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Finite Difference Weights, Spectral Differentiation, and Superconvergence

Let $z_{1},z_{2},...,z_{N}$ be a sequence of distinct grid points. A finite difference formula approximates the $m$-th derivative $f^{(m)}(0)$ as $\sum w_{k}f(z_{k})$, with $w_{k}$ being the weights. We derive an algorithm for finding the weights $w_{k}$ which is an improvement of an algorithm of Fornberg (\emph{Mathematics of Computation}, vol. 51 (1988), p. 699-706). This algorithm uses fewer arithmetic operations than that of Fornberg by a factor of $4/(5m+5)$ while being equally accurate. The algorithm that we derive computes finite difference weights accurately even when $m$, the order of the derivative, is as high as 16. In addition, the algorithm generalizes easily to the efficient computation of spectral differentiation matrices. The order of accuracy of the finite difference formula for $f^{(m)}(0)$ with grid points $hz_{k}$, $1\leq k\leq N$, is typically $\mathcal{O}(h^{N-m})$. However, the most commonly used finite difference formulas have an order of accuracy that is higher than the typical. For instance, the centered difference approximation $(f(h)-2f(0)+f(-h))/h^{2}$ to $f"(0)$ has an order of accuracy equal to 2 not 1. Even unsymmetric finite difference formulas can exhibit such superconvergence or boosted order of accuracy, as shown by the explicit algebraic condition that we derive. If the grid points are real, we prove a basic result stating that the order of accuracy can never be boosted by more than 1.