The stable evaluation of multivariate simplex splines
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- by Thomas A. Grandine PDF
- Math. Comp. 50 (1988), 197-205 Request permission
Abstract:
This paper gives a general method for the stable evaluation of multivariate simplex splines, based on the well-known recurrence relation of Micchelli [12]. This paper deals with two problems which arise in the implementation of the recurrence relation. First, the coefficients in the recurrence are shown to be efficiently computable via the dual simplex method of linear programminig. Secondly, the problem of evaluation along mesh boundaries is discussed in detail.References
- Robert G. Bland, New finite pivoting rules for the simplex method, Math. Oper. Res. 2 (1977), no. 2, 103–107. MR 459599, DOI 10.1287/moor.2.2.103 Carl de Boor, "Splines as linear combinations of B-splines," in Approximation Theory II (G.G. Lorentz, C. K. Chui, and L. L. Schumaker, eds.), Academic Press, New York, 1976, pp. 1-47.
- Carl de Boor, A practical guide to splines, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR 507062
- C. de Boor, Topics in multivariate approximation theory, Topics in numerical analysis (Lancaster, 1981) Lecture Notes in Math., vol. 965, Springer, Berlin-New York, 1982, pp. 39–78. MR 690430 Carl de Boor & Klaus Höllig, "B-splines from parallelepipeds," J. Analyse Math., v. 42, 1982/83, pp. 99-115. W. Dahmen & C. A. Micchelli, "Numerical algorithms for least squares approximation by multivariate B-splines," in Numerical Methods of Approximation Theory, vol. 6 (Collatz, Meinardus, and Werner, eds.), Birkhäuser, Basel, 1981.
- George B. Dantzig, Linear programming and extensions, Princeton University Press, Princeton, N.J., 1963. MR 0201189
- Hakop Hakopian, Multivariate spline functions, $B$-spline basis and polynomial interpolations, SIAM J. Numer. Anal. 19 (1982), no. 3, 510–517. MR 656466, DOI 10.1137/0719033
- K. Höllig, A remark on multivariate $B$-splines, J. Approx. Theory 33 (1981), no. 2, 119–125. MR 643907, DOI 10.1016/0021-9045(81)90081-2 Olvi Mangasarian, Linear Programming Lecture Notes, Manuscript, 1978. R. H. J. Gmelig Meyling, An Algorithm for Constructing Configurations of Knots for Bivariate B-splines, Mathematisch Instituut, Universiteit van Amsterdam, Report 85-06, 1985. C. A. Micchelli, "On a numerically efficient method for computing multivariate B-splines," in Multivariate Approximation Theory (W. Schempp and K. Zeller, eds.), ISNM 51, Birkhäuser, Basel, 1979.
- Charles A. Micchelli, A constructive approach to Kergin interpolation in $\textbf {R}^{k}$: multivariate $B$-splines and Lagrange interpolation, Rocky Mountain J. Math. 10 (1980), no. 3, 485–497. MR 590212, DOI 10.1216/RMJ-1980-10-3-485
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Math. Comp. 50 (1988), 197-205
- MSC: Primary 65D07; Secondary 41A15, 41A63
- DOI: https://doi.org/10.1090/S0025-5718-1988-0917827-2
- MathSciNet review: 917827