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., p '(x) ~ £j_jL< ^ (x)f(x¡), a criterion for optimal stability is minimization of £*»« \L¡ (x)\. Let L = L(n, k, Xj,..., xn; x or x0)=S^=1l¿¿ '(x or x0)|. For x, and fixed x = Xq in [ — 1,1], one problem is to find the n x-'s to give Lq = L^(n, k, xq) = min L. When the truncation error is negligible for any Xq within [-1,1], a second problem is to find x0 = x* to obtain L* = L*(n, k) = min ¿0 = min min L. A third much simpler problem, for x¡ equally spaced, x.' = — 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 Lq and L* whçn fc= 1, and on L when fc = 1 and 2 by direct calculation from available tables of L¡ '(x). The relation of Lq, 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.