The most common approach to the numerical integration of function f(x) I have seen in articles is the Gauss quadrature formula, even if the function is analytic on the interval in which it needs to be integrated. However, many special functions I have observed are calculated through their Taylor series expansions. Why and when is it wiser to use the quadrature formula instead of the Taylor series expansion? Or in other words, when does the Taylor series fail (considering the closed interval inside the circle of convergence), even though it is proven that the integration of analytic functions is polynomial time computable? Edit: the question could be reformulated the following way - when it is more wise to use Taylor series expansion for numerical integration instead of quadrature?
2026-03-27 03:43:08.1774582988
Direct integration of Taylor series and Gauss Quadrature
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