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Mathematics of Computation

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Minimum norm differentiation formulas with improved roundoff error bounds


Author: David K. Kahaner
Journal: Math. Comp. 26 (1972), 477-485
MSC: Primary 65D25
DOI: https://doi.org/10.1090/S0025-5718-1972-0309279-9
MathSciNet review: 0309279
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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\vert{w_i}\vert$. 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.


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1972-0309279-9
Keywords: Numerical differentiation, minimum norm differentiation, least squares differentiation, roundoff error analysis
Article copyright: © Copyright 1972 American Mathematical Society