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Numerical integration on $ S^0$

In this paper, integrals on the unit sphere are computed numerically via the Gauss-Legendre quadrature in $ \theta$ direction and the trapezoidal quadrature rule in $ \phi$ direction in the following manner:
$\displaystyle \int_{S^0} f(\hat{\mbox{$\mathbf k $}}) dS_{\hat{\mbox{$\mathbf k $}}}$ $\displaystyle =$ $\displaystyle \int_{\phi=0}^{2\pi} \int_{\theta=0}^{\pi}
f(\theta,\phi ) \sin\theta d\theta d\phi$  
  $\displaystyle =$ $\displaystyle \int_{\phi=0}^{2\pi} \int_{x=-1}^{1}
f(\cos^{-1}(x),\phi) dx d\phi$  
  $\displaystyle =$ $\displaystyle \sum_{J=0}^{2p} \sum_{I=1}^{p+1}
w^{p}_I \eta^p_J f(\lambda^p_I,\phi^p_J)$  

where $ f(\theta,\phi )$ is a function, $ \lambda^p_I$ and $ w^{p}_I$ are the arccosine of the $ I$th abscissa and $ I$th weight of the $ (p+1)$-point Gauss-Legendre quadrature, $ \phi_J=2\pi J/(2p+1)$ and $ \eta^p_J=2\pi/(2p+1)$.

In the numerical evaluation of the integral in Eq. (9), the coefficients of the local expansion are computed at the finite sample points on the unit sphere, i.e. a finite set of $ \hat{\mbox{$\mathbf k $}}$ given by $ \lambda_I^p$ and $ \phi^p_J$. Also, we need to prepare multipole moments for the same set of $ \hat{\mbox{$\mathbf k $}}$ to obtain the coefficients of local expansion using Eqs. (10) and (11).


next up previous
Next: Translation of multipole and Up: FM-BIEM Previous: Multipole moments and coefficients
2001-12-14