Schmidt, Jochen W.; Heß, Walter Positivity of cubic polynomials on intervals and positive spline interpolation. (English) Zbl 0642.41007 BIT 28, No. 2, 340-352 (1988). Summary: A criterion for the positivity of a cubic polynomial on a given interval is derived. By means of this result a necessary and sufficient condition is given under which cubic C 1-spline interpolants are nonnegative. Further, since such interpolants are not uniquely determined, for selecting one of them the geometric curvature is minimized. The arising optimization problem is solved numerically via dualization. Cited in 65 Documents MSC: 41A15 Spline approximation 65D07 Numerical computation using splines 41A10 Approximation by polynomials Keywords:cubic polynomial; cubic C 1-spline interpolants; optimization problem Software:pchip PDFBibTeX XMLCite \textit{J. W. Schmidt} and \textit{W. Heß}, BIT 28, No. 2, 340--352 (1988; Zbl 0642.41007) Full Text: DOI References: [1] W. Burmeister, W. Heß and J. W. Schmidt,Convex spline interpolants with minimal curvature, Computing 35 (1985), 219–229. · Zbl 0564.65005 · doi:10.1007/BF02260507 [2] S. Dietze and J. W. Schmidt,Determination of shape preserving spline interpolants with minimal curvature via dual programs, TU Dresden Informationen 07-06-85 (1985) and J. Approx. Theory 51 (1987). · Zbl 0662.41008 [3] F. N. Fritsch and R. E. Carlson,Monotone piecewise cubic interpolation, SIAM J. Numer. Anal. 17 (1980), 238–246. · Zbl 0423.65011 · doi:10.1137/0717021 [4] E. Neuman,Uniform approximation by some Hermite interpolating splines, J. Comput. Appl. Math. 4 (1978), 7–9. · Zbl 0388.41007 · doi:10.1016/0771-050X(78)90013-X [5] G. Opfer and H. J. Oberle,The derivation of cubic splines with obstacles by methods of optimization and optimal control, Numer. Math. (to appear). · Zbl 0628.41012 [6] J. W. Schmidt,On shape preserving spline interpolation: existence theorems and determination of optimal splines, Banach Center Publ., Semester XXVII (to appear) and TU Dresden Informationen 07-01-87 (1987). [7] J. W. Schmidt and W. Heß,Positive interpolation with rational quadratic splines, Computing 38 (1987), 261–267. · Zbl 0676.41017 · doi:10.1007/BF02240100 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.