Optimum-point formulas for osculatory and hyperosculatory interpolation

Abstract:

Formulas are given for n-point osculatory and hyperosculatory (as well as ordinary) polynomial interpolation for $f(x)$, over (-1, 1), in terms of $f({x_i})$, $f’({x_i})$ and $f”({x_i})$ at the irregularly-spaced Chebyshev points ${x_i} = - \cos \{ (2i - 1)\pi /2n\}$, $i = 1,\ldots ,n$. The advantage over corresponding formulas for ${x_i}$ equally spaced is in the squaring and cubing, in the respective osculatory and hyperosculatory formulas, of the approximate ratio of upper bounds for the remainder in ordinary interpolation using Chebyshev and equal spacing (e.g., for n = 10, the 15 per cent ratio for ordinary interpolation becoming 2.4 per cent and 0.37 per cent for osculatory and hyperosculatory interpolation). The upper bounds for the remainders in these optimum n-point r-ply confluent formulas (here r = 1 and 2) are around $2^r$ times those of the optimum $\{ (r + 1)n\}$-point non-confluent formulas. But these present confluent formulas may require fewer computations for irregular arguments when $f(x)$ satisfies a simple first or second-order differential equation. To facilitate computation, for n = 2(1)10, auxiliary quantities ${a_i}$, ${b_i}$ and ${c_i}$, $i = 1,\ldots ,n$, independent of x, are tabulated exactly or to 15S, not precisely for the optimum points, but for those Chebyshev arguments rounded to 2D ("near-optimum” points). At the very worst (n = 9, hyperosculatory) this change about doubles the remainder, which is still less than $(\tfrac {1}{50})$ th of the remainder in the corresponding equally-spaced formula.