Some problems in optimally stable Lagrangian differentiation
Mathematics of Computation
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
... 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. numerical differentiation, criteria for optimal stability, minimin and minimax solutions, Chebyshev points, Chebyshev polynomials, central difference formulas.