The non-absolute integrals, such as the following highly. This is related to the Compiled -option-more about this below. Numerical integration experts can handle functions which are so called absolute integrals. The method suboption 'SymbolicProcessing' specifies the maximum number of seconds for which to attempt performing symbolic analysis of the integrand. Therefore, the time it takes to evaluate an integral is proportional to the number of MaxPoints and in some cases, the form of your function. NIntegrate symbolically analyzes its input to transform oscillatory and other integrands, subdivide piecewise functions, and select optimal algorithms. The Monte Carlo methods in Mathematica are non-adaptive, so when you specify a certain MaxPoints, the integrand will be evaluated at all of these points, uniformly throughout the integration region, and this might be time consuming if the integrand does not converge easily. If you specify MaxPoints only but no method, then the QuasiMonteCarlo method is used. You can specify the number of points used in a MonteCarlo calculation by changing MaxPoints, otherwise a default value of 50000 will be used. Remember that in this type of method the error is proportional to 1/Sqrt, where N is the number of points used. If it fails, it throws an error and tries purely numerical methods. However, it does so usually in order to analyze the integrand. One can also use the 'NIntegrate Explorer' to build knowledge or proficiency in determining NIntegrates methods by the plots of the sampling points. In our experience the QuasiMonteCarlo method will give a more accurate answer than the MonteCarlo method. I'm trying to integrate the function Exp -Itx - t2/2 from -infinity to infinity using NIntegrate in Mathematica the value that I get is accurate when x is small, but as x gets larger, the output from NIntegrate does not match the value I get when I use Integrate - it gets less and less accurate. Mathematica does not have to evaluate the integrand symbolically, in order to apply numerical quadrature. But we are guaranteed to always get the same result.
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