - 692
• Main Authors: Casciola, G., Romani, L.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 15.01.2009; Elsevier
• Subjects: Approximations and expansions; Automatic selection of weights; Constrained approximation; Exact sciences and technology; Global analysis, analysis on manifolds; Interior-point Newton-type method; Least-squares data-fitting; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Rational Hermite interpolation; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; 90C51; Interior-point Newton-type method; 90C30; Least-squares data-fitting; 65D10; 41A20; Automatic selection of weights; Rational Hermite interpolation; 65D17; Constrained approximation; 41A29; Numerical approximation; Least-squares data-fitting; Constrained approximation; Rational Hermite interpolation; Automatic selection of weights; Interior-point Newton-type method; 65D10; 65D17; 41A20; 41A29; 90C51; 90C30; Numerical analysis; Rational interpolation; Fitting; Applied mathematics; Least squares method; Hermite interpolation; Interior point method; Fast algorithm; Newton method
• ISSN: 0377-0427; 1879-1778; 1879-1778
• DOI: 10.1016/j.cam.2008.02.005

While the mathematics of constrained least-squares data-fitting is neat and clear, implementing a rapid and fully automatic fitter that is able to generate a fair curve approximating the shape described by an ordered sequence of distinct data subject to certain interpolation requirements, is far more difficult. The novel idea presented in this paper allows us to solve this problem with efficient performance by exploiting a class of very flexible and easy-to-control piecewise rational Hermite interpolants that make it possible to identify the desired solution with only a few computations. The key step of the fitting procedure is represented by a fast Newton-type algorithm which enables us to automatically compute the weights required by each rational piece to model the shape that best fits the given data. Numerical examples illustrating the effectiveness and efficiency of the new method are presented.