It is shown that in Lagrangian numerical differentiation formulas, the coefficients are explicitly expressed by means of cycle indicator polynomials of symmetric group. Moreover, asymptotic expansions of the remainders are also explicitly represented as a fixed number of interpolation nodes approaching infinitely to the point at which the derivative is evaluated. This implies that complete explicit formulas for local Lagrangian numerical differentiation can be obtained.
The remainder estimates of numerical divided difference formula are given for the functions of lower and higher smoothness, respectively. Then several divided difference formulas with super-convergence are derived with their remainder expressions.
Wang Xinghua,Wang Heyu & Ming-Jun Lai Department of Mathematics, Zhejiang University, Hangzhou 310028, China
The best quadrature formula has been found in the following sense: for a function whose norm of the second derivative is bounded by a given constant and the best quadrature formula for the approximate evaluation of integration of that function can minimize the worst possible error if the values of the function and its derivative at certain nodes are known.The best interpolation formula used to get the quadrature formula above is also found.Moreover,we compare the best quadrature formula with the open compound corrected trapezoidal formula by theoretical analysis and stochastic experiments.
Some asymptotic representations for the truncation error for the Lagrangian numerical differentiation are presented, when the ratio of the distance between each interpolation node and the differentiated point to step-parameter h is known. Furthermore, if the sampled values of the function at these interpolation nodes have perturbations which are bounded by ε, a method for determining step-parameter h by means of perturbation bound ε and order n of interpolation is provided to saturate the order of approximation. And all the investigations in this paper can be generalized to the set of quasi-uniform nodes.
Explicit representations for the Hermite interpolation and their derivatives of any order are provided.Furthermore,suppose that the interpolated function f has continuous derivatives of sufficiently high order on some sufficiently small neighborhood of a given point x and any group of nodes are also given on the neighborhood.If the derivatives of any order of the Hermite interpolation polynomial of f at the point x are applied to approximating the corresponding derivatives of the function f(x),the asymptotic representations for the remainder are presented.
