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A new algorithm for approximating the least concave majorant

Martin Franců; Ron Kerman; Gord Sinnamon — 2017

Czechoslovak Mathematical Journal

The least concave majorant, $\hat{F}$ , of a continuous function $F$ on a closed interval, $I$ , is defined by $\hat{F} (x) = inf {G (x) : G \geq F, G concave}, x \in I .$ We present 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 $F \in 𝒞^{4} (I)$ , 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.

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