计算机科学 ›› 2018, Vol. 45 ›› Issue (3): 83-91.doi: 10.11896/j.issn.1002-137X.2018.03.014
钱江,王凡,郭庆杰
QIAN Jiang, WANG Fan and GUO Qing-jie
摘要: 首先,基于新的二元非张量积型逆差商递推算法,分别建立奇数与偶数个插值节点上的二元连分式插值格式,并得到被插函数的两类恒等式。接着,利用连分式三项递推关系式,分别确定渐近式的分子和分母的次数,即特征定理,并给出推导分子、分母的递推算法。同时,研究表明所提连分式的分子、分母次数分别小于相应的二元Thiele型插值连分式分子、分母次数,这主要是因为所提连分式插值减少了对冗余的插值节点的采用。然后,从计算复杂性的角度出发,所提二元有理函数插值的计算量小于相同插值节点上的径向基函数插值的计算量。最后,数值算例表明所提二元连分式插值方法有效且可行,同时也揭示了即使插值节点集合不变,所提插值连分式的表达式也会随着插值节点顺序的改变而改变。
[1] BUHMANN M D.Radial BasisFunctions[M].U.K.:Cam-bridge University Press,2003. [2] WENDLAND H.Scattered Data Approximation[M].U.K.:Cambridge University Press,2005. [3] WANG R H,SHI X Q,LUO Z X,et al.Multivariate splineFunctions and Their Applications[M].Beijing:Science Press/Kluwer Academic Publishers,2001. [4] LAI M J.Convex preserving scattered data interpolation usingbivariate C1 cubic splines[J].Journal of Computation & Appllied Mathematics,2000,9(1/2):249-258. [5] ZHOU T H,LAI M J.Scattered data interpolation by bivariate splines with higher approximation order[J].Journal of Computational and Applied Mathematics,2013,2(4):125-140. [6] ZHU C G,WANG R H.Lagrange interpolation by bivariatesplines on cross-cut partitions[J].Journal of Computational & Applied Mathematics,2006,5(1):326-340. [7] LI C J,WANG R H.Bivariate cubic spline space and bivariate cubic NURBS surfaces[C]∥Proceedings of Geometric Modeling and Processing.IEEE,2004:115-123. [8] QIAN J,WANG F.On the approximation of the derivatives of spline quasi-interpolation in cubic spline space S1,23(Δ(2)mn) [J].Numerical Mathematics Theory Methods & Applications,2014,7(1):1-22. [9] QIAN J,WANG R H,LI C J.The bases of the Non-uniform cubic spline space S1,23(Δ(2)mn)[J].Numerical Mathematics Theory Methods & Applications,2012,5(4):635-652. [10] QIAN J,WANG R H,ZHU C G,et al.On spline quasi-interpolation in cubic spline space S1,23(Δ(2)mn)[J].Science Sinica(Mathe-matica),2014,44(7):769-778.(in Chinese) 钱江,王仁宏,朱春钢,等.二元三次样条空间S1,23(Δ(2)mn)的样条拟插值[J].中国科学:数学,2014,44(7):769-778. [11] WANG R H,LI C J.Bivariate quartic spline spaces and quasi-interpolation operators[J].Journal of Computational & Applied Mathematics,2006,190(1/2):325-338. [12] 王仁宏.数值逼近[M].北京:高等教育出版社,1999. [13] CUYT A,VERDONK B.Multivariate Reciprocal differnces for branched Thiele continued fraction expressions[J].Journal Computational & Applied Mathematics,1988,21(2):145-160. [14] CUYT A,VERDONK B.A review of branched continuedfraction thoery for the construction of multivariate rational approximation[J].Applied Numerical Mathematics,1988,4(2-4):263-271. [15] TAN J Q,FENG Y Y.Newton-Thiele’s rational interpolants[J].Numerical Algorithms,2000,24(1/2):141-157. [16] TAN J Q,TANG S.Composite schemes for multivariate blen-ding rational interpolation[J].Journal Computational & Applied Mathematics,2002,144(1/2):263-275. [17] WANG R H,QIAN J.On branched continued fractions rational interpolation over pyramid-typed grids[J].Numerical Algorithms,2010,54(1):47-72. [18] WANG R H,QIAN J.Bivariate polynomial and continued fraction interpolation over ortho-triples[J].Applied Mathematics & Computation,2011,217(19):7620-7635. [19] SALZER H E.Some new divided difference algorithm for two variables[M]∥Langer R E,Ed.On Numerical Approximation.1959:61-98. [20] QIAN J,WANG F,ZHU C G.Scattered data interpolationbased upon bivariate recursive polynomials.http://www.researchgate.net/publication/317642310-scattered-data-interpolation-based-upon_bivariate_recursive_polyhomials. [21] QIAN J,ZHENG S J,WANG F,et al.Bivariate polynomial interpolation over nonrectangular meshes[J].Numerical Mathematics Theory Methods & Applications,2016,9(4):549-578. [22] QIAN J,WANG F,FU Z J,et al.Recursive schemes for scattered data interpolation via bivariate continued fractions[J].Journal of Mathematical Research with Applications,2016,36(5):583-607. [23] 檀结庆.连分式理论及其应用[M].北京:科学出版社,2008. [24] SAUER T.Numerical Analysis (Second Edition)[M].Beijing:China Machine Press,2012. |
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