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?
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?
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.