Some problems in optimally stable Lagrangian differentiation

Abstract:

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., ${f^{(k)}}(x) \sim \Sigma _{i = 1}^nL_i^{n(k)}(x)f({x_i})$, a criterion for optimal stability is minimization of $\Sigma _{i = 1}^n|L_i^{n(k)}(x)|$. Let $L \equiv L(n,k,{x_1}, \ldots ,{x_n};x\;{\text {or}}\;{x_0}) = \Sigma _{i = 1}^n|L_i^{n(k)}(x\;{\text {or}}\;{x_0})|$. For ${x_i}$ and fixed $x = {x_0}$ in $[ - 1,1]$, one problem is to find the n ${x_i}$’s to give ${L_0} \equiv {L_0}(n,k,{x_0}) = \min L$. When the truncation error is negligible for any ${x_0}$ within $[ - 1,1]$, a second problem is to find ${x_0} = {x^\ast }$ to obtain ${L^\ast } \equiv {L^\ast }(n,k) = \min {L_0} = \min \min L$. A third much simpler problem, for ${x_i}$ equally spaced, ${x_1} = - 1,{x_n} = 1$, is to find $\bar x$ to give $\bar L \equiv \bar L(n,k) = \min L$. For lower values of n, some results were obtained on ${L_0}$ and ${L^\ast }$ when $k = 1$, and on $\bar L$ when $k = 1$ and 2 by direct calculation from available tables of $L_i^{n(k)}(x)$. The relation of ${L_0},{L^\ast }$ and $\bar L$ to equally spaced points, Chebyshev points, Chebyshev polynomials ${T_m}(x)$ for $m \leqslant n - 1$, minimax solutions, and central difference formulas, considering also larger values of n, is indicated sketchily.