Optimum-Point Formulas for Osculatory and Hyperosculatory Interpolation

Formulas are given for n-point osculatory and hyperosculatory (as well as ordinary) polynomial interpolation for f(x), over ( —1, 1), in terms of fixi), f'(xi) and f"(x/) at the irregularly-spaced Chebyshev points x¡ = —cos {(2i — l)ir/2n}, i = 1, • • ■ , n. The advantage over corresponding formulas for Xi 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 2r times those of the optimum {(r + l)n}-point non-confluent formulas. But these present confluent formulas may require fewer computations for irregular arguments when/(.r) satisfies a simple first or second-order differential equation. To facilitate computation, for n = 2(1)10, auxiliary quantities a¿, 6¿ and d, i — 1, • • • , 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 (-^th of the remainder in the corresponding equally-spaced formula. 1. Advantage Over Equal-Interval Formulas. Formulas are given here for n-point osculatory and hyperosculatory polynomial interpolation for f(x), from prescribed values of f(x) with its first, or first and second derivatives at the irregularly-spaced Chebyshev points z„_i+i = cos \(2i — l)7r/2nj, i = 1, 2, • • • , n, instead of equally-spaced points. In this notation, Xi = — £„_,+i and Xi increases with i. For the sake of completeness, the ordinary Lagrangian interpolation formulas are also given for these Chebyshev points. All n-point ordinary, osculatory and hyperosculatory formulas given here are exact for f(x) a polynomial of degree n — 1, 2n — 1 and 3n — 1 respectively. The advantage of Chebyshev-point over equal-interval polynomial interpolation formulas is apparent from the factor n(a:) ss nr_i(x — x/) in the remainder term, which is LT(:r)/(7i>(ij)/n! for n-point ordinary Lagrangian interpolation, {LT(x)]2/<2n)(£)/(2n) ! for n-point osculatory interpolation and fn(a;)}3/(3")(ê)/(3n) ! for n-point hyperosculatory interpolation. At the moment, in order to compare Chebyshev-point with equal-interval formulas, let the range of z be ( — 1, 1), since the relative improvement of the former over the latter is unchanged under any linear transformation. For xf at the Chebyshev points, | U(x) \ ^ (§)"~\ which is a fraction of the upper bound of | H(x) | for equally-spaced x/s. However, that fraction is not impressively small, decreasing rather slowly with increasing n (except Received August 28, 1961.