Convex interval interpolation using a three-term staircase algorithm

Abstract

Motivated by earlier considerations of interval interpolation problems as well as a particular application to the reconstruction of railway bridges, we deal with the problem of univariate convexity preserving interval interpolation. To allow convex interpolation, the given data intervals have to be in (strictly) convex position. This property is checked by applying an abstract three-term staircase algorithm, which is presented in this paper. Additionally, the algorithm provides strictly convex ordinates belonging to the data intervals. Therefore, the known methods in convex Lagrange interpolation can be used to obtain interval interpolants. In particular, we refer to methods based on polynomial splines defined on grids with additional knots. Classification (1991): 65D05, 65D07, 41A15 Mathematics Subject

Composition based staircase algorithm and constrained interpolation with boundary conditions

“…An extension of the staircase algorithm to tridiagonal systems is developed by Mulansky and Schmidt [8]. Following a suggestion by the referee of that paper, see also the Acknowledgements, it is mentioned there that the weakly coupled system can be advantagously interpreted as a chain of relations.…”

“…We are very grateful to Carl de Boor. Acting as a referee of our paper [8], he suggested the fruitful interpretation of the solution of weakly coupled systems as the composition of relations.…”