Composition based staircase algorithm and constrained interpolation with boundary conditions

Summary.

Weakly coupled systems of inequalities arise frequently in the consideration of so-called direct methods for shape preserving interpolation. In this paper, a composition based staircase algorithm for bidiagonal systems subject to boundary conditions is developed. Using the compositions of the corresponding relations instead of their projections, we are able to derive a necessary and sufficient solvability criterion. Further, all solutions of the system can be constructed in a backward pass. To illustrate the general approach, we consider in detail the problem of convex interpolation by cubic $C^1$ splines. For this problem, an algorithm of the complexity O(n) in the number n of data points is obtained.