Dynamical control of computations using the Trapezoidal and Simpson's rules

Jean-Marie Chesneaux 1 Fabienne Jézéquel 1
1 ANP - Algorithmique numérique et parallélisme
LIP6 - Laboratoire d'Informatique de Paris 6
Abstract : If $I_n$ is the approximation of a definite integral $\int_{a}^{b}f(x)dx$ with step $\frac{b-a}{2^n}$ using the trapezoidal rule (respectively Simpson's rule), if $C_{a,b}$ denotes the number of significant digits common to $a$ and $b$, we show, in this paper, that $C_{I_{n},I_{n+1}} = C_{I_ {n},I}+\log_{10}(\frac{4}{3})+\mathcal{O}(\frac{1}{4^n})$ (respectively $C_{I_{n},I_{n+1}} = C_{I_ {n},I}+\log_{10}(\frac{16}{15})+\mathcal{O}(\frac{1}{16^n})$). According to the previous theorems, using the CADNA library which allows on computers to estimate the round-off error effect on any computed result, we can compute dynamically the optimal value of $n$ to approximate $I$ and we are sure that the exact significant digits of $I_n$ are in common with the significant digits of $I$.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-01150535
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Submitted on : Monday, May 11, 2015 - 2:20:08 PM
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Jean-Marie Chesneaux, Fabienne Jézéquel. Dynamical control of computations using the Trapezoidal and Simpson's rules. Journal of Universal Computer Science, Springer, 1998, 4 (1), pp.2-10. ⟨10.3217/jucs-004-01-0002⟩. ⟨hal-01150535⟩

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