A new algorithm for approximating the least concave majorant

Martin Franců; Ron Kerman; Gord Sinnamon

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 4, page 1071-1093
  • ISSN: 0011-4642

Abstract

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The least concave majorant, F ^ , of a continuous function F on a closed interval, I , is defined by F ^ ( x ) = inf { G ( x ) : G ≥ F , G concave } , x ∈ 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 ∈ 𝒞 4 ( I ) , it can be well-approximated on I by a clamped cubic spline S . We show that S ^ is then a good approximation to F ^ . We give two examples, one to illustrate, the other to apply our algorithm.

How to cite

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Franců, Martin, Kerman, Ron, and Sinnamon, Gord. "A new algorithm for approximating the least concave majorant." Czechoslovak Mathematical Journal 67.4 (2017): 1071-1093. <http://eudml.org/doc/294413>.

@article{Franců2017,
abstract = {The least concave majorant, $\hat\{F\}$, of a continuous function $F$ on a closed interval, $I$, is defined by \[ \hat\{F\} (x) = \inf \lbrace G(x)\colon G \ge F,\ G \text\{ concave\}\rbrace ,\quad x \in I. \] 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 $F \in \mathcal \{C\}^4(I)$, 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.},
author = {Franců, Martin, Kerman, Ron, Sinnamon, Gord},
journal = {Czechoslovak Mathematical Journal},
keywords = {least concave majorant; level function; spline approximation},
language = {eng},
number = {4},
pages = {1071-1093},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A new algorithm for approximating the least concave majorant},
url = {http://eudml.org/doc/294413},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Franců, Martin
AU - Kerman, Ron
AU - Sinnamon, Gord
TI - A new algorithm for approximating the least concave majorant
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 4
SP - 1071
EP - 1093
AB - The least concave majorant, $\hat{F}$, of a continuous function $F$ on a closed interval, $I$, is defined by \[ \hat{F} (x) = \inf \lbrace G(x)\colon G \ge F,\ G \text{ concave}\rbrace ,\quad x \in I. \] 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 $F \in \mathcal {C}^4(I)$, 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.
LA - eng
KW - least concave majorant; level function; spline approximation
UR - http://eudml.org/doc/294413
ER -

References

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