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How exactly does this work? How is the special set of n points determined, and why does this work?


I was reading about gauss quadrature in wikipedia. It says that we need n point to exactly yield the integral of 2n-1 degree polynomial.


Is this a similar idea to polynomial interpolation (i.e. a set of n + 1 points uniquely defines a polynomial of degree at most n)?


The examples given show the quadrature points as evenly spaced along the x-axis. Is this always the case or are these just carefully chosen examples?