Some problems in optimally stable Lagrangian differentiation

In many practical problems in numerical differentiation of a function f(x) that is known, observed, measured, or found experimentally to limited accuracy, the computing error is often much more significant than the truncating error. In numerical differentiation of the n-point Lagrangian interpolation polynomial, i.e., f(k)(x) n L7()(x)f(xk ) a criterion for optimal stability is minimization of E'in Li W I l.Let L L(n, k, x1,.. Xn; x or xo) iL IL7(k)(x or xo)I. For xi and fixed x = xo in [-1, 11, one problem is to find the n xi's to give Lo Lo(n, k, xo) = min L. When the truncation error is negligible for any x0 within [-1, 11 , a second problem is to find x0 x * to obtain L *L*(n, k) = min Lo = min min L. A third much simpler problem, for xi equally spaced, x 1= --1, xn = 1, is to find x to give L-L(n, k) min L. For lower values of n, some results were obtained on Lo and L* whqn k 1, and on L when k= 1 and 2 by direct calculation from available tables of Li (k)(x). The relation of Low L * and L to equally spaced points, Chebyshev points, Chebyshev polynomials Tm(x) for m < n 1, minimax solutions, and central difference formulas, considering also larger values of n, is indicated sketchily.