Abstract
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 cubicC 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.
Similar content being viewed by others
References
W. Burmeister, W. Heß and J. W. Schmidt,Convex spline interpolants with minimal curvature, Computing 35 (1985), 219–229.
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).
F. N. Fritsch and R. E. Carlson,Monotone piecewise cubic interpolation, SIAM J. Numer. Anal. 17 (1980), 238–246.
E. Neuman,Uniform approximation by some Hermite interpolating splines, J. Comput. Appl. Math. 4 (1978), 7–9.
G. Opfer and H. J. Oberle,The derivation of cubic splines with obstacles by methods of optimization and optimal control, Numer. Math. (to appear).
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).
J. W. Schmidt and W. Heß,Positive interpolation with rational quadratic splines, Computing 38 (1987), 261–267.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Schmidt, J.W., Heß, W. Positivity of cubic polynomials on intervals and positive spline interpolation. BIT 28, 340–352 (1988). https://doi.org/10.1007/BF01934097
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01934097