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Mathematics of Computation

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Some problems in optimally stable Lagrangian differentiation

Author: Herbert E. Salzer
Journal: Math. Comp. 28 (1974), 1105-1115
MSC: Primary 65D25
MathSciNet review: 0368391
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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\vert L_i^{n(k)}(x)\vert$. Let $ L \equiv L(n,k,{x_1}, \ldots ,{x_n};x\;{\text{or}}\;{x_0}) = \Sigma _{i = 1}^n\vert L_i^{n(k)}(x\;{\text{or}}\;{x_0})\vert$. 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.

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Keywords: Numerical differentiation, Lagrangian interpolation, optimally stable numerical differentiation, criteria for optimal stability, minimin and minimax solutions, Chebyshev points, Chebyshev polynomials, central difference formulas
Article copyright: © Copyright 1974 American Mathematical Society