Numerical Evaluation of Integral

ViceCityYayo

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I am trying to evaluate the integral numerically using 96 point Gaussian quadrature, but the singularity is causing me issues. Any suggestions to handle this?
Thanks.
 

HallsofIvy

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The singularities occur at the endpoints, x= -1 and x= 1. Technically that is equal to \(\displaystyle \lim_{\alpha\to -1}\int_{\alpha}^0 f(x)dx+ \lim_{\beta\to 1}\int_0^\beta f(x)dx\). (The "0" can be any number between -1 and 1.)

Since a numerical integral will be an approximation anyway, take the integral from slightly larger than -1 to slightly less than 1.
 

ViceCityYayo

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Jan 13, 2021
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Thanks for the reply. Maybe I am mistaken, but since this is a double integral, and with Gauss-Legendre quadrature we have evaluation with the same values of each variable, this causes the denominator to be 0 at each evaluation. So even if the singularity is skipped over, then we still have a divergent result. Is it valid, as an approximation, to simply choose slightly different values, eg. I = I + w(i)*w(j)*f[x(i), 0.999...999*x(j)]? Also, are you familiar with any 2D quadrature rules that use different evaluation points intrinsically? Thanks.
 
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