基于块的混合切触插值
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摘要
本文主要讨论了基于块的混合切触插值问题,其主要内容包括基于块的Lagrange-Salzer混合切触有理插值和基于块的Newton型混合切触插值。
利用分块的思想将连分式切触插值与Lagrange多项式相结合,构造了一种基于块的Lagrange-Salzer混合切触有理插值。该有理插值具有更好的灵活性,传统的Salzer连分式插值则是它的一个特例,同时数值例子表明该插值的有效性。
借助于Newton插值的插值格式,构造了一种基于块的Newton型混合切触插值。由于采用了分块的方法,该插值提供了多种插值框架可供选择,其中扩展的Newton插值则是本文的一个特例。同时讨论了二元情况,给出的数值例子表明了该插值的有效性。
The summaries of this dissertation are the researches on the block based blending osculatory interpolation, which include block based Lagrange-Salzer blending osculatory rational interpolation and block based Newton-like blending osculatory interpolation.
we combine continued fraction osculatory interpolation with the Lagrange's polynomial by dividing blocks and construct a kind of block based Lagrange-Salzer blending osculatory rational interpolation. The new construction method is more flexible and the traditional Salzer continued fraction interpolation is a special case. Some numerical examples are given to illustrate the effectiveness of the method in this paper.
With the Newton's interpolating formation, we construct a kind of block based Newton-like blending osculatory interpolation. The interpolation provides us with many flexible interpolation schemes for choices which include the expansive Newton's polynomial interpolation as its special case. A bivariate analogy is also discussed and numerical examples are given to show the effectiveness of the interpolation.
利用分块的思想将连分式切触插值与Lagrange多项式相结合,构造了一种基于块的Lagrange-Salzer混合切触有理插值。该有理插值具有更好的灵活性,传统的Salzer连分式插值则是它的一个特例,同时数值例子表明该插值的有效性。
借助于Newton插值的插值格式,构造了一种基于块的Newton型混合切触插值。由于采用了分块的方法,该插值提供了多种插值框架可供选择,其中扩展的Newton插值则是本文的一个特例。同时讨论了二元情况,给出的数值例子表明了该插值的有效性。
The summaries of this dissertation are the researches on the block based blending osculatory interpolation, which include block based Lagrange-Salzer blending osculatory rational interpolation and block based Newton-like blending osculatory interpolation.
we combine continued fraction osculatory interpolation with the Lagrange's polynomial by dividing blocks and construct a kind of block based Lagrange-Salzer blending osculatory rational interpolation. The new construction method is more flexible and the traditional Salzer continued fraction interpolation is a special case. Some numerical examples are given to illustrate the effectiveness of the method in this paper.
With the Newton's interpolating formation, we construct a kind of block based Newton-like blending osculatory interpolation. The interpolation provides us with many flexible interpolation schemes for choices which include the expansive Newton's polynomial interpolation as its special case. A bivariate analogy is also discussed and numerical examples are given to show the effectiveness of the interpolation.
引文
[1]黄有谦,李岳生.数值逼近,北京:高等教育出版社,1987
[2]胡敏,檀结庆.保持轮廓清晰的有理.线性彩色图象内插放大方法,系统仿真学报,16(12)(2004),2857-2859.(Color image magnification with reserved sharp contour based on rational-linear interpolation,Journal of System Simulation,16(12)(2004),2857-2859.)
[3]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
[4]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
[5]王仁宏.数值逼近,北京:高等教育出版社,1999
[6]王仁宏.有理逼近的理论与方法,逼近论会议论文集,杭州大学出版社,1978
[7]王仁宏,朱功勤.有理函数逼近及其应用,北京:科学出版社,2004
[8]吴天毅.差商非构造性定义,工科数学,2(1991)
[9]吴天毅.构造Hermite插值多项式的差商方法,天津职业技术师范学院学报,2(1998),15-19
[10]朱功勤,黄有群.插值(切触)分式表的构造,计算数学,3(1983),310-317
[11]朱功勤,黄有群.二元有理插值逐步算法,合肥工业大学学报(自然科学版),1(2000),97-104
[12]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
[13]G.Claessens.A new algorithm for osculatory rational interpolation,Numer.Math.,27(1976),77-83
[14]A.Cuyt,B.Verdonk.Multivariate rational interpolation Computing.34(1985),41-61
[15]A.Cuyt,B.Verdonk.Multivariate reciprocal divided differences for branched Thiele continued fraction expansions,J.Comput.Appl.Math.,21(1988),145-160
[16]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
[17]A.Cuyt.A recursive computation scheme for multivariate rational interpolants,SIAM,J.Num.Anal,24(1987),228-238
[18]A.Cuyt.Multivariate qd-like algorithm,BIT,28(1988),98-112
[19]P.R.Graves-Morris.Vector valued rational interpolants Ⅱ,IMA J.Numer.Anal.,4(1984),209-224
[20]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
[21]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
[22]W.B. Jones, W.J. Thron. Continued fractions, Analytic Theory and Application, Addison-Weslay, Reading, MA, 1980
[23] J.B Wang, C.Q Gu. Vector valued Thiele-Werner-type osculatory rational interpolants, J. Comput. Appl. Math., 163(2004), 241-252
[24] S.W. Kahng. Osculatory interpolation, Math. Comput., 23(1969),621- 629
[25] David Kincaid, Ward Cheney. Numerical analysis: Mathematics of scientific computing (Third Edition). Thomson Learning, Inc. 2003
[26] Erik Meijering. A Chronology of Interpolation:From Ancient Astronomy to Modern, Signal and Image Processing, Proceedings of IEEE, vol.19 (3), 2002
[27] C. Hermite. Sur la formule d'interpolation de Lagrange, Journal fur die Reine und Angewandte Mathematik, vol, 84(1)(1878), 70-79
[28]Ch. Meray. Observations sur la legitimite de l'interpolation, AnnalesScientifiques de l'Ecole Normale Superieure, ser. 3, 1(1884), 165-176
[29] G.D. Birkhoff. General mean value and remainder theorems with applications to mechanical differentiation and quadrature, Trans. Amer. Math. Soc, 7(1)(1906), 107-136
[30] H.E. Salzer. Note on osculatory rational interpolation, Math. Comput., 16 (1962), 486-491
[31] W. Siemaszko. Branched continued fractions for double power series, J. Comput. Appl. Math., 6(1980), 121-125
[32] W. Siemaszko. On some conditions for convergence of branched continued fractions, LNM,888(1981), 363-370
[33] W. Siemaszko.Thiele-type branched continued fractions for two variable functions, J. Comput. Appl. Math., 9(1983), 137-153
[34] I. Newton, Letter to Oldenburg(24 October 1676),in The Correspondence of Isaac Newton, H.W. Turnbull, Ed. Cambridge, U.K.: Cambridge Univ. Press, 1960, vol. II,pp.110-161
[35] J.L. Lagrange. Lecons elementaires sur les mathematiques donnees a l'ecole normale, in (Euvres de Lagrange, J.-A. Serret, Ed. Paris,France:Gauthier-Villars, 7(1877), 183-287
[36] J. Stoer, R. Bulirsch. Introduction to Numerical Analysis,Second Edition, Springer-Verlag, 1992
[37] J.Q. Tan. Bivariate rational interpolants with rectangle-hole structure, Comput.Math., 17(1)(1999), 1-14
[38] J.Q. Tan. Bivariate blending rational interpolants, Approx. Theory & Its Appl., 15(2)1999,74-83
[39] J.Q. Tan. Computation of vector valued blending rational interpolation, Numer. Math. A J. Chinese Univ.,12(1) (2003), 91-98
[40] J.Q. Tan, Y. Fang. Newton-Thiele's rational interpolants, Numerical Algorithms, 24(2000), 141-157
[41] J.Q. Tan, S. Tang. Composite schemes for multivariate blending rational interpolants, Comput. Appl. Math., 144(2002), 263-275
[42] J.Q. Tan, G.Q. Zhu. Bivariate vector valued rational interpolants by branched continued fractions, Numer. Math. A. J. Chin. Univ., 4(1) (1995), 37-43
[43] 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
[44]S.Tang,M.Sheng.A scheme for bivariate blending osculatory rational interpolation,Journal of Information and Computational Science,4(2005),789-798
[45]H.Werner.A reliable method for rational interpolation,in:L.Wuytack,eds.,Pade approximation and its applications,Springer,Berlin,1979,257-277
[46]L.Wuytack.On the osculatory rational interpolation problem,Math.Comput.,29(1975),837-843
[47]P.Wynn.Vector continued fractions,Linear Algebra and its Applications,1(1968),357-395
[48]Q.J.Zhao,J.Q.Tan.Block based Lagrange-Thiele-like blending rational interpolation,Journal of Information and Computational Science,3(1)(2006),167-177
[49]Q.J.Zhao,J.Q.Tan.Block based Newton-like blending interpolation,Journal of Computational Mathematics,24(4)(2006),515-526
[50]Q.J.Zhao,J.Q.Tan.Block-based Thiele-like blending rational interpolation,J.Comput.Appl.Math.,195(2006),312-325
[51]S.Tang,Y.Liang.Bivariate Blending Thiele-Werner's Osculatory Rational Interpolation,Numer.Math.J.Chinese Univ.(English Ser.),3(16)(2007),271-288
[2]胡敏,檀结庆.保持轮廓清晰的有理.线性彩色图象内插放大方法,系统仿真学报,16(12)(2004),2857-2859.(Color image magnification with reserved sharp contour based on rational-linear interpolation,Journal of System Simulation,16(12)(2004),2857-2859.)
[3]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
[4]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
[5]王仁宏.数值逼近,北京:高等教育出版社,1999
[6]王仁宏.有理逼近的理论与方法,逼近论会议论文集,杭州大学出版社,1978
[7]王仁宏,朱功勤.有理函数逼近及其应用,北京:科学出版社,2004
[8]吴天毅.差商非构造性定义,工科数学,2(1991)
[9]吴天毅.构造Hermite插值多项式的差商方法,天津职业技术师范学院学报,2(1998),15-19
[10]朱功勤,黄有群.插值(切触)分式表的构造,计算数学,3(1983),310-317
[11]朱功勤,黄有群.二元有理插值逐步算法,合肥工业大学学报(自然科学版),1(2000),97-104
[12]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
[13]G.Claessens.A new algorithm for osculatory rational interpolation,Numer.Math.,27(1976),77-83
[14]A.Cuyt,B.Verdonk.Multivariate rational interpolation Computing.34(1985),41-61
[15]A.Cuyt,B.Verdonk.Multivariate reciprocal divided differences for branched Thiele continued fraction expansions,J.Comput.Appl.Math.,21(1988),145-160
[16]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
[17]A.Cuyt.A recursive computation scheme for multivariate rational interpolants,SIAM,J.Num.Anal,24(1987),228-238
[18]A.Cuyt.Multivariate qd-like algorithm,BIT,28(1988),98-112
[19]P.R.Graves-Morris.Vector valued rational interpolants Ⅱ,IMA J.Numer.Anal.,4(1984),209-224
[20]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
[21]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
[22]W.B. Jones, W.J. Thron. Continued fractions, Analytic Theory and Application, Addison-Weslay, Reading, MA, 1980
[23] J.B Wang, C.Q Gu. Vector valued Thiele-Werner-type osculatory rational interpolants, J. Comput. Appl. Math., 163(2004), 241-252
[24] S.W. Kahng. Osculatory interpolation, Math. Comput., 23(1969),621- 629
[25] David Kincaid, Ward Cheney. Numerical analysis: Mathematics of scientific computing (Third Edition). Thomson Learning, Inc. 2003
[26] Erik Meijering. A Chronology of Interpolation:From Ancient Astronomy to Modern, Signal and Image Processing, Proceedings of IEEE, vol.19 (3), 2002
[27] C. Hermite. Sur la formule d'interpolation de Lagrange, Journal fur die Reine und Angewandte Mathematik, vol, 84(1)(1878), 70-79
[28]Ch. Meray. Observations sur la legitimite de l'interpolation, AnnalesScientifiques de l'Ecole Normale Superieure, ser. 3, 1(1884), 165-176
[29] G.D. Birkhoff. General mean value and remainder theorems with applications to mechanical differentiation and quadrature, Trans. Amer. Math. Soc, 7(1)(1906), 107-136
[30] H.E. Salzer. Note on osculatory rational interpolation, Math. Comput., 16 (1962), 486-491
[31] W. Siemaszko. Branched continued fractions for double power series, J. Comput. Appl. Math., 6(1980), 121-125
[32] W. Siemaszko. On some conditions for convergence of branched continued fractions, LNM,888(1981), 363-370
[33] W. Siemaszko.Thiele-type branched continued fractions for two variable functions, J. Comput. Appl. Math., 9(1983), 137-153
[34] I. Newton, Letter to Oldenburg(24 October 1676),in The Correspondence of Isaac Newton, H.W. Turnbull, Ed. Cambridge, U.K.: Cambridge Univ. Press, 1960, vol. II,pp.110-161
[35] J.L. Lagrange. Lecons elementaires sur les mathematiques donnees a l'ecole normale, in (Euvres de Lagrange, J.-A. Serret, Ed. Paris,France:Gauthier-Villars, 7(1877), 183-287
[36] J. Stoer, R. Bulirsch. Introduction to Numerical Analysis,Second Edition, Springer-Verlag, 1992
[37] J.Q. Tan. Bivariate rational interpolants with rectangle-hole structure, Comput.Math., 17(1)(1999), 1-14
[38] J.Q. Tan. Bivariate blending rational interpolants, Approx. Theory & Its Appl., 15(2)1999,74-83
[39] J.Q. Tan. Computation of vector valued blending rational interpolation, Numer. Math. A J. Chinese Univ.,12(1) (2003), 91-98
[40] J.Q. Tan, Y. Fang. Newton-Thiele's rational interpolants, Numerical Algorithms, 24(2000), 141-157
[41] J.Q. Tan, S. Tang. Composite schemes for multivariate blending rational interpolants, Comput. Appl. Math., 144(2002), 263-275
[42] J.Q. Tan, G.Q. Zhu. Bivariate vector valued rational interpolants by branched continued fractions, Numer. Math. A. J. Chin. Univ., 4(1) (1995), 37-43
[43] 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
[44]S.Tang,M.Sheng.A scheme for bivariate blending osculatory rational interpolation,Journal of Information and Computational Science,4(2005),789-798
[45]H.Werner.A reliable method for rational interpolation,in:L.Wuytack,eds.,Pade approximation and its applications,Springer,Berlin,1979,257-277
[46]L.Wuytack.On the osculatory rational interpolation problem,Math.Comput.,29(1975),837-843
[47]P.Wynn.Vector continued fractions,Linear Algebra and its Applications,1(1968),357-395
[48]Q.J.Zhao,J.Q.Tan.Block based Lagrange-Thiele-like blending rational interpolation,Journal of Information and Computational Science,3(1)(2006),167-177
[49]Q.J.Zhao,J.Q.Tan.Block based Newton-like blending interpolation,Journal of Computational Mathematics,24(4)(2006),515-526
[50]Q.J.Zhao,J.Q.Tan.Block-based Thiele-like blending rational interpolation,J.Comput.Appl.Math.,195(2006),312-325
[51]S.Tang,Y.Liang.Bivariate Blending Thiele-Werner's Osculatory Rational Interpolation,Numer.Math.J.Chinese Univ.(English Ser.),3(16)(2007),271-288