Minimum norm differentiation formulas with improved roundoff error bounds

Abstract

Numerical differentiation formulas of the form Σ i = 1 N w i f ( x i ) ≈ f ( m ) ( a ) , α ≦ x i ≦ β \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 Σ i = 1 N | w i | \Sigma _{i = 1}^N|{w_i}| . We consider formulas that have minimum norm Σ i = 1 N w i 2 \Sigma _{i = 1}^Nw_i^2 and converge to f ( m ) ( a ) {f^{(m)}}(a) as β − α → 0 \beta - \alpha \to 0 . The resulting roundoff error bounds can be several orders of magnitude less than corresponding bounds for high order differences.