Barycentric Hermite Interpolation
Barycentric Hermite Interpolation
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DOI：
10.1137/110833221
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发表时间：
2011-05
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影响因子：
--
通讯作者：
Burhan A. Sadiq;D. Viswanath
中科院分区：
文献类型：
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作者：
Burhan A. Sadiq;D. Viswanath
设$z_{1}，\ldots，z_{K}$是不同的网格点。如果$f_{k，0}$是函数在网格点$z_{k}$的预定值，$f_{k，r}$是$r$阶导数的预定值，对$1\leq n_{k}-1$，Hermite插值项是唯一的N-1次多项式($N=n_{1}+\cdots+n_(K)$)，它对规定的函数值和函数导数进行插补。我们得到了Butcher等人最近提出的Hermite插值方法的另一种推导。[Numer.算法，56(2011)，第319-347页]。我们推导的一个优点是，它导致了一种更新重心权重的有效方法。如果在其中一个插值点指定了额外的导数，我们将展示如何仅使用$\数学{O}(N)$操作来更新重心系数。即使在合流牛顿级数的背景下，更新系数的相对有效和通用的方法似乎也是未知的。如果该方法实现得当，它会计算重心魏氏……
Let $z_{1},\ldots,z_{K}$ be distinct grid points. If $f_{k,0}$ is the prescribed value of a function at the grid point $z_{k}$ and $f_{k,r}$ the prescribed value of the $r$th derivative, for $1\leq r\leq n_{k}-1$, the Hermite interpolant is the unique polynomial of degree $N-1$ ($N=n_{1}+\cdots+n_{K}$) which interpolates the prescribed function values and function derivatives. We obtain another derivation of a method for Hermite interpolation recently proposed by Butcher et al. [Numer. Algorithms, 56 (2011), pp. 319--347]. One advantage of our derivation is that it leads to an efficient method for updating the barycentric weights. If an additional derivative is prescribed at one of the interpolation points, we show how to update the barycentric coefficients using only $\mathcal{O}(N)$ operations. Even in the context of confluent Newton series, a comparably efficient and general method to update the coefficients appears not to be known. If the method is properly implemented, it computes the barycentric wei...