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Non-uniform exponential tension splines. (English) Zbl 1130.65019
Algorithms are constructed and implemented with the purpose of computing univariate splines in tension. The tension is fixed via certain parameters which need not be the same along the spline which is to be evaluated. Both splines and their derivatives can be computed with these algorithms. They are based on knot insertion methods such as the Oslo algorithm. Many examples of splines, exponential tension splines of various smoothness are presented and the accuracy of the iterative method is analysed too.

MSC:
65D07 Numerical computation using splines
41A50 Best approximation, Chebyshev systems
41A15 Spline approximation
Software:
TSPACK
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References:
[1] Barsky, B.A.: Exponential and polynomial methods for applying tension to an interpolating spline curve. Comput. Vis. Graph. Image Process. 27, 1–18 (1984) · Zbl 0601.65007 · doi:10.1016/0734-189X(84)90078-1
[2] Bister, D., Prautzsch, H.: A New Approach to Tchebycheffian B-Splines. In: Méhauteé, A.L., Rabut, C., Schumaker, L.L. (eds.) Curve and Surfaces in Geometric Design, pp. 35–43. Nashville, TN (1997) · Zbl 0938.65020
[3] Bosner, T.: Knot insertion algorithms for weighted splines. In: Drmač, Z., Marušić, M., Tutek Z. (eds.) Proceedings of the Conference on Applied Mathematics and Scientific Computing, pp. 151–160. (2005) · Zbl 1069.65009
[4] Bosner, T.: Knot insertion algorithms for Chebyshev splines. Ph.D. thesis, Dept. of Mathematics, University of Zagreb (2006) · Zbl 1069.65009
[5] Bosner, T., Rogina, M.: Numerically Stable Algorithm for Cycloidal Splines. Annali dell’Università di Ferrara (2007) (to appear) · Zbl 1182.41024
[6] Burrill, C.W.: Measure, Integration, and Probability. McGraw-Hill Book Company (1972) · Zbl 0248.28001
[7] Cohen, E., Lyche, T., Riesenfeld, R.: Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphic. Comput. Graph. Image Process. 14, 87–111 (1980) · doi:10.1016/0146-664X(80)90040-4
[8] Costantini, P., Lyche, T., Manni, C.: On a class of weak Tchebycheff systems. Numer. Math. 101, 333–354 (2005) · Zbl 1085.41002 · doi:10.1007/s00211-005-0613-6
[9] Foley, T.A.: Interpolation with interval and point tension controls using cubic weighted \(\nu\)-splines. ACM Trans. Math. Softw. 13(1), 68–96 (1987) · Zbl 0626.65008 · doi:10.1145/23002.23004
[10] Goldman, R.N., Lyche, T. (eds.): Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces. SIAM (1993) · Zbl 0772.00011
[11] Horvat, V., Rogina, M.: Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations. J. Comput. Appl. Math. 140, 381–402 (2002) · Zbl 0998.65133 · doi:10.1016/S0377-0427(01)00517-9
[12] Kadalbajoo, M.K., Patidar, K.C.: Numerical solution of singularly perturbed two-point boundary value problems by spline in tension. Appl. Math. Comput. 131, 299–320 (2002) · Zbl 1030.65087 · doi:10.1016/S0096-3003(01)00146-1
[13] Koch, P.E., Lyche, T.: Exponential B-splines in tension. In: Chui, C.K., Schumaker, L.L., Ward, J.D. (eds.) Approximation Theory VI, vol. 2, pp. 361–364. (1989) · Zbl 0754.41006
[14] Koch, P.E., Lyche, T.: Construction of exponential tension B-splines of arbitrary order. In: Laurent, P.J., Le Méhauté, A., Schumaker, L.L. (eds.) Curves and Surfaces, pp. 255–258. Boston (1991) · Zbl 0736.41013
[15] Kulkarni, R., Laurent, P.-J.: Q-splines. Numer. Algorithms 1, 45–73 (1991) · Zbl 0797.65003 · doi:10.1007/BF02145582
[16] Kvasov, B.I.: Algorithms for shape preserving local approximation with automatic selection of tension parameters. Comput. Aided Geom. Des. 17, 17–37 (2000) · Zbl 0939.68122 · doi:10.1016/S0167-8396(99)00037-0
[17] Kvasov, B.I.: Shape–Preserving Spline Approximation. World Scientific, Singapore (2000) · Zbl 0960.41001
[18] Lyche, T.: A recurrence relation for Chebyshevian B-splines. Constr. Approx. 1, 155–173 (1985) · Zbl 0583.41011 · doi:10.1007/BF01890028
[19] Marušić, M.: Stable calculation by splines in tension. Grazer Math. Ber. 328, 65–76 (1996) · Zbl 0878.65009
[20] Marušić, M.: A fourth/second order accurate collocation method for singularly perturbed two-point boundary value problems using tension splines. Numer. Math. 88, 135–158 (2001) · Zbl 0989.65082 · doi:10.1007/PL00005437
[21] Marušić, M., Rogina, M.: A collocation method for singularly perturbed two-point boundary value problems with splines in tension. Adv. Comput. Math. 6(1), 65–76 (1996) · Zbl 0876.65058 · doi:10.1007/BF02127696
[22] Mazure, M.-L.: Blossoming: a geometrical approach. Constr. Approx. 15, 33–68 (1999) · Zbl 0924.65010 · doi:10.1007/s003659900096
[23] Mazure, M.-L.: Blossoms and optimal bases. Adv. Comput. Math. 20, 177–203 (2004) · Zbl 1042.65016 · doi:10.1023/A:1025855123163
[24] Mazure, M.-L., Pottmann, H.: Tchebycheff curves. In: Gasca, M., Micchelli, C.A. (eds.) Total Positivity and Its Applications, pp. 187–218. Kluwer Academic Pub. (1996) · Zbl 0902.41018
[25] Mülbach, G., Tang, Y.: Computing ECT–B-splines recursively. Numer. Algorithms 41, 35–78 (2006) · Zbl 1090.65015 · doi:10.1007/s11075-005-9005-3
[26] Pottmann, H.: The geometry of Tchebycheffian splines. Comput. Aided Geom. Des. 10, 181–210 (1993) · Zbl 0777.41016 · doi:10.1016/0167-8396(93)90036-3
[27] Pottmann, H., Wagner, M.G.: Helix splines as an example of affine Tchebycheffian splines. Adv. Comput. Math. 2, 123–142 (1994) · Zbl 0832.65008 · doi:10.1007/BF02519039
[28] Prenter, P.M.: Piecewise L-Splines. Numer. Math. 18, 243–253 (1971) · Zbl 0227.65012 · doi:10.1007/BF01397084
[29] Renka, R.J.: Interpolatory tension splines with automatic selection of tension factors. SIAM J. Sci. Stat. Comput. 8(3), 393–415 (1987) · Zbl 0629.65009 · doi:10.1137/0908041
[30] Renka, R.J.: Algorithm 716 TSPACK: tension spline curve–fitting package. ACM Trans. Math. Softw. 19(1), 81–94 (1993) · Zbl 0889.65007 · doi:10.1145/151271.151277
[31] Rentrop, P.: An algorithm for the computation of the exponential spline. Numer. Math. 35, 81–93 (1980) · Zbl 0461.65009 · doi:10.1007/BF01396372
[32] Rogina, M.: Basis of splines associated with some singular differential operators. BIT 32, 496–505 (1992) · Zbl 0765.65013 · doi:10.1007/BF02074883
[33] Rogina, M.: Nove rekurentne relacije za Čebiševljeve spline funkcije i njihove primjene. Ph.D. thesis, Dept. of Mathematics, University of Zagreb (1994)
[34] Rogina, M.: On construction of fourth order Chebyshev splines. Math. Commun. 4, 83–92 (1999) · Zbl 0943.65022
[35] Rogina, M.: Algebraic proof of the B-Spline derivative formula. In: Drmač, Z., Marušić, M., Tutek, Z. (eds.) Proceedings of the Conference on Applied Mathematics and Scientific Computing, pp. 273–282. (2005) · Zbl 1069.41011
[36] Rogina, M., Bosner, T.: On calculating with lower order Chebyshev splines. In: Laurent, P.J., Sabloniere, P., Schumaker, L.L. (eds.) Curves and Surfaces Design, pp. 343–353. Nashville (2000) · Zbl 1036.65010
[37] Rogina, M., Bosner, T.: A de Boor type algorithm for tension splines. In: Cohen, A., Merrien, J.-L., Schumaker, L.L. (eds.) Curve and Surface Fitting, pp. 343–352. Brentwood (2003) · Zbl 1036.65010
[38] Schumaker, L.L.: On Tchebycheffian spline functions. J. Approx. Theory 18, 278–303 (1976) · Zbl 0339.41004 · doi:10.1016/0021-9045(76)90021-6
[39] Schumaker, L.L.: Spline Functions: Basic Theory. Wiley, New York (1981) · Zbl 0449.41004
[40] Schumaker, L.L.: On recursions for generalized splines. J. Approx. Theory 36, 16–31 (1982) · Zbl 0529.41011 · doi:10.1016/0021-9045(82)90067-3
[41] Schumaker, L.L.: On hyperbolic splines. J. Approx. Theory 38, 144–166 (1983) · Zbl 0512.41008 · doi:10.1016/0021-9045(83)90121-1
[42] Schweikert, D.: An interpolation curve using splines in tension. J. Math. Phys. 45, 312–317 (1966) · Zbl 0146.14102
[43] Stynes, M.: Steady-state convection-diffusion problems. Acta Numer. 14, 445–508 (2005) · Zbl 1115.65108 · doi:10.1017/S0962492904000261
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