Reliable computation of a multiple integral involved in the neutron star theory
Résumé
The following multiple integral is involved in the neutron star theory:
\[\tau(\epsilon,\upsilon)=\frac{1}{\omega(\epsilon)}\int_{0}^{\pi/2}d\theta sin(\theta) \int_{0}^{\infty}dnn^{2} \int_{0}^{\infty}dph(n, p, \theta, \epsilon, \upsilon)\]
where
\[h(n,p,\theta,\epsilon,\upsilon)=\psi(z)\phi(n-\epsilon-z)+\psi(-z)\phi(n-\epsilon+z)-\psi(z)\phi(n+\epsilon-z)-\psi(z)\phi(n+\epsilon+z)\]
and
\[z=\sqrt{p^{2}+(\upsilon \sin(\theta))^{2}},\psi(x)=\frac{1}{\exp x + 1},\phi(x)=\frac{x}{\exp x - 1}.\]
$\omega(\epsilon)$ is a normalization function.
The aim is to get a table for $\tau(\epsilon,\upsilon)$ for some values of $(\epsilon,\upsilon)$ in $[10^{−4},10^{4}]\times[10^{−4},10^{3}]$ and then to interpolate for the others. We present a new strategy, using the Gauss–Legendre quadrature formula, which allows to have one code available whatever the values of vv and εε are. We guarantee the accuracy of the final result including both the truncation error and the round-off error using Discrete Stochastic Arithmetic.