【作者】 郝又平；

【导师】 赵前进；

【作者基本信息】 安徽理工大学 ， 应用数学， 2011， 硕士

【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。更多还原

【Abstract】 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.更多还原