You might have heard of "2nd order" or "4th order methods to calculate an integral. This means that the error drops off with the number of sampling points like N^-2 or N^-4, respectively. But Gaussian quadrature has spectral accuracy, which transcends this measure. Error goes down like an exponential of N.
Another neat feature is polynomials of degree 2N-1 or less are integrated exactly with this method. So if you have just 10 sampling points, you can calculate exactly the integral of a 19th order polynomial (times, perhaps, some known weighting function).
Right!? Such is the magic of orthogonal polynomials. If one chooses the integration grid to lie at zeros of the appropriate orthogonal polynomial, information is gained.
You might have heard of "2nd order" or "4th order methods to calculate an integral. This means that the error drops off with the number of sampling points like N^-2 or N^-4, respectively. But Gaussian quadrature has spectral accuracy, which transcends this measure. Error goes down like an exponential of N.
Another neat feature is polynomials of degree 2N-1 or less are integrated exactly with this method. So if you have just 10 sampling points, you can calculate exactly the integral of a 19th order polynomial (times, perhaps, some known weighting function).