A New Algorithm for Approximating the Least Concave Majorant

TL;DR

This paper introduces an algorithm inspired by the Jarvis March to efficiently approximate the least concave majorant of a differentiable piecewise polynomial function, with applications to cubic spline approximations.

Contribution

The paper presents a novel algorithm for approximating the least concave majorant of differentiable functions, particularly using cubic spline approximations, extending previous methods.

Findings

The algorithm effectively approximates the least concave majorant for piecewise polynomial functions.

Cubic spline approximation of $F$ leads to a good approximation of its least concave majorant.

Two examples demonstrate the algorithm's practical application and effectiveness.

Abstract

The least concave majorant, $\hat F$, of a continuous function $F$ on a closed interval, $I$, is defined by \[ \hat F (x) = \inf \left\{ G(x): G \geq F, G \mbox{ concave}\right\},\; x \in I. \] We present here an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on $I$. Given any function $F \in \mathcal{C}^4(I)$, it can be well-approximated on $I$ by a clamped cubic spline $S$. We show that $\hat S$ is then a good approximation to $\hat F$. We give two examples, one to illustrate, the other to apply our algorithm.