Summary
This paper is concerned with the problem of convexity-preservng (orc-preserving) interpolation by using Exponential Splines in Tension (or EST's). For this purpose the notion of ac-preserving interpolant, which is usually employed in spline-in-tension interpolation, is refined and the existence ofc-preserving EST's is established for the so-calledc-admissible data sets. The problem of constructing ac-preserving and visually pleasing EST is then treated by combining a generalized Newton-Raphson method, due to Ben-Israel, with a step-length technique which serves the need for “visual pleasantness”. The numerical performance of the so formed iterative scheme is discussed for several examples.
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References
Atkins, D.A., Berger, S.A., Webster, W.C., Tapia, R.: Mathematical ship lofting. J. Ship Res.10, 203–222 (1966)
Ben-Israel, A.: On iterative methods for solving nonlinear least squares problems over convex sets. Israel J. Math.5, 211–224 (1967)
Bézier, P.: Numerical Control, Translated from the French Edition, Editions Masson et Cie. Paris (1970), London: Wiley (1972)
Buczkowski, L.: Mathematical construction, approximation and design of the ship body form. J. Ship Res.13, 185–206 (1969)
de Boor, C.: A practical guide to splines. Appl. Math. Sciences No 27. Berlin Heidelberg New York, Springer (1978)
Cline, A.K.: Scalar and planar valued curve fitting using splines under tension. Comm. ACM17, 218–223 (1974)
Encarnacao, J., Schlechtendahl, E.G.: Computer Aided Design. Berlin Heidelberg New York: Springer (1983)
Ford, W.T., Roulier, J.A.: On interpolation and approximation by polynomials with monotone derivatives. J. Approximation Theory10, 123–130 (1974)
Fritsch, F.N., Carlson, R.E.: Monotone piecewise cubic interpolation. SIAM J. Numer. Anal.17, 238–246 (1980)
Golub, G.H.: Numerical methods for solving linear least least squares problems. Numer. Math.7, 206–216 (1965)
Gregory, J.A., Delbourgo, R.: Piecewise rational quadratic interpolation to monotonic data. IMA J. Numer. Anal.2, 123–130 (1982)
Ioffe, A.D., Tihomirov, V.M.: Theory of extremal problems. Translated from the Russian Edition, Moskow: Nauka (1974) Amsterdam North-Holland (1979)
Lynch, R.W.: A method for chosing a tension factor for spline under tension interpolation, M.S. Thesis, University of Texas at Austin (1982)
McAllister, D.F., Roulier, J.A.: Interpolation by convex quadratic splines. Math. Comput.32, 1154–1162 (1978)
McAllister, D.F., Passow, E., Roulier, J.A.: Algorithms for computing shape preserving spline interpolations to data. Math. Comput.3, 717–725 (1977)
Passow, E.: Piecewise monotone spline interpolation. J. Approximation Theory12, 240–241 (1974)
Passow, E., Raymon, L.: The degree of piecewise monotone interpolation. Proc. Amer. Math. Soc.48, 409–412 (1975)
Passow, E.: An improved estimate of the degree of monotone interpolation. J. Approximation Theory17, 115–118 (1976)
Passow, E.: Monotone quadratic spline interpolation. J. Approximation Theory19, 143–147 (1977)
Pruess, S.: Properties of splines in tension. J. Approx. Theory17, 86–96 (1976)
Pruess, S.: Alternatives to the exponential spline in tension. Math. Comput.33, 1273–1281 (1979)
Rentrop, P.: An algorithm for the computation of the exponential spline. Numer. Math.35, 81–93 (1980)
Rubinstein, Z.: On polynomial δ-type functions and approximation by monotonic polynomials. J. Approximation Theory3, 1–6 (1970)
Schweikert, D.: An interpolation curve using a spline in tension. J. Math. Phys.45, 312–317 (1966)
Späth, H.: Exponential spline interpolation. Computing4, 225–233 (1969)
Young, S.W.: Piecewise monotone polynomial interpolation. Bull. Amer. Math.73, 642–643 (1967)
Sakai, M., López de Silanes, M.C.: A simple rational spline and its application to monotonic interpolation to monotonic data. Numer. Math.50, 171–182 (1986)
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Sapidis, N.S., Kaklis, P.D. & Loukakis, T.A. A method for computing the tension parameters in convexity-preserving spline-in-tension interpolation. Numer. Math. 54, 179–192 (1989). https://doi.org/10.1007/BF01396973
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DOI: https://doi.org/10.1007/BF01396973