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  "body": "\n## Summary\n\nGaussian quadrature is a [[quadrature rule|ADSVMCY77A]] with $n$ nodes in which both the nodes and the weights are free.\nThey are chosen to make the rule exact for all polynomials of degree up to $2n - 1$.\nThis is the highest [[degree of exactness|ZYT1JNT1Y0]] that an $n$-point rule can reach.\nThe nodes are the zeros of the degree $n$ [[orthogonal polynomial|8TXASARQN1]] for the weight of the integral, and the weights are positive.\nFor smooth integrands it converges very fast, and it is one of the standard tools of numerical integration.\n\n## Explanations\n\n### L1 human\n\nA simple rule such as the trapezoidal rule fixes the sample points in advance and only chooses the weights.\nGaussian quadrature also lets the sample points move, which doubles the number of free parameters.\nWith these extra parameters it can integrate polynomials of about twice the degree exactly.\nFor example, the two-point rule $\\int_{-1}^{1} f(x)\\,dx \\approx f(-1/\\sqrt{3}) + f(1/\\sqrt{3})$ is exact for every cubic polynomial.\n",
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