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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $n \\ge 1$.\nA *quadrature rule* on $[a, b]$ with $n$ nodes consists of distinct nodes $x_1, \\dots, x_n \\in [a, b]$ and real weights $w_1, \\dots, w_n$.\nIt assigns to each [[continuous function|0X3WGX95FE]] $f \\colon [a, b] \\to \\R$ the number\n$$Q(f) = \\sum_{i=1}^{n} w_i\\, f(x_i),$$\nwhich is used as an approximation of the [[definite integral|021WEM680W]] $I(f) = \\int_a^b f(x)\\, dx$.\n\n## Remarks\n\nA quadrature rule is a [[numerical integration|7HYF6YG0RS]] method, and $Q$ is a linear functional on the continuous functions.\nMore generally, a rule for a weight function $\\omega \\ge 0$ approximates $\\int_a^b f(x)\\, \\omega(x)\\, dx$ by the same kind of sum.\nExample: the midpoint rule has $n = 1$, $x_1 = (a + b)/2$, $w_1 = b - a$; for $f = 1$ it gives $b - a = I(1)$.\nNon-example: $f \\mapsto (b - a) \\max_{[a, b]} f$ is not a quadrature rule, because it is not of the form $\\sum w_i f(x_i)$ with fixed nodes; it is not even linear.\nA rule is called interpolatory if $Q(f)$ is the integral of the polynomial that interpolates $f$ at the nodes.\nA rule with nonnegative weights is stable, since $\\abs{Q(f)} \\le \\big(\\sum_i w_i\\big) \\max \\abs{f}$.\n",
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