Minimum norm differentiation formulas with improved roundoff error bounds

Abstract:

Numerical differentiation formulas of the form $\Sigma _{i = 1}^N{w_i}f({x_i}) \approx {f^{(m)}}(a),\alpha \leqq {x_i} \leqq \beta$, are considered. The roundoff error of such formulas is bounded by a value proportional to $\Sigma _{i = 1}^N|{w_i}|$. We consider formulas that have minimum norm $\Sigma _{i = 1}^Nw_i^2$ and converge to ${f^{(m)}}(a)$ as $\beta - \alpha \to 0$. The resulting roundoff error bounds can be several orders of magnitude less than corresponding bounds for high order differences.