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] |
| Copyright of Computational Mathematics & Modeling is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 195094532 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: An efficient forward-marching construction of the interpolating splines. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Cheaito%2C+Nasri%22">Cheaito, Nasri</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> ncheaito@ul.edu.lb</i><br /><searchLink fieldCode="AR" term="%22Mourad%2C+Ayman%22">Mourad, Ayman</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> ayman.mourad@ul.edu.lb</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Computational+Mathematics+%26+Modeling%22">Computational Mathematics & Modeling</searchLink>. Jun2026, Vol. 37 Issue 2, p393-408. 16p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Splines%22">Splines</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Smoothness+of+functions%22">Smoothness of functions</searchLink><br /><searchLink fieldCode="DE" term="%22Linear+systems%22">Linear systems</searchLink><br /><searchLink fieldCode="DE" term="%22Taylor's+series%22">Taylor's series</searchLink><br /><searchLink fieldCode="DE" term="%22Iterative+methods+%28Mathematics%29%22">Iterative methods (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: 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] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Computational Mathematics & Modeling is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s10598-026-09693-9 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 16 StartPage: 393 Subjects: – SubjectFull: Splines Type: general – SubjectFull: Algorithms Type: general – SubjectFull: Smoothness of functions Type: general – SubjectFull: Linear systems Type: general – SubjectFull: Taylor's series Type: general – SubjectFull: Iterative methods (Mathematics) Type: general – SubjectFull: Numerical analysis Type: general Titles: – TitleFull: An efficient forward-marching construction of the interpolating splines. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Cheaito, Nasri – PersonEntity: Name: NameFull: Mourad, Ayman IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 1046283X Numbering: – Type: volume Value: 37 – Type: issue Value: 2 Titles: – TitleFull: Computational Mathematics & Modeling Type: main |
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