Comp. Appl. Math.
A comparative analysis of local cubic splines
· R. Mijiddorj
Received: 29 September 2016 / Revised: 23 May 2017 / Accepted: 15 May 2018
© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract In this note, we develop a local construction of cubic splines and make a com-
parative analysis of local integro cubic splines. We also derive explicit formulae for a local
integro cubic spline and its ﬁrst two derivatives. These formulae are short and four-point ones
that require less computational cost compared to an integro cubic spline quasi-interpolant.
Keywords Cubic splines · Approximation properties · End conditions
Mathematics Subject Classiﬁcation 65D05 · 65D07
In last years, the local construction of splines has attracted a lot of attention from researchers.
For example, the local integro cubic spline was constructed in Zhanlav and Mijiddorj (2010).
Quite recently, the integro cubic spline quasi-interpolant was developed in Boujraf et al.
(2015) and some comparisons were made with the results obtained in Zhanlav and Mijiddorj
(2010). As seen from Boujraf et al. (2015) and Zhanlav and Mijiddorj (2010), the approxi-
mation order of these two integro splines is equal and is O(h
). The difference between these
two splines is the construction of algorithms.
It should be mentioned that numerical results given in the work (Zhanlav and Mijiddorj
2010) show that the maximum errors are obtained near the end points as a consequence of
using recurrence relations unsuitable for ﬁnding boundary coefﬁcients in B-spline represen-
tation. In this note, we develop a new approach to completely construct explicit formulae
Communicated by Luz de Teresa.
Institute of Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia
Department of Informatics, Mongolian National University of Education, Ulaanbaatar, Mongolia