Seven-diagonal run-through method for solving B-spline approximation problems

Geoinformatics

UDC:

004.9:51-74

DOI:

10.22389/0016-7126-2026-1027-1-9-16

1 Vasiliev N.P.

2 Vagizov M.R.

Year:

2026

№:

1027

Pages:

9-16

St. Petersburg State Forest Technical University named after S. M. Kirov

1,

Russian state hydrometeorological university

2,

Abstract:

The B-spline approximation problem comes down to solving a system of linear equations with a symmetric coefficient matrix, which enables using the Cholesky method. This matrix is also a band one. If the degree of the splines is three (the most common case), the band width is seven. To solve systems with the said matrices, the Thomas algorithm is used. This is the name by which the method is known for a tridiagonal band. In this paper, a seven-diagonal version is investigated, and recurrence formulas are obtained. They are tested with the problem of smoothing altitude measurements using satellite geolocation. Compared to the Cholesky scheme, the technique is applicable to uncertain and asymmetric matrices and is more efficient in terms of computation speed and computer memory consumption. The results of numerical studies of the proposed scheme’s stability for ill-conditioned problems are presented

Keywords:

approximation, banded matrices, B-splines, matrix run-through method, satellite geolocation, seven-diagonal Thomas algorithm, smoothing satellite geolocation measurements

References:

1. Bubnov I. G. Stroitel'naya mekhanika korablya. Ch. I. Saint-Petersburg: tip. Mor. m-va, 1912, 330 p.

2. Vasiliev N.P., Vagizov M.R. (2024) Using B-splines for alignment of satellite geolocation elevation data. Geodezia i Kartografia, 85(12), pp. 54-59. (In Russian). DOI: 10.22389/0016-7126-2024-1014-12-54-59.

3. Vasil'ev N. P., Vagizov M. R. O neobkhodimosti opredeleniya proizvodnykh na granitsakh setki splainovykh modelei geopolei. Informatsiya i kosmos, 2024, no. 1, pp. 139–145.

4. Vyatkin S. I., Dolgovesov B. S. Gibridnyi rendering funktsional'no zadannykh poverkhnostei, voksel'no-baziruemogo rel'efa mestnosti i rasseyannogo sveta s primeneniem graficheskikh akseleratorov. Vestnik komp'yuternykh i informatsionnykh tekhnologii, 2018, no. 9 (171), pp. 32–38. DOI: 10.14489/vkit.2018.09.pp.032-038.

5. Madorskii V. M. O priblizhennom reshenii nelineinoi differentsial'noi zadachi vtorogo poryadka. Vestnik Brestskogo universiteta. Ser. 4. Fizika. Matematika, 2013, no. 2, pp. 80–86.

6. Samarskii A. A., Nikolaev E. S. Metody resheniya setochnykh uravnenii. Moskva: Nauka, 1978, 590 p.

7. Schoenberg I. J. (1946) Contributions to the problem of approximation of equidistant data by analytic functions. Part A. On the problem of smoothing or graduation. A first class of analytic approximation formulae. Quarterly of Applied Mathematics, no. 4, pp. 45–99. DOI: 10.1090/QAM/15914.

8. Schoenberg I. J. (1946) Contributions to the problem of approximation of equidistant data by analytic functions. Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae. Quarterly of Applied Mathematics, no. 4, pp. 112–141. DOI: 10.1090/QAM/16705.

9. De Boor C. (1972) On calculating with B-splines. Journal of Approximation Theory, no. 6, pp. 50–62. DOI: 10.1016/0021-9045(72)90080-9.

10. Chakalov L. (1938) On a certain presentation of the Newton divided differences in interpolation theory and its applications. Godishnik na Sofijskiya Universitet, Fiziko-Matematicheski Fakultet, no. 34, pp. 353–394.

11. Cox M. G. (1972) The numerical evaluation of B-splines. Journal of Applied Mathematics, Volume 10, no. 2, pp. 134–149.

12. Curry H. B. (1947) Review. Mathematical Tables and other Aids to Computation, 2. pp. 167–169, 211–213.

13. Curry H. B., Schoenberg I. J. (1947) On spline distributions and their limits: The Polya distribution functions. Bulletin of the American Mathematical Society, no. 53,

14. Curry H. B., Schoenberg I. J. (1966) On Polya frequency functions IV: the fundamental spline functions and their limits. Journal d`Analyse Mathématique, no. 17, pp. 71–107.

15. Favard J. (1940) Sur l`interpolation. Journal de Mathématiques Pures et Appliquées, no. 19, pp. 281–306.

16. Kunoth A., Lyche T., Sangalli G., Serra-Capizzano S. (2018) Splines and PDEs: From Approximation Theory to Numerical Linear Algebra. Springer, Cham, 318 p. DOI: 10.1007/978-3-319-94911-6.

Citation:

Vasiliev N.P.,

Vagizov M.R.,

(2026) Seven-diagonal run-through method for solving B-spline approximation problems. Geodesy and cartography = Geodeziya i Kartografiya, 87(1), pp. 9-16. (In Russian). DOI: 10.22389/0016-7126-2026-1027-1-9-16