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An efficient forward-marching construction of the interpolating splines.

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Title: An efficient forward-marching construction of the interpolating splines.
Authors: Cheaito, Nasri1 (AUTHOR) ncheaito@ul.edu.lb, Mourad, Ayman1 (AUTHOR) ayman.mourad@ul.edu.lb
Source: Computational Mathematics & Modeling. Jun2026, Vol. 37 Issue 2, p393-408. 16p.
Subjects: Splines, Algorithms, Smoothness of functions, Linear systems, Taylor's series, Iterative methods (Mathematics), Numerical analysis
Abstract: Splines are widely used for smooth interpolation of a set of data points ,. The classical approaches require solving a linear system to determine the second derivatives at the knots. This linear system arises from enforcing derivatives continuity and satisfying boundary conditions. This process, while efficient for moderate-sized data, can become cumbersome in cases where the system needs to be re-solved frequently or embedded in iterative frameworks. In this paper, we develop a recursive algorithm for the construction of the interpolating splines of degree q ≥ 2. The method is based on generating q auxiliary splines with prescribed derivatives at the initial knot x0, by using Taylor expansions at knots and the propagation of derivative information, after which the target interpolating spline will be obtained as an appropriate linear combination of these auxiliary splines. This approach avoids solving any global linear system, leading to an efficient forward-marching construction. In addition, and in order to avoid instabilities in the calculation of the splines coefficients, we propose an algorithm called "Forward-and-Update" that allows to deal with any number of data points by controlling the derivatives at all knots. The advantage of the proposed approach is its conceptual simplicity and its potential for efficient implementation in streaming or adaptive contexts. Moreover, it offers new insight into the structural properties of interpolating splines, particularly in how local constraints can be extended to global smoothness through suitable linear combinations. The numerical implementation of the proposed method has shown its outperformance compared to the classical approach. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:Splines are widely used for smooth interpolation of a set of data points ,. The classical approaches require solving a linear system to determine the second derivatives at the knots. This linear system arises from enforcing derivatives continuity and satisfying boundary conditions. This process, while efficient for moderate-sized data, can become cumbersome in cases where the system needs to be re-solved frequently or embedded in iterative frameworks. In this paper, we develop a recursive algorithm for the construction of the interpolating splines of degree q ≥ 2. The method is based on generating q auxiliary splines with prescribed derivatives at the initial knot x0, by using Taylor expansions at knots and the propagation of derivative information, after which the target interpolating spline will be obtained as an appropriate linear combination of these auxiliary splines. This approach avoids solving any global linear system, leading to an efficient forward-marching construction. In addition, and in order to avoid instabilities in the calculation of the splines coefficients, we propose an algorithm called "Forward-and-Update" that allows to deal with any number of data points by controlling the derivatives at all knots. The advantage of the proposed approach is its conceptual simplicity and its potential for efficient implementation in streaming or adaptive contexts. Moreover, it offers new insight into the structural properties of interpolating splines, particularly in how local constraints can be extended to global smoothness through suitable linear combinations. The numerical implementation of the proposed method has shown its outperformance compared to the classical approach. [ABSTRACT FROM AUTHOR]
ISSN:1046283X
DOI:10.1007/s10598-026-09693-9