多元Lagrange插值适定结点组及插值基的构造
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摘要
多元插值是目前热门的研究领域之一，本文首先对现有的多元多项式插值方法作了一个介绍与评述，并应用C.de Boor引进的多元差商的概念给出了高维空间中张量积型结点组上的一种插值余项公式。
多元多项式插值不是一元情形的简单推广，它必须首先解决插值的适定性问题。这方面的工作首先应提到梁学章教授，他通过代数曲线将二元Lagrange插值适定性问题转化为一个几何问题。本文第三章中我们进一步研究了Chung和Yao给出的GC条件，并发展了梁的理论。进一步，利用代数几何中关于理想和代数集的理论，研究了代数超曲面上插值适定结点组的几何结构，给出了构造代数超曲面上插值适定结点组的添加代数超曲面法，从而弄清了多元Lagrange插值适定结点组的几何结构。
由于Grobner基的提出，使得用代数方法计算多元Lagrange插值基成为可能。第四章中我们给出了由Grobner基方法及CoCoA编程方法计算多元Lagrange插值基的算法和例子，并给出了插值结点组为适定情形时插值基的构造形式。关于不适定情形时插值基的构造问题更为复杂，这方面的研究很少，本文对此进行了初步讨论，并举了一些具体的例子。
Polynomial interpolation in several variables is a subjectwhich is currently an active area of research. First, we introduce anddiscuss the various methods of multivariate polynomial interpolationin the literature.In particular,using the multivariate divided differenceintroduce by C.de Boor,we obtain a error formula for tensor productinterpolation in R~s.
Multivariate polynomial interpolation is not the simple generaliza-tion of univariate context.First of all,one must solve the well-posedproblem.One of the work is Liang's theorem,which transfer the bi-variate Lagrange interpolation into a geometrical problem by meansof algebraic curves.In chapter three,we discuss Chung and Yao's GCcondition and improve Liang's theorem.Furthermore,using the resultof variety in algebraic geometry, we study the geometrical structure ofproperly posed set of nodes(PPSN) for interpolation on algebraic hy-perplane, and give a Hyperplane Superposition Process to constructthe PPSN for interpolation on algebraic hyperplane,therefore we makeclear the geometrical structure of PPSN for multivariate Lagrange in-terpolation.
The base calculating multivariate Lagrange interpolation with al-gebraic method since GrSbner bases suggesting that, can be usedbecomes possibility.In chapter four,we give out the algorithm thatGrobner method and CoCoA programming method calculating multi-variate Lagrange interpolation bases and several examples. The formof the multivariate Lagrange interpolation bases on PPSN is given.Theform of multivariate Lagrange interpolation bases on unproperly posedof nodes is more complex. The research about it is poor.So we dis- cuss this character primarily, and cite several examples to make outits complexity.
多元多项式插值不是一元情形的简单推广，它必须首先解决插值的适定性问题。这方面的工作首先应提到梁学章教授，他通过代数曲线将二元Lagrange插值适定性问题转化为一个几何问题。本文第三章中我们进一步研究了Chung和Yao给出的GC条件，并发展了梁的理论。进一步，利用代数几何中关于理想和代数集的理论，研究了代数超曲面上插值适定结点组的几何结构，给出了构造代数超曲面上插值适定结点组的添加代数超曲面法，从而弄清了多元Lagrange插值适定结点组的几何结构。
由于Grobner基的提出，使得用代数方法计算多元Lagrange插值基成为可能。第四章中我们给出了由Grobner基方法及CoCoA编程方法计算多元Lagrange插值基的算法和例子，并给出了插值结点组为适定情形时插值基的构造形式。关于不适定情形时插值基的构造问题更为复杂，这方面的研究很少，本文对此进行了初步讨论，并举了一些具体的例子。
Polynomial interpolation in several variables is a subjectwhich is currently an active area of research. First, we introduce anddiscuss the various methods of multivariate polynomial interpolationin the literature.In particular,using the multivariate divided differenceintroduce by C.de Boor,we obtain a error formula for tensor productinterpolation in R~s.
Multivariate polynomial interpolation is not the simple generaliza-tion of univariate context.First of all,one must solve the well-posedproblem.One of the work is Liang's theorem,which transfer the bi-variate Lagrange interpolation into a geometrical problem by meansof algebraic curves.In chapter three,we discuss Chung and Yao's GCcondition and improve Liang's theorem.Furthermore,using the resultof variety in algebraic geometry, we study the geometrical structure ofproperly posed set of nodes(PPSN) for interpolation on algebraic hy-perplane, and give a Hyperplane Superposition Process to constructthe PPSN for interpolation on algebraic hyperplane,therefore we makeclear the geometrical structure of PPSN for multivariate Lagrange in-terpolation.
The base calculating multivariate Lagrange interpolation with al-gebraic method since GrSbner bases suggesting that, can be usedbecomes possibility.In chapter four,we give out the algorithm thatGrobner method and CoCoA programming method calculating multi-variate Lagrange interpolation bases and several examples. The formof the multivariate Lagrange interpolation bases on PPSN is given.Theform of multivariate Lagrange interpolation bases on unproperly posedof nodes is more complex. The research about it is poor.So we dis- cuss this character primarily, and cite several examples to make outits complexity.
引文
[1] 梁学章．关于多元函数的插值与逼近：[硕士学位论文]．吉林长春：吉林大学数学系，1965
[2] 崔利宏．多元Lagrang插值与多元Kergin插值：[博士学位论文]．吉林长春：吉林大学数学系，2003
[3] 梁学章，李强．多元逼近．北京：国防工业出版社，2005
[4] R. J. Walker.Algebraic Curves, Princeton,NJ:Princeton University Press,1950
[5] X.Z.Liang,C.M.Lu.Properly posed set of nodes for bivariate interpolation, Approximation Theory Ⅸ, Vol.2, Computation Aspect,Vanderbit University Press, 1998,189-196
[6] De. Boor and Amos Ron, On multivariate polynomial interpolation,Constructive Approximation.1990,Vol.6, 287-302.
[7] De. Boor and A.Ron.The least solution for the polynomial interpolation problem, Math.Z, 1992,vol.210,347-378
[8] K.C.Chung and T.H.Yao.On lattices admitting unique Lag-rang interpolation, SISAM J.Nu-mer,Anal, 1977,NO. 14,735-741
[9] R. A.Lorentz.Multivariate Hermite interpolation by algebraic polynomials: A survery, Journal of Computational and Applied Mathematics.2000,Vol.122.167-201
[10] C.A.Micchelli.On a numerical efficient method for computing multivariate B-splines, in Multivariate Approximation Theory, W.Schempp and K.Zeller (eds),Birkhauser (Basel),1979,211-248
[11] Carl de Boor. A multivariate divided difference, Approximation theory Ⅷ, World Scientific, Singapora, 1995,87-96
[12] C.A.Micchelli and Milman,P.A formula for kergin interpolation R~k, J.Approx Theory,29,1980,294-296
[13] T.Sauer,Y.Xu.Regular points for Lagrange on the unit disc,Number.Algorithm, 1996,Vol.12,287-296
[14] M. Gasca and J.I.Maeztu.On Lagrange and Hermite interpolation in R~k, Numer Math,1982,Vol 39,1-14
[15] 刘木兰．Grobner基理论及其应用．北京：科学出版社，2000
[16] 杨路，张景中，候晓荣．非线性代数方程组与定理的机器证明，上海：上海科技教育出版社．1996
[17] 王仁宏，梁学章．多元函数逼近，北京：科学出版社，1988
[18] X.Z.Liang,C.M.Lu and R.Z.Feng.Properly posed sets of nodes for multivariate Lagrange interpolation in, SIAM,Numer.Anal,2001,No.2 Vol.39,578-595
[19] David Cox, John Little, Donal O'sha. Ideals, Varieties and Algorithms,Springer-Verlag, New York,1992
[20] M.G.Marinari.H.M.Moeller,T.Mora.Grobner bases of ideals given by dual bases, in processings of ISSAC,1991
[21] 陈玉洁．多元多项式插值：[硕士学位论文]．湖南长沙：国防科学大学数学系，2003
[2] 崔利宏．多元Lagrang插值与多元Kergin插值：[博士学位论文]．吉林长春：吉林大学数学系，2003
[3] 梁学章，李强．多元逼近．北京：国防工业出版社，2005
[4] R. J. Walker.Algebraic Curves, Princeton,NJ:Princeton University Press,1950
[5] X.Z.Liang,C.M.Lu.Properly posed set of nodes for bivariate interpolation, Approximation Theory Ⅸ, Vol.2, Computation Aspect,Vanderbit University Press, 1998,189-196
[6] De. Boor and Amos Ron, On multivariate polynomial interpolation,Constructive Approximation.1990,Vol.6, 287-302.
[7] De. Boor and A.Ron.The least solution for the polynomial interpolation problem, Math.Z, 1992,vol.210,347-378
[8] K.C.Chung and T.H.Yao.On lattices admitting unique Lag-rang interpolation, SISAM J.Nu-mer,Anal, 1977,NO. 14,735-741
[9] R. A.Lorentz.Multivariate Hermite interpolation by algebraic polynomials: A survery, Journal of Computational and Applied Mathematics.2000,Vol.122.167-201
[10] C.A.Micchelli.On a numerical efficient method for computing multivariate B-splines, in Multivariate Approximation Theory, W.Schempp and K.Zeller (eds),Birkhauser (Basel),1979,211-248
[11] Carl de Boor. A multivariate divided difference, Approximation theory Ⅷ, World Scientific, Singapora, 1995,87-96
[12] C.A.Micchelli and Milman,P.A formula for kergin interpolation R~k, J.Approx Theory,29,1980,294-296
[13] T.Sauer,Y.Xu.Regular points for Lagrange on the unit disc,Number.Algorithm, 1996,Vol.12,287-296
[14] M. Gasca and J.I.Maeztu.On Lagrange and Hermite interpolation in R~k, Numer Math,1982,Vol 39,1-14
[15] 刘木兰．Grobner基理论及其应用．北京：科学出版社，2000
[16] 杨路，张景中，候晓荣．非线性代数方程组与定理的机器证明，上海：上海科技教育出版社．1996
[17] 王仁宏，梁学章．多元函数逼近，北京：科学出版社，1988
[18] X.Z.Liang,C.M.Lu and R.Z.Feng.Properly posed sets of nodes for multivariate Lagrange interpolation in, SIAM,Numer.Anal,2001,No.2 Vol.39,578-595
[19] David Cox, John Little, Donal O'sha. Ideals, Varieties and Algorithms,Springer-Verlag, New York,1992
[20] M.G.Marinari.H.M.Moeller,T.Mora.Grobner bases of ideals given by dual bases, in processings of ISSAC,1991
[21] 陈玉洁．多元多项式插值：[硕士学位论文]．湖南长沙：国防科学大学数学系，2003