插值的一般框架研究
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摘要
本文主要讨论了二元插值的一般框架问题,其主要内容包括二元插值一般框架的一个注记、对称型有理插值的框架以及基于块的混合有理插值的一般结构。
通过引进新的参数将檀结庆和方毅研究的二元插值的一般框架做进一步的推广和改进,使之可以用来处理带不可达点和逆差商不存在的插值问题,并将其推广至多元情形和基于块的混合有理插值情形,包含近年来人们研究的多种插值格式,如修正Thiele型连分式混合插值,基于块的牛顿型混合有理插值,基于块的Thiele型混合有理插值等。
通过引进新的参数将檀结庆和方毅研究的二元对称型插值的一般框架做进一步的推广和改进,包含近年来赵前进、王家正、Kh.I.Kuchmins'ka,S.M.Vonza等人所研究的几种对称插值格式,并给出了几种新形式的对称型混合有理插值格式。
将S.W.Kahng研究的插值的一种显示表达式进行改进和推广,构造了基于块的一元插值函数的一般结构,这种插值结构不仅包含了S.W.Kahng研究的显示公式,而且包含基于块的牛顿型混合有理插值,基于块的Thiele型混合有理插值以及Thiele-Werner型切触有理插值,此外还可以通过对参数的选取获得其它类型的插值函数。此外还推广S.W.Kahng研究的显示公式到多元情形,研究了二元情形的插值框架,插值定理,对偶插值格式,包括的特殊插值格式,并给出了这种插值框架的误差估计。
The summaries of this dissertation are the researches on general frames of bivariate interpolant, which include note on general frames of bivariate interpolant, general frames of symmetry rational interpoaltion and general structures of black based interpolational function。
A new general frame is established by introducing multiple parameters , which is extensions and improvements of those for the general frames studied by Tan and Fang, and can be used to deal with the interpolant problem when inverse differences are nonexistent or meeting unattainable points。We extend the general frames to multivariable case and block based blending rational interpolation case。Modify Thiele-type continued fraction blending interpolation, block based blending rational interpolation are special cases of the general frames。Numerical examples are given to show the effectiveness of the results。
A new symmetry general frame is established by introducing multiple parameters, which is extensions and improvements of those for the general frames studied by Tan and Fang, the symmety interpolation formulae studied by Kh.I.Kuchmins'ka, S.M.Vonza, Zhao Qianjin, Wang Jiazheng etc. are special cases of the symmetry general frame, and give several symmetry blending rational interpolations of new form.
We extend and improved the results of S.W.Kahng, construct general structure of one variable interpolation function, which not only include the results of S.W.Kahng. but also embody block based Newton-like blending rational interpolation, block based Thiele-like blending rational interpolation, Thiele-Werner osculatory rational interpolation and some other class interpolation function。We also extend the general structure to multivariate case, disscuss general structure of bivariate interpolation function, interpolation theorem, special case, dual interpolation frames and error estimation。Clearly, our method offers many flexible interpolation schemes for choices。
通过引进新的参数将檀结庆和方毅研究的二元插值的一般框架做进一步的推广和改进,使之可以用来处理带不可达点和逆差商不存在的插值问题,并将其推广至多元情形和基于块的混合有理插值情形,包含近年来人们研究的多种插值格式,如修正Thiele型连分式混合插值,基于块的牛顿型混合有理插值,基于块的Thiele型混合有理插值等。
通过引进新的参数将檀结庆和方毅研究的二元对称型插值的一般框架做进一步的推广和改进,包含近年来赵前进、王家正、Kh.I.Kuchmins'ka,S.M.Vonza等人所研究的几种对称插值格式,并给出了几种新形式的对称型混合有理插值格式。
将S.W.Kahng研究的插值的一种显示表达式进行改进和推广,构造了基于块的一元插值函数的一般结构,这种插值结构不仅包含了S.W.Kahng研究的显示公式,而且包含基于块的牛顿型混合有理插值,基于块的Thiele型混合有理插值以及Thiele-Werner型切触有理插值,此外还可以通过对参数的选取获得其它类型的插值函数。此外还推广S.W.Kahng研究的显示公式到多元情形,研究了二元情形的插值框架,插值定理,对偶插值格式,包括的特殊插值格式,并给出了这种插值框架的误差估计。
The summaries of this dissertation are the researches on general frames of bivariate interpolant, which include note on general frames of bivariate interpolant, general frames of symmetry rational interpoaltion and general structures of black based interpolational function。
A new general frame is established by introducing multiple parameters , which is extensions and improvements of those for the general frames studied by Tan and Fang, and can be used to deal with the interpolant problem when inverse differences are nonexistent or meeting unattainable points。We extend the general frames to multivariable case and block based blending rational interpolation case。Modify Thiele-type continued fraction blending interpolation, block based blending rational interpolation are special cases of the general frames。Numerical examples are given to show the effectiveness of the results。
A new symmetry general frame is established by introducing multiple parameters, which is extensions and improvements of those for the general frames studied by Tan and Fang, the symmety interpolation formulae studied by Kh.I.Kuchmins'ka, S.M.Vonza, Zhao Qianjin, Wang Jiazheng etc. are special cases of the symmetry general frame, and give several symmetry blending rational interpolations of new form.
We extend and improved the results of S.W.Kahng, construct general structure of one variable interpolation function, which not only include the results of S.W.Kahng. but also embody block based Newton-like blending rational interpolation, block based Thiele-like blending rational interpolation, Thiele-Werner osculatory rational interpolation and some other class interpolation function。We also extend the general structure to multivariate case, disscuss general structure of bivariate interpolation function, interpolation theorem, special case, dual interpolation frames and error estimation。Clearly, our method offers many flexible interpolation schemes for choices。
引文
[1]王仁宏.数值逼近,高等教育出版社,1999
[2]王仁宏,梁学章.多元函数逼近,北京:科学出版社,1998
[3]王仁宏,朱功勤.有理函数逼近及其应用,北京:科学出版社,2004
[4]徐献瑜,李家楷.徐国良.Padé逼近概论,上海:上海科学技术出版社,1990
[5]朱功勤,檀结庆.矩形网格上二元向量有理插值的对偶性,计算数学,3(1995),311-320
[6]朱功勤,顾传青.二元Thiele型向量有理插值,计算数学,3(1990),293-301
[7]朱功勤,顾传青.檀结庆.多元有理逼近方法,北京:中国科学技术出版社,1996
[8]苏家铎,黄有度.“切触有理插值的一个新算法”,高等学校计算数学学报,2(1987),170-176.
[9]朱功勤,黄有群.插值(切触)分式表的构造,计算数学,3(1983),310-317
[10]檀结庆,唐烁.向量值三重分叉连分式插值的算法,数值计算与计算机应用,21(1996),146-149.
[11]徐利治,王仁宏,周蕴时.函数逼近的理论与方法,上海:上海科技教育出版社,1983.
[12]王家正.Stieltjes-Newton型有理插值,应用数学与计算数学学报20(2)2006.77-82.
[13]檀结庆,朱功勤.二元向量分叉连分式插值的矩阵算法,高等学校计算数学学报,3(1996),250-254.
[14]王金波.Thiele-Werner型连分式复向量有理插值若干问题及应用,博士学位论文,上海大学,2003.
[15]赵前进.混合有理插值及其在图像处理中的应用,合肥工业大学博士论文,2006.
[16]王仁宏.数值有理逼近,上海,上海科学技术出版社,1980.
[17]檀结庆.连分式理论及其应用,北京,科学出版社,2007.
[18]朱晓临.“(向量)有理函数插值的研究及其应用”,中国科技大学博士学位论文,2002.9.
[19]G.Claessens.A new algorithm for osculatory rational interpolation,Numer.Math.,27(1976),77-83.
[20]A.Cuyt,B.Verdonk.Multivariate rational interpolation.Computing,34(1985),41-61.
[21]A.Cuyt.General order multivariate rational Hermite interpolants,Monograph,1986,University of Antwerp(UIA).
[22]A.Cuyt,B.Verdonk.Multivariate reciprocal differences for branched Thiele continued fraction expansions, J. Comput. Appl. Math. , 21(1988), 145-160.
[23] A. Cuyt, B.Verdonk. A review of branched continued fraction theory for the construction of multivariate rational approximants, Appl. Numer. Math. , 4(1988), 263-271.
[24] P.R. Graves-Morris. Vector valued rational interpolants I , Numer. Math., 42(1983), 331-348.
[25] P.R. Graves-Morris. Vector valued rational interpolants II, IMA J. Numer. Anal., 4(1984), 209-224.
[26] P.R. Graves-Morris, C. D. Jenkins. Vector valued rational interpolants III, Constr. Approx., 2 (1986), 263-289.
[27] M. Hu, J.Q. Tan. Image compression and reconstruction based on bivariate interpolation by continued fractions, Proceedings of second international conference on image and graphics, Wei sui eds. 87-92, SPIE vol.4875, 2002
[28] L. Iorentzen, H. Waadeland. Continued fractions with applications, Elsevier science publishers B.V., 1992.
[29] H.E. Salzer. Note on osculatory rational interpolation, Math. Comput., 16 (1962), 486-491.
[30] W. Siemazko. Thiele-type branched continued fractions for two variable functions, J. Comput. Appl. Math., 9(1983), 137-153.
[31] J.Q. Tan. Algorithms for lacunary vector valued rational interpolants, Numer. Math. A. J. Chinese Univ., 2(1998), 169-182.
[32] J.Q. Tan. Bivariate blending rational interpolants, Approx. Theory and its Appl., 21(1999), 74-83.
[33] J.Q. Tan, Y. Fang. Newton-Thiele's rational interpolants, Numerical Algorithms, 24(2000), 141-157.
[34] J.Q. Tan, S. Tang. Bivariate composite vector valued interpolation, Math. Comput . , 69 (2000), 1521-1532.
[35] J.Q. Tan, S. Tang. Composite schemes for multivariate blending rational interpolants, Comput. Appl. Math., 144(2002), 263-275.
[36] J.Q. Tan, G.Q. Zhu. Bivariate vector valued rational interpolants by branched continued fractions, Numer. Math. J. Chin. Univ., 4(1995), 37-43.
[37] S. Tang, M. Sheng. A scheme for bivariate blending osculatory rational interpolation, Journal of Information and Computational Science, 4(2005), 789-798.
[38] L. Wuytack. On the osculatory rational interpolation problem, Math. Comput. , 29(1975), 837-843.
[39] G.Q. Zhu, J.Q. Tan. A note on matrix-valued rational interpolants, J. Comput. Math. Appl., 110(1999), 129-140.
[40] H. Werner. A reliable method for rational interpolation, in: L.Wuytack, eds., Pade approximation and its applications, Springer, Berlin, 1979, 257-277.
[41] J.Q. Tan. Computation of vector valued blending rational interpolation, Numer. Math. A J. Chinese Univ., 12(1) (2003), 91-98.
[42] Kh.I.Kuchmins'ka. On function approximation by continued and branched continued fractions, Mat. Met.Fiz. Meh.Polya., 1980, 12:3-10.
[43] V.YA. Skorobogat'ko. Theory of Branched continued fractions and its applications in computational mathematics, Nauka.Moscow. 1983.
[44] S.W. Kahng. Osculatory interpolation, Math. Comput. , 23(1969),621-629.
[45] J.Q. Tan, G.Q. Zhu. General framework for vector-valued interpolants, in: Proc. of the 3rd China-Japan Seminar on Numerical Mathematics, Z. Shi ed., Science Press, Beijing/New York, 1998, 273-278.
[46] W. Siemaszko. Branched continued fractions for double power series, J. Comput. Appl. Math., 6(1980), 121-125.
[47] Kh.I.Kuchmins'ka, W. Siemaszko, Rational Approximation and Interpolation of Functions by Branched Continued Fractions, Rational Approximation and its Applications in Mathematics and Physics, Lectures Notes in Mathematics, vol 1237,24-40.
[48] S. Tang, Y. Liang. Bivariate Blending Thiele-Werner's Osculatory Rational Interpolation. Numer. Math. J. Chinese Univ. (English Ser.), 2007, 3(16): 271-288.
[49] P. Wynn. Vector continued fractions, Linear Algebra and its Applications, 1(1968), 357-395.
[50] P.R. Graves-Morris. Efficient reliable rational interpolation, in: H.van Rossum and M.de Bruin, eds., Pade approximation and its applications, Springer, Berlin, 1984,29-63 .
[51] W.B. Jones, W.J. Thron. Continued fractions, Analytic Theory and Application, Addison-Weslay, Reading, MA, 1980.
[52] Erik Meijering, A Chronology of Interpolation: From Ancient Astronomy to Modern, Signal and Image Processing, Proceedings of IEEE, vol.19 (3), 2002.
[53] Tan Jieqing, Fang Yi .General Frames for Bivariate Interpolation J . Math. Res. & Exp. 19(1999):681-687.
[54] Q.J. Zhao., J.Q. Tan. Block-based Newton-like Blending Rational Interpolation, J. Comp. Math, 24(4) (2006), 515-526.
[55] J.A.Murphy, M.R.O'Donohoe, A Two-variable generalization of the Stieltjes-type continued fraction, J. Comput. Appl. Math. 4(3)1978, 181-190.
[56] Kh.I.Kuchmins'ka, S.M.Vonza. On Newton-Thiele-like Interpolating formula. Commun. Anal.Theory Continued fractions, 8, 2000.74-79.
[57] Kh.I.Kuchmins'ka, O.M.Sus, S.M.Vonza. Approximation Properties of Two-dimensional Continued Fractions, Ukrainian mathematical Journal. 55(1)2003, 36-54.
[58] M.M.Pahirya, T.S.Svyda. Problem of interpolation of functions by two-dimensional continued fractions, Ukrainian mathematical Journal. 58(6)2006, 954-966.
[59] Q.J. Zhao., J.Q. Tan. Block-based Thiele-like Blending Rational Interpolation, J. Comput. Appl. Math 2006, 195:312-325.
[60] J.B. Wang, C.Q. Gu. Vector valued Thiele-Werner-type osculatory rational interplants, J.Comput. Appl. Math. 163,2004,2041-252.
[61] X.L. Zhu, G.Q. Zhu. A method for directly finding the denominator value of rational interpolants, J. Comput. Appl. Math .2002,148: 341-348.
[62] X.L. Zhu. G.Q. Zhu. A Study of the Existence of Vector Valued Rational Interpolation, J. Infor. Comput.Science. 2005, 2:631-640.
[63] P.R.Grave Morris, CD. Jenkins. Generalised inverse vector valued rational interpolation,In: pade Approximation and its Applications BadHonnef, 1983 (H.Werner, H, J.bunger eds) Lectures Notes in Mathematics, vol 1071, Heidelberg: Springer Verlag. 144-156.
[64] Tang S., Zou L. A note on general frames for bivariate interpolation. J. Math. Res. & Exp, To appear.
[65] Q.J. Zhao., J.Q. Tan. The Limiting Case of Blending Differences for Bivariate Blending Continued Fraction Expansions, Northeastern Mathematical Journal, 22 (4) ,2006, 404-414.
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[67] J.Q. Tan. Vector valued rational interpolants over rectangular grids with holes, University of Antwerp, Report no.95-10.
[68] Tang Shuo, Zou Le, Li Chensheng. Block Based Newton-like Blending Osculatory Rational Interpolation, Submmited
[69] 邹乐,唐烁,关联连分式插值与逼近,己投稿
[70] Tang Shuo, Zou Le. A New Approach to Bivariate Blending Rational Interpolants, Submmited,
[71]Zou Le,Tang Shuo.Block Based Associated-like Continued Fractions Blending Rational Interpolation(已完成).
[72]梁艳,多元混合切触有理插值,合肥工业大学硕士论文.2007.6.
[73]盛敏,基于连分式的多元混合切触有理插值,合肥工业大学硕士论文.2006.6.
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[77]S.M.Vonza.Newton-tile-type interpolational formula in the form of twodimensional continued fraction with non-equivalent variables,Mat.Met.Fiz.Meh.Polya.,2004,47:67-72.
[2]王仁宏,梁学章.多元函数逼近,北京:科学出版社,1998
[3]王仁宏,朱功勤.有理函数逼近及其应用,北京:科学出版社,2004
[4]徐献瑜,李家楷.徐国良.Padé逼近概论,上海:上海科学技术出版社,1990
[5]朱功勤,檀结庆.矩形网格上二元向量有理插值的对偶性,计算数学,3(1995),311-320
[6]朱功勤,顾传青.二元Thiele型向量有理插值,计算数学,3(1990),293-301
[7]朱功勤,顾传青.檀结庆.多元有理逼近方法,北京:中国科学技术出版社,1996
[8]苏家铎,黄有度.“切触有理插值的一个新算法”,高等学校计算数学学报,2(1987),170-176.
[9]朱功勤,黄有群.插值(切触)分式表的构造,计算数学,3(1983),310-317
[10]檀结庆,唐烁.向量值三重分叉连分式插值的算法,数值计算与计算机应用,21(1996),146-149.
[11]徐利治,王仁宏,周蕴时.函数逼近的理论与方法,上海:上海科技教育出版社,1983.
[12]王家正.Stieltjes-Newton型有理插值,应用数学与计算数学学报20(2)2006.77-82.
[13]檀结庆,朱功勤.二元向量分叉连分式插值的矩阵算法,高等学校计算数学学报,3(1996),250-254.
[14]王金波.Thiele-Werner型连分式复向量有理插值若干问题及应用,博士学位论文,上海大学,2003.
[15]赵前进.混合有理插值及其在图像处理中的应用,合肥工业大学博士论文,2006.
[16]王仁宏.数值有理逼近,上海,上海科学技术出版社,1980.
[17]檀结庆.连分式理论及其应用,北京,科学出版社,2007.
[18]朱晓临.“(向量)有理函数插值的研究及其应用”,中国科技大学博士学位论文,2002.9.
[19]G.Claessens.A new algorithm for osculatory rational interpolation,Numer.Math.,27(1976),77-83.
[20]A.Cuyt,B.Verdonk.Multivariate rational interpolation.Computing,34(1985),41-61.
[21]A.Cuyt.General order multivariate rational Hermite interpolants,Monograph,1986,University of Antwerp(UIA).
[22]A.Cuyt,B.Verdonk.Multivariate reciprocal differences for branched Thiele continued fraction expansions, J. Comput. Appl. Math. , 21(1988), 145-160.
[23] A. Cuyt, B.Verdonk. A review of branched continued fraction theory for the construction of multivariate rational approximants, Appl. Numer. Math. , 4(1988), 263-271.
[24] P.R. Graves-Morris. Vector valued rational interpolants I , Numer. Math., 42(1983), 331-348.
[25] P.R. Graves-Morris. Vector valued rational interpolants II, IMA J. Numer. Anal., 4(1984), 209-224.
[26] P.R. Graves-Morris, C. D. Jenkins. Vector valued rational interpolants III, Constr. Approx., 2 (1986), 263-289.
[27] M. Hu, J.Q. Tan. Image compression and reconstruction based on bivariate interpolation by continued fractions, Proceedings of second international conference on image and graphics, Wei sui eds. 87-92, SPIE vol.4875, 2002
[28] L. Iorentzen, H. Waadeland. Continued fractions with applications, Elsevier science publishers B.V., 1992.
[29] H.E. Salzer. Note on osculatory rational interpolation, Math. Comput., 16 (1962), 486-491.
[30] W. Siemazko. Thiele-type branched continued fractions for two variable functions, J. Comput. Appl. Math., 9(1983), 137-153.
[31] J.Q. Tan. Algorithms for lacunary vector valued rational interpolants, Numer. Math. A. J. Chinese Univ., 2(1998), 169-182.
[32] J.Q. Tan. Bivariate blending rational interpolants, Approx. Theory and its Appl., 21(1999), 74-83.
[33] J.Q. Tan, Y. Fang. Newton-Thiele's rational interpolants, Numerical Algorithms, 24(2000), 141-157.
[34] J.Q. Tan, S. Tang. Bivariate composite vector valued interpolation, Math. Comput . , 69 (2000), 1521-1532.
[35] J.Q. Tan, S. Tang. Composite schemes for multivariate blending rational interpolants, Comput. Appl. Math., 144(2002), 263-275.
[36] J.Q. Tan, G.Q. Zhu. Bivariate vector valued rational interpolants by branched continued fractions, Numer. Math. J. Chin. Univ., 4(1995), 37-43.
[37] S. Tang, M. Sheng. A scheme for bivariate blending osculatory rational interpolation, Journal of Information and Computational Science, 4(2005), 789-798.
[38] L. Wuytack. On the osculatory rational interpolation problem, Math. Comput. , 29(1975), 837-843.
[39] G.Q. Zhu, J.Q. Tan. A note on matrix-valued rational interpolants, J. Comput. Math. Appl., 110(1999), 129-140.
[40] H. Werner. A reliable method for rational interpolation, in: L.Wuytack, eds., Pade approximation and its applications, Springer, Berlin, 1979, 257-277.
[41] J.Q. Tan. Computation of vector valued blending rational interpolation, Numer. Math. A J. Chinese Univ., 12(1) (2003), 91-98.
[42] Kh.I.Kuchmins'ka. On function approximation by continued and branched continued fractions, Mat. Met.Fiz. Meh.Polya., 1980, 12:3-10.
[43] V.YA. Skorobogat'ko. Theory of Branched continued fractions and its applications in computational mathematics, Nauka.Moscow. 1983.
[44] S.W. Kahng. Osculatory interpolation, Math. Comput. , 23(1969),621-629.
[45] J.Q. Tan, G.Q. Zhu. General framework for vector-valued interpolants, in: Proc. of the 3rd China-Japan Seminar on Numerical Mathematics, Z. Shi ed., Science Press, Beijing/New York, 1998, 273-278.
[46] W. Siemaszko. Branched continued fractions for double power series, J. Comput. Appl. Math., 6(1980), 121-125.
[47] Kh.I.Kuchmins'ka, W. Siemaszko, Rational Approximation and Interpolation of Functions by Branched Continued Fractions, Rational Approximation and its Applications in Mathematics and Physics, Lectures Notes in Mathematics, vol 1237,24-40.
[48] S. Tang, Y. Liang. Bivariate Blending Thiele-Werner's Osculatory Rational Interpolation. Numer. Math. J. Chinese Univ. (English Ser.), 2007, 3(16): 271-288.
[49] P. Wynn. Vector continued fractions, Linear Algebra and its Applications, 1(1968), 357-395.
[50] P.R. Graves-Morris. Efficient reliable rational interpolation, in: H.van Rossum and M.de Bruin, eds., Pade approximation and its applications, Springer, Berlin, 1984,29-63 .
[51] W.B. Jones, W.J. Thron. Continued fractions, Analytic Theory and Application, Addison-Weslay, Reading, MA, 1980.
[52] Erik Meijering, A Chronology of Interpolation: From Ancient Astronomy to Modern, Signal and Image Processing, Proceedings of IEEE, vol.19 (3), 2002.
[53] Tan Jieqing, Fang Yi .General Frames for Bivariate Interpolation J . Math. Res. & Exp. 19(1999):681-687.
[54] Q.J. Zhao., J.Q. Tan. Block-based Newton-like Blending Rational Interpolation, J. Comp. Math, 24(4) (2006), 515-526.
[55] J.A.Murphy, M.R.O'Donohoe, A Two-variable generalization of the Stieltjes-type continued fraction, J. Comput. Appl. Math. 4(3)1978, 181-190.
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