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  "body": "\n## Definition\n\nLet $(a, b)$ be an interval, bounded or not, and let $w$ be a weight function on $(a, b)$.\nThis means that $w$ is integrable, $w \u003e 0$ almost everywhere, and every moment $\\int_a^b x^k w(x)\\,dx$ is finite.\nLet $n \\ge 1$.\nConsider a [[quadrature rule|ADSVMCY77A]]\n$$\\int_a^b f(x)\\, w(x)\\,dx \\approx \\sum_{i=1}^{n} w_i f(x_i)$$\nwith distinct nodes $x_1, \\dots, x_n$ in $(a, b)$ and real weights $w_1, \\dots, w_n$.\nIt is an $n$-point *Gaussian quadrature* rule for $w$ if it is an equality for every polynomial $f$ of degree at most $2n - 1$.\n\n## Remarks\n\nFor each $w$ and $n$ such a rule exists and is unique.\nIts nodes are the zeros of the degree $n$ [[orthogonal polynomial|8TXASARQN1]] $p_n$ for the inner product $\\langle f, g \\rangle = \\int_a^b f g\\, w\\,dx$, and its weights are positive.\nNo $n$-point rule is exact for all polynomials of degree $2n$: the polynomial $\\prod_{i}(x - x_i)^2$ has a positive integral but vanishes at every node.\nSo $2n - 1$ is the largest possible [[degree of exactness|ZYT1JNT1Y0]].\n\nExample: $w = 1$ on $(-1, 1)$ and $n = 1$.\nThe rule $\\int_{-1}^{1} f \\approx 2 f(0)$ is exact for $f = 1$ (both sides are $2$) and for $f = x$ (both sides are $0$), so it is the one-point Gaussian rule, the midpoint rule.\n\nNon-example: the trapezoidal rule $\\int_{-1}^{1} f \\approx f(-1) + f(1)$ has two nodes but fails for $f = x^2$, since the left side is $2/3$ and the right side is $2$.\nSo it is not exact up to degree $3$ and is not a Gaussian rule.\n\nThe rule for $w = 1$ on $(-1, 1)$ is [[Gauss–Legendre quadrature|RGGXDRCJVE]].\n",
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