Computation of an infinite integral using Romberg's method

Abstract : A numerical computation in crystallography involves the integral $g(a)=\int_{0}^{+\infty}[(\exp^{x}+\exp^{-x})^{a}-\exp^{ax}-\exp^{-ax}]dx$, 0<a<2. A first approximation value for $g(\frac{5}{3})=4.45$ has been given. This result has been obtained by a classical method of numerical integration. It has been followed in an other paper by a second one 4.6262911 obtained from a theoretical formula which seems to lead to a more reliable result. The difficulty when one wants to use a numerical method is the choice of parameters on which the method depends, in this case, the size of the integration interval for instance and the number of steps in Romberg's method. We present a new approach of numerical integration which dynamically allows to take into account both the round-off error and the truncation error and leads to reliable results for every value of a.
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Fabienne Jézéquel, Jean-Marie Chesneaux. Computation of an infinite integral using Romberg's method. Numerical Algorithms, Springer Verlag, 2004, 36 (3), pp.265-283. ⟨10.1023/B:NUMA.0000040066.63826.46⟩. ⟨hal-01146534⟩

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