When the term Gaussian Quadrature appears in most Literatures, does it actually refer to Gauss-Legendre Quadrature.
In other words, do they implicitly admit that they use the Legendre orthogonal polynomials by default?
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3$\begingroup$ I'm not sure what answer you expect; in any particular paper, you will have to read the paper to find out what orthogonal polynomials they use, how could this site provide an answer without specifying a particular paper? $\endgroup$– Carlo BeenakkerCommented Nov 12, 2015 at 18:35
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$\begingroup$ When they say Gauss hey usually mean Gauss-Legendre (by default). Legendre was very unhappy because of this, and many other similar cases when his name is omitted btw. $\endgroup$– Alexandre EremenkoCommented Nov 12, 2015 at 21:02
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$\begingroup$ Well, I mean, in many places such as textbooks or lectures, they merely tell us that they will or we should use Gaussian quadrature without explicitly mentioning the kind, and in other places they might also say that by doing this, we can get the exact solution if the integrand happens to be a (2N-1)-degree polynomial. So I wonder which of these methods they actually refer to by default when they say Gaussian quadrature. $\endgroup$– user123Commented Nov 13, 2015 at 0:58
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It is standard to use the term Gaussian Quadrature to refer broadly to any approximation
$$\int_a^b \omega (w)f(x)dx=\sum_{i=1}^n w_if(x_i)$$
that is optimanl in the sense of Gauss's fundamental theorem of quadrature, regardless of the associated polynomial.
In particular, unless the context makes it somehow clear, I would not jump to the conclusion that Gaussian Quadrature means Gauss-Legendre Quadrature, i.e. that $\omega (w)=1$.
For example Wikipedia and Wolfram are consistent with this use.
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$\begingroup$ But when w(x) = (1-x^2), it can also be Lobatto quadrature, which however can only get the exact solution for polynomials of degree (2n-3). Does it belong to the Gaussian Quadrature family? $\endgroup$– user123Commented Nov 13, 2015 at 12:46
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$\begingroup$ I would consider it a variation, but I guess you can also include it as a Gaussian quadrature. $\endgroup$– MyshkinCommented Nov 13, 2015 at 12:58