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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $x_1, \\dots, x_n$ be the nodes of the $n$-point [[Gauss–Legendre rule|RGGXDRCJVE]] on $[-1, 1]$.\nThe $(2n + 1)$-point *Gauss–Kronrod quadrature formula* is a [[quadrature rule|ADSVMCY77A]]\n$$\\int_{-1}^{1} f(x)\\,dx \\approx \\sum_{i=1}^{n} a_i f(x_i) + \\sum_{j=1}^{n+1} b_j f(y_j)$$\nin which:\n\n1. the nodes $x_1, \\dots, x_n$ are the Gauss–Legendre nodes;\n2. the $n + 1$ new nodes $y_1, \\dots, y_{n+1}$ and the $2n + 1$ weights $a_i, b_j$ are free parameters;\n3. these parameters are chosen so that the rule is exact for every polynomial of degree at most $3n + 1$.\n\n## Remarks\n\nThe rule has $3n + 2$ free parameters, which matches the $3n + 2$ conditions of exactness for $1, x, \\dots, x^{3n+1}$.\nThe new nodes are the zeros of the Stieltjes polynomial $E_{n+1}$, the polynomial of degree $n + 1$ with $\\int_{-1}^{1} E_{n+1}(x) P_n(x) x^k\\,dx = 0$ for $k = 0, \\dots, n$, where $P_n$ is the Legendre polynomial.\nFor the Legendre weight these zeros are real, lie in $(-1, 1)$, and interlace with the Gauss nodes.\nIn practice $\\abs{G_n - K_{2n+1}}$, the difference of the Gauss value and the Kronrod value, serves as an error estimate.\n\nExample: $n = 1$.\nThe Gauss node is $x_1 = 0$, and the rule $\\int_{-1}^{1} f \\approx \\frac{8}{9} f(0) + \\frac{5}{9} f(-\\sqrt{3/5}) + \\frac{5}{9} f(\\sqrt{3/5})$ keeps it and adds two nodes.\nIt is exact for $1$ ($\\frac89 + \\frac{10}{9} = 2$), for $x^2$ ($\\frac{10}{9} \\cdot \\frac35 = \\frac23$), for $x^4$ ($\\frac{10}{9} \\cdot \\frac{9}{25} = \\frac25$), and for odd powers by symmetry, so up to degree $3n + 1 = 4$.\nIn this smallest case the Kronrod rule coincides with the $3$-point Gauss–Legendre rule.\n\nNon-example: for $n = 2$ the $5$-point Gauss–Legendre rule is not a Gauss–Kronrod formula.\nIts nodes are $0$ and the zeros of $63x^4 - 70x^2 + 15$, approximately $\\pm 0.538$ and $\\pm 0.906$, so they do not contain the $2$-point Gauss nodes $\\pm 1/\\sqrt{3} \\approx \\pm 0.577$.\n",
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