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A new approach to general interpolation formulae for bivariate interpolation. (English) Zbl 1470.41008

Summary: General interpolation formulae for bivariate interpolation are established by introducing multiple parameters, which are extensions and improvements of those studied by J. Tan and Y. Fang [J. Math. Res. Expo. 19, No. 4, 681–687 (1999; Zbl 0982.41007)]. The general interpolation formulae include general interpolation formulae of symmetric branched continued fraction, general interpolation formulae of univariate and bivariate interpolation, univariate block based blending rational interpolation, bivariate block based blending rational interpolation and their dual schemes, and some interpolation form studied by many scholars in recent years. We discuss the interpolation theorem, algorithms, dual interpolation, and special cases and give many kinds of interpolation scheme. Numerical examples are given to show the effectiveness of the method.

MSC:

41A20 Approximation by rational functions
41A05 Interpolation in approximation theory
65D05 Numerical interpolation

Citations:

Zbl 0982.41007
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References:

[1] Tan, J.-Q.; Fang, Y., General frames for bivariate interpolation, Journal of Mathematical Research and Exposition, 19, 4, 681-687 (1999) · Zbl 0982.41007
[2] Cuyt, A.; Verdonk, B., Multivariate rational interpolation, Computing, 34, 1, 41-61 (1985) · Zbl 0553.41004 · doi:10.1007/BF02242172
[3] Cuyt, A.; Verdonk, B., Multivariate reciprocal differences for branched Thiele continued fraction expansions, Journal of Computational and Applied Mathematics, 21, 2, 145-160 (1988) · Zbl 0638.65014 · doi:10.1016/0377-0427(88)90264-6
[4] Kučminskaja, H. I., Approximation of functions by continued and branching continued fractions, Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, 12, 3-10 (1980) · Zbl 0439.41006
[5] Murphy, J. A.; O’Donohoe, M. R., A two-variable generalization of the Stieltjes-type continued fraction, Journal of Computational and Applied Mathematics, 4, 3, 181-190 (1978) · Zbl 0407.40002 · doi:10.1016/0771-050X(78)90002-5
[6] Kuchmins’ka, K.; Vozna, S., On Newton-Thiele-like interpolating formula, Communications in the Analytic Theory of Continued Fractions, 8, 74-79 (2000)
[7] Kuchmins’ka, Kh. I.; Sus, O. M.; Vozna, S. M., Approximation properties of two-dimensional continued fractions, Ukrainian Mathematical Journal, 55, 1, 30-44 (2003) · Zbl 1035.40003 · doi:10.1023/A:1025016501397
[8] Pahirya, M. M.; Svyda, T. S., Problem of interpolation of functions by two-dimensional continued fractions, Ukrainian Mathematical Journal, 58, 5, 954-966 (2006) · Zbl 1124.41011
[9] Zhao, Q.-J.; Tan, J.-Q., The limiting case of blending differences for bivariate blending continued fraction expansions, Northeastern Mathematical Journal, 22, 4, 404-414 (2006) · Zbl 1148.41014
[10] Wang, J. Z., Stieltjes-Newton’s rational interpolants, Communication on Applied Mathematics and Computation, 20, 2, 77-82 (2006)
[11] Wang, R.-H.; Qian, J., Bivariate polynomial and continued fraction interpolation over ortho-triples, Applied Mathematics and Computation, 217, 19, 7620-7635 (2011) · Zbl 1286.65016 · doi:10.1016/j.amc.2011.02.057
[12] Zhao, Q.-J.; Tan, J.-Q., Block based Newton-like blending interpolation, Journal of Computational Mathematics, 24, 4, 515-526 (2006) · Zbl 1111.65011
[13] Zhao, Q.-J.; Tan, J., Block-based Thiele-like blending rational interpolation, Journal of Computational and Applied Mathematics, 195, 1-2, 312-325 (2006) · Zbl 1098.65009 · doi:10.1016/j.cam.2005.03.089
[14] Kahng, S. W., Generalized Newton’s Interpolation Functions and Their Applications to Chebyahev Approximations (1967), Lockheed Electronics Company Report
[15] Kahng, S. W., Osculatory interpolation, Mathematics of Computation, 23, 621-629 (1969) · Zbl 0182.49501 · doi:10.1090/S0025-5718-1969-0247732-7
[16] Tan, J. Q., Theory of Continued Fractions and Its Applications (2007), Beijing, China: Science Publishers, Beijing, China
[17] Tang, S.; Zou, L., A note on general frames for bivariate interpolation, Journal of Mathematical Research and Exposition, 29, 4, 700-706 (2009) · Zbl 1212.41005
[18] Li, C. W.; Zhu, X. L.; Lin, W. R.; Chen, H. H., Block-based bivariate blending rational interpolation, Journal of Hefei University of Technology, 31, 3, 484-488 (2008) · Zbl 1199.41009
[19] Zou, L.; Tang, S., Notes on general frames of symmetric interpolation, Communication on Applied Mathematics and Computation, 25, 1, 111-118 (2011)
[20] Zou, L.; Tang, S., General structures of block based interpolational function, Communications in Mathematical Research, 28, 3, 193-208 (2012) · Zbl 1274.41021
[21] Salzer, H. E.; Langer, R. E., Some new divided difference algorithm for two variables, On Numerical Approximation (1959), Madison, Wis, USA: The University of Wisconsin Press, Madison, Wis, USA
[22] Tan, J. Q.; Zhu, G. Q.; Shi, Z., General framework for vector-valued interpolants, Proceedings of the 3rd China-Japan Seminar on Numerical Mathematics, Science Press
[23] Zhu, G.-Q.; Tan, J.-Q., A note on matrix-valued rational interpolants, Journal of Computational and Applied Mathematics, 110, 1, 129-140 (1999) · Zbl 0936.41003 · doi:10.1016/S0377-0427(99)00203-4
[24] Wang, J. B.; Gu, C. Q., Vector valued Thiele-Werner-type osculatory rational interpolants, Journal of Computational and Applied Mathematics, 163, 1, 241-252 (2004) · Zbl 1048.65011 · doi:10.1016/j.cam.2003.08.069
[25] Siemaszko, W., Thiele-type branched continued fractions for two-variable functions, Journal of Computational and Applied Mathematics, 9, 2, 137-153 (1983) · Zbl 0515.41015 · doi:10.1016/0377-0427(83)90037-7
[26] Zou, L.; Tang, S., New approach to bivariate blending rational interpolants, Chinese Quarterly Journal of Mathematics, 26, 1, 280-284 (2011) · Zbl 1240.41011
[27] Vozna, S. M., Newton-Thiele-type interpolational formula in the form of two dimensional continued fraction with non-equivalent variables, Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, 47, 67-72 (2004) · Zbl 1084.40003
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