Comonotone shape-preserving spline histopolation
In this paper, the monotonicity preservation in histopolation problems is studied. Consider a mesh \( a = x_0 < x_1 < \ldots < x_n = b \) and let \( z_i, i = 1, \ldots , n \) be real numbers. The authors are interested in the construction of a \( C^1 \) function \( S \) on \([a,b]\) that for \( x \in [x_{i-1}, x_i]E, i = 1, \ldots , n \) is either a quadratic polynomial or a linear/linear rational function satisfying the histopolation conditions \[ \int^{x_i}_{x_{i-1}}ES(x) dx = z_i (x_i - x_{i-1}), \qquad i = 1, \ldots , n \] and boundary conditions \( S' (x_0) = \alpha, S' (x_n) = \beta \) or \( S(x_0) = \alpha, S(x_n) = \beta. \) The subintervals \( [x_{i-1}, x_i] \) are chosen to be rational or quadratic such that \( S(x) \) is comonotone shape preserving. It is shown that a unique comonotone histopolating spline exists for any histogramm having weak alternation of data on quadratic sections. For the general case a modification of the algorithm is proposed which consists in replacing some quadratic intervals by rational ones.
scientific article; zbMATH DE number 5117672