重心有理Hermite插值方法
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摘要
插值法是逼近论中的一种基本方法。多项式插值是整个数值逼近的基础,但是高次插值产生的Runge现象限制了它的应用。有理插值的收敛速度较多项式快,它适合于逼近有极点的函数,但是有理插值(如传统的Thiele型切触插值连分式)可能有逆差商不存在、无法避免极点和不可达点以及无法控制极点的位置等问题。放宽对分子分母的次数限制,在一定条件下构造的重心有理Hermite插值函数不仅满足插值条件,而且还可以避免极点。重心有理插值和重心有理Hermite插值具有较好的数值稳定性。本文基于一元pade型逼近和一元重心有理Hermite插值,构造了高精度复合重心有理Hermite插值方法,基于切触插值连分式和一元重心有理Hermite插值,构造了一种新的复合重心有理Hermite插值方法,基于一元重心有理Hermite插值构造了二元重心有理Hermite插值格式,基于pade型逼近和二元重心有理Hermite插值,构造了复合二元重心有理Hermite插值方法。选取不同的插值权可以得到不同的一元(或二元)重心有理Hermite插值函数,通过适当地选取插值权,可以使重心有理Hermite插值没有极点以及不可达点。本文针对如何选取插值权使得插值误差最小的关键问题,给出了计算最优插值权的优化算法。文中给出了数值实例表明新方法的有效性。
Interpolation is one of the fundamental techniques of approximation theory. Polynomial interpolation is the basic of the whole numerical approximations, however, the high order interpolation which bring into the Runge phenomenon restricts its applications. Rational interpolation is suitable for approximating the function which has poles and it possesses a faster convergence than polynomial. However some rational interpolation, for example, the classical Thiele type's osculatory continued fraction interpolation may be revealed the following problems:The reverse divided differences are not existent, poles and unattainable points can not be avoided and the locations of the poles can not be controlled and so on. Loosing the restrictions on the degree of denominator and numerator, the barycentric rational Hermite interpolation which is constructed on certain conditions can not only satisfy the interpolation condition, but also avoid the unwanted poles. The barycentric rational interpolation and barycentric rational Hermite interpolation possess a quite better stability. In this paper, the composite barycentric rational Hermite interpolation with high-accuracy is constructed based on univariate pade-type approximation and univariate barycentric rational Hermite interpolation; New composite barycentric rational Hermite interpolation is constructed based on osculatory rational continued fraction interpolation and univariate barycentric rational Hermite interpolation; A new bivariate barycentric rational Hermite interpolation is constructed based on univariate barycentric rational Hermite interpolation; A composite bivariate barycentric rational Hermite interpolation is constructed based on pade-type approximation and bivariate barycentric rational Hermite interpolation. Choosing the different weights one may obtain the different univariate or bivariate barycentric rational Hermite interpolation. The poles and the unattainable points of the barycentric rational Hermite interpolation may be avoided through doing a proper choice for the interpolation weights. In this paper, the optimization model which is used for computing the optimal interpolation weights is given with respect to how to choose the weights so that the interpolation error is minimal. Lots of numerical examples are present to show the effectiveness of our new methods.
Interpolation is one of the fundamental techniques of approximation theory. Polynomial interpolation is the basic of the whole numerical approximations, however, the high order interpolation which bring into the Runge phenomenon restricts its applications. Rational interpolation is suitable for approximating the function which has poles and it possesses a faster convergence than polynomial. However some rational interpolation, for example, the classical Thiele type's osculatory continued fraction interpolation may be revealed the following problems:The reverse divided differences are not existent, poles and unattainable points can not be avoided and the locations of the poles can not be controlled and so on. Loosing the restrictions on the degree of denominator and numerator, the barycentric rational Hermite interpolation which is constructed on certain conditions can not only satisfy the interpolation condition, but also avoid the unwanted poles. The barycentric rational interpolation and barycentric rational Hermite interpolation possess a quite better stability. In this paper, the composite barycentric rational Hermite interpolation with high-accuracy is constructed based on univariate pade-type approximation and univariate barycentric rational Hermite interpolation; New composite barycentric rational Hermite interpolation is constructed based on osculatory rational continued fraction interpolation and univariate barycentric rational Hermite interpolation; A new bivariate barycentric rational Hermite interpolation is constructed based on univariate barycentric rational Hermite interpolation; A composite bivariate barycentric rational Hermite interpolation is constructed based on pade-type approximation and bivariate barycentric rational Hermite interpolation. Choosing the different weights one may obtain the different univariate or bivariate barycentric rational Hermite interpolation. The poles and the unattainable points of the barycentric rational Hermite interpolation may be avoided through doing a proper choice for the interpolation weights. In this paper, the optimization model which is used for computing the optimal interpolation weights is given with respect to how to choose the weights so that the interpolation error is minimal. Lots of numerical examples are present to show the effectiveness of our new methods.
引文
[1]王仁宏.数值有理逼近[M]上海,科学技术出版社,1980.
[2]徐利治、王仁宏、周蕴时.函数逼近的理论与方法[M],上海：上海科技教育出版社,1983.
[3]朱晓临.(向量)有理函数插值的研究及其应用[D].合肥：中国科学技术大学,1998.
[4]檀结庆.连分式理论及其应用[M],北京：科学出版社,2006.
[5]王仁宏.有理函数逼近及其应用[M],北京：科学出版社,2006.
[6]李辰盛,唐烁.基于块的Lagrange-Salzer混合切触有理插值[J],合肥工业大学学报(自然科学版),Vol.31 No.7 Jul.2008.
[7]Viscovatov B. De la methode generale pour reduire toutes sortes de quantites en fractions continues[J].Mem. Acad. Imperiale Sci. St-Petersburg 1,1803-1806:226-247.
[8]Luc Wuytack Salzer H E. Note on osculatory rational interpolation[J]. Mathematics of Computation,1962,16:486-491.
[9]Siemaszko W. Thiele-type branch continued fractions for two variate functions [J]. Comput. Appl. Math.,1983,9:137-153.
[10]Cuyt A., Verdonk B. Multivaite rational interpolation [J]. Computing,1985,34:141-161.
[11]Schneider C. Werner W. Some new aspects of rational interpolation [J]. Math. Comp.,1986, 175(47):285-299.
[12]Cuyt A, Wuytack L. Nonlinear Mothods in Numerical Analysis[M]. Amsterdam: North-Holland,1987.
[13]Berrut J.-P. Rational functions for guaranteed and experimentally well-conditioned global interpolation [J]. Comput. Math. Appl.,1988,15 (1):1-16.
[14]Schneider C. Werner W. Hermite Interpolation:The Barycentric Approach [J]. Computing., 1991,46:35-51.
[15]Nielson G. M. Scattered Data Modeling [J]. IEEE Computer Graphics and Applications, 13(1)(1993):60-70.
[16]Berrut J.-P, Mittelmann H., Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval[J], Comput. Math. Appl.1997,33(6):77-86.
[17]Berrut J.-P. Mittelmann H., Matrices for the direct determination of the barycentric weights of rational interpolation[J]. Comput. Appl. Math.,1997,78:355-370.
[18]Tan J. Bivariate blending rational interpolants [J]. Approx Theory & Its Appl,1999,15(2): 74-83.
[19]Baltensperger R., Berrut J.-P., Noel B. Exponential converge of a linear rational interpolant between transformed Chebyshev points[J]. Math.Comp.,1999,68:1109-1120.
[20]Berrut J.-P., A matrix for determining lower complexity barycentric representations of rational interpolants [J]. Numerical Algorithms,2000, Volume 24, Numbers 1-2, Pages 17-29.
[21]Tan J, Fang Y. Newton-Thiele's rational interpolants [J]. Numerical Algorithms,2000,24: 141-157.
[22]Berrut J.-P. Mittelmann H., Rational interpolation through the optimal attachment of poles to the interpolation polynomial[J]. Numer. Algorithms,2000,23:315-328.
[23]Gasca M, Sauer T. On the history of multivariate polynomial interpolation [J]. Comput. Appl.Math.,2000,122:23-35.
[24]Tan J Q. The limiting case of Thiele's interpolating continued fraction expansion[J]. Comput. Math.,2001,19(4):433-444.
[25]Baltensperger R., Berrut J. P. The linear rational collocation method[J]. Comput. Appl. Math., 2001,134(1-2):243-258.
[26]Berrut J. P., Baltensperger R. The linear rational pseudospectral method for boundary value problems[J]. BIT.,2001,41(5):868-879.
[27]Berrut J. P., Mittelmann H. D. The linear rational pseudospectral method with iteratively optimized poles for two-point boundary value problems[J]. SIAM. J. Sci. Comput.,2001, 23(3):961-975.
[28]Jieqing Tan and Shuo Tang. Composite schemes for multivariate blending rational interpolation [J]. Com Appl Math.2002,144(1-2):263-275.
[29]Baltensperger, R., Berrut, J.-P., Dubey, Y.:The linear rational pseudospectral method with preassigned poles [J]. Numer. Algorithms 33,53-63 (2003).
[30]Tan J, and Jiang P. A Neville-like method via continued fractions [J]. Comp. Appl. Math.,2004, 163(1):219-232.
[31]Berrut J P. Trefethen L N., Barycentric Lagrange Interpolation [J]. SIAM. Rev.,2004,46: 501-517.
[32]Higham N.J. The numerical stability of barycentric Lagrange interpolation [J]. IMA J. Numer. Anal.,2004,24 (4):547-556.
[33]Berrut J.-P., Baltensperger R., Mittclmann, H.D. Recent developments in barycentric rational interpolation[J]. In:de bruin, M.G., Mache, D.H., Szabados, (eds) Trends and Applications in Constructive Approximation. Interpolation Series of Numerical Mathematics [C]. Birkhauser, Basel,2005,151:27-51.
[34]Shuo Tang, Yan Liang. The construction of bivariate branched continued fraction osculatory rational interpolation[J], Journal of Information and Computational Science,2006,3(4):877-885.
[35]Zhao Q, Tan J. Block based Thiele-like blending rational interpolation [J]. Comput. Appl. Math.,2006,195:312-325.
[36]Michael S. Floater, Kai Hormann. Barycentric rational interpolation with no poles and high rates of approximation [J]. Numer. Math.2007,107:315-331.
[37]Shuo Tang, Yan Liang, Bivariate blending Thiele-Werner's osculatory rational interpolation[J], Numer. Math. Chinese Univ. (English Ser.), issue 3, vol.16:271-288,2007.
[38]Luc Knockaert, Senior Member. A Simple accurate algorithm for barycentric rational interpolation [J]. IEEE Signal processing letters,2008,15:156-157.
[39]Jean-Paul Berrut, First applications of a formula for the error of finite sinc interpolation [J], Numerische Mathematik,2009, Volume 112, Number 3, Pages 341-361.
[40]Richard Baltensperger, Barycentric rational interpolation with asymptotically monitored poles [J], Numerical Algorithms, Online FirstTM,19 August 2010.
[41]Jesus M. Carnicer, Weighted interpolation for equidistant nodes [J], Numerical Algorithms, 2010, Volume 55, Numbers 2-3, Pages 223-232.
[2]徐利治、王仁宏、周蕴时.函数逼近的理论与方法[M],上海：上海科技教育出版社,1983.
[3]朱晓临.(向量)有理函数插值的研究及其应用[D].合肥：中国科学技术大学,1998.
[4]檀结庆.连分式理论及其应用[M],北京：科学出版社,2006.
[5]王仁宏.有理函数逼近及其应用[M],北京：科学出版社,2006.
[6]李辰盛,唐烁.基于块的Lagrange-Salzer混合切触有理插值[J],合肥工业大学学报(自然科学版),Vol.31 No.7 Jul.2008.
[7]Viscovatov B. De la methode generale pour reduire toutes sortes de quantites en fractions continues[J].Mem. Acad. Imperiale Sci. St-Petersburg 1,1803-1806:226-247.
[8]Luc Wuytack Salzer H E. Note on osculatory rational interpolation[J]. Mathematics of Computation,1962,16:486-491.
[9]Siemaszko W. Thiele-type branch continued fractions for two variate functions [J]. Comput. Appl. Math.,1983,9:137-153.
[10]Cuyt A., Verdonk B. Multivaite rational interpolation [J]. Computing,1985,34:141-161.
[11]Schneider C. Werner W. Some new aspects of rational interpolation [J]. Math. Comp.,1986, 175(47):285-299.
[12]Cuyt A, Wuytack L. Nonlinear Mothods in Numerical Analysis[M]. Amsterdam: North-Holland,1987.
[13]Berrut J.-P. Rational functions for guaranteed and experimentally well-conditioned global interpolation [J]. Comput. Math. Appl.,1988,15 (1):1-16.
[14]Schneider C. Werner W. Hermite Interpolation:The Barycentric Approach [J]. Computing., 1991,46:35-51.
[15]Nielson G. M. Scattered Data Modeling [J]. IEEE Computer Graphics and Applications, 13(1)(1993):60-70.
[16]Berrut J.-P, Mittelmann H., Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval[J], Comput. Math. Appl.1997,33(6):77-86.
[17]Berrut J.-P. Mittelmann H., Matrices for the direct determination of the barycentric weights of rational interpolation[J]. Comput. Appl. Math.,1997,78:355-370.
[18]Tan J. Bivariate blending rational interpolants [J]. Approx Theory & Its Appl,1999,15(2): 74-83.
[19]Baltensperger R., Berrut J.-P., Noel B. Exponential converge of a linear rational interpolant between transformed Chebyshev points[J]. Math.Comp.,1999,68:1109-1120.
[20]Berrut J.-P., A matrix for determining lower complexity barycentric representations of rational interpolants [J]. Numerical Algorithms,2000, Volume 24, Numbers 1-2, Pages 17-29.
[21]Tan J, Fang Y. Newton-Thiele's rational interpolants [J]. Numerical Algorithms,2000,24: 141-157.
[22]Berrut J.-P. Mittelmann H., Rational interpolation through the optimal attachment of poles to the interpolation polynomial[J]. Numer. Algorithms,2000,23:315-328.
[23]Gasca M, Sauer T. On the history of multivariate polynomial interpolation [J]. Comput. Appl.Math.,2000,122:23-35.
[24]Tan J Q. The limiting case of Thiele's interpolating continued fraction expansion[J]. Comput. Math.,2001,19(4):433-444.
[25]Baltensperger R., Berrut J. P. The linear rational collocation method[J]. Comput. Appl. Math., 2001,134(1-2):243-258.
[26]Berrut J. P., Baltensperger R. The linear rational pseudospectral method for boundary value problems[J]. BIT.,2001,41(5):868-879.
[27]Berrut J. P., Mittelmann H. D. The linear rational pseudospectral method with iteratively optimized poles for two-point boundary value problems[J]. SIAM. J. Sci. Comput.,2001, 23(3):961-975.
[28]Jieqing Tan and Shuo Tang. Composite schemes for multivariate blending rational interpolation [J]. Com Appl Math.2002,144(1-2):263-275.
[29]Baltensperger, R., Berrut, J.-P., Dubey, Y.:The linear rational pseudospectral method with preassigned poles [J]. Numer. Algorithms 33,53-63 (2003).
[30]Tan J, and Jiang P. A Neville-like method via continued fractions [J]. Comp. Appl. Math.,2004, 163(1):219-232.
[31]Berrut J P. Trefethen L N., Barycentric Lagrange Interpolation [J]. SIAM. Rev.,2004,46: 501-517.
[32]Higham N.J. The numerical stability of barycentric Lagrange interpolation [J]. IMA J. Numer. Anal.,2004,24 (4):547-556.
[33]Berrut J.-P., Baltensperger R., Mittclmann, H.D. Recent developments in barycentric rational interpolation[J]. In:de bruin, M.G., Mache, D.H., Szabados, (eds) Trends and Applications in Constructive Approximation. Interpolation Series of Numerical Mathematics [C]. Birkhauser, Basel,2005,151:27-51.
[34]Shuo Tang, Yan Liang. The construction of bivariate branched continued fraction osculatory rational interpolation[J], Journal of Information and Computational Science,2006,3(4):877-885.
[35]Zhao Q, Tan J. Block based Thiele-like blending rational interpolation [J]. Comput. Appl. Math.,2006,195:312-325.
[36]Michael S. Floater, Kai Hormann. Barycentric rational interpolation with no poles and high rates of approximation [J]. Numer. Math.2007,107:315-331.
[37]Shuo Tang, Yan Liang, Bivariate blending Thiele-Werner's osculatory rational interpolation[J], Numer. Math. Chinese Univ. (English Ser.), issue 3, vol.16:271-288,2007.
[38]Luc Knockaert, Senior Member. A Simple accurate algorithm for barycentric rational interpolation [J]. IEEE Signal processing letters,2008,15:156-157.
[39]Jean-Paul Berrut, First applications of a formula for the error of finite sinc interpolation [J], Numerische Mathematik,2009, Volume 112, Number 3, Pages 341-361.
[40]Richard Baltensperger, Barycentric rational interpolation with asymptotically monitored poles [J], Numerical Algorithms, Online FirstTM,19 August 2010.
[41]Jesus M. Carnicer, Weighted interpolation for equidistant nodes [J], Numerical Algorithms, 2010, Volume 55, Numbers 2-3, Pages 223-232.