In 1918 Bernstein [2] published a result concerning the divergence of Lagrange interpolation based on equidistant nodes. This result, which now has a prominent place in the study of the appoximation of functions by interpolation polynomials, may be described as follows. Throughout this paper let / (* ) = |x| (—1 < x < 1) and Xk,n = 1 + 2(fcl ) / ( n l ) (Jfe = 1,2,... ,n; n = 1 ,2 ,3 , . . . ) ...

Numerical operations on and among rational matrices are traditionally handled by direct manipulation with their scalar entries. A new numerically attractive alternative is proposed here that is based on rational matrix interpolation. The procedure begins with evaluation of rational matrices in several complex points. Then all the required operations are performed consecutively on constant matri...

This paper presents cubic spline interpolation based trajectory planning method which is aiming to achieve smooth biped robot walking trajectory. We first characterize the bipedal walking cycle and point out some major issues that need to be addressed to plan a continuous swing leg trajectory by using the concept of cubic polynomial and cubic spline interpolation. By applying these interpolatio...

A recursive algorithm for Hermite interpolation of bivariate data over triangular grids is presented. This interpolation algorithm has a dynamic programming flavor and it computes a single polynomial that interpolates the full set of data. The data we interpolate are partial derivatives and mixed partials up to some fixed order at the nodes of the grid. The interpolant is a polynomial with mini...

We discuss several applications of the Leja point method for univariate polynomial interpolation. First we show how more or less arbitrary interpolation points sets can be stabilized by adding some points from the Leja sequence generated beginning with the given set. We then show how the Leja point idea can be used to generate good point sets for Hermite–Lagrange polynomial interpolation. Then ...

f(xj) = p(xj), j = 1, . . . , n. This is sometimes called Lagrange interpolation to distinguish it from Hermite interpolation, see below. For the basics, see Chapter 7 in Heath. As in Heath, we let Pn be the (function) space of polynomials of degree at most n. We let C([a, b]) denote the set of continuous functions on the interval [a, b]. Similarly, we let C([a, b]) be the n times continuously ...

A multivariate kth divided diierence is proposed and shown to lead to new error formull for polynomial interpolation on some standard multivariate point sets. x1. Introduction In trying to understand what might be a reasonable way to describe the error in interpolation by polynomials on l R d , I came upon the following considerations: The standard error formula g(x) = g(a) + Z 1 0 D x?a g(a + ...